2te 2 ie ttiot Beses3 | > oe ty - , eS pid J Phys bes 4 ‘ ah th ttriyhit iy bt PREPAID ARTO Nal L f Ms Hts dy tien Bice bets Fiphtotey pres i? hie, teh baday | atk Ap Y a MiAsitedal of ; HAS ote Mitton Hel aanty if Be ays cate aa : bie iit oH ‘ patel 1 es isnt eral Abehtags vF . i os) ? ty mn - bcati es Vs i a i igi: + Aus tat ae NA pays i a . ee ti * S? ht ; tit it An alee A AMbeks pre ii afl ey ) eae lattes mat ete an a He oats if an Whaat 4 Mie } ea Hens aa ie sali as a Hs Hi ey a 4 fa nary, ine } iy fit H SoFEaeet ae s\ a Die ae f tly: Ney oe ee ae Ap oh * Baie Ronee ae i} i Sinaia eae cmt m a ERs re aie 14} _ As nt iA ey ae aye Hs ne nh mein ii Ris emi noe a ryt y} st : a 4 i F ‘ i ah it ui if : ony ) H sey ae : HE aaa alien le ree ul halve (Lape i . ; af pele ti saloon inf i, a “ hg a a tates ie Vi “ aie - et ah ey ty a nee aa a Fe) itll A Ceinees oe act he ag 4) S hes jFildse: i 554 ose SCIENTIFIC LIBRARY g Ct & S i Q AIDA YAIQRIOIR GRIN IR GOXS BICMIOSRNO NOT IVIRIONOPGON SAsaka 11— 86265 ING OFFIOB GOVERNMENT PRINT: i \ a i ‘" qi ‘7 BY { ; (] f, THE 0 aa LONDON, EDINBURGH, anp DUBLIN PHILOSOPHICAL MAGAZINE : AND JOURNAL OF SCIENCE. . CONDUCTED BY SIR OLIVER JOSEPH LODGE, D.Sc., LL.D., F.B.S. _ SIR JOSEPH JOHN THOMSON, O.M., M.A., Sc.D., LL.D., F.R.S. JOHN JOLY, M.A., D.Sc., F.R.S., F.G.S. tis GEORGE CAREY FOSTER, B.A., LL.D., F.RB.S. f AND WILLIAM FRANCIS, F.L.S. 7 / ‘Nec aranearum sane textus ideo melior quia ex se fila gignunt, nec noster vilior quia ex alienis libamus ut apes.” Just. Lips. Polit. lib. i. cap. 1. Not. a cee VOL. XXXIV.—SIXTH SERIES. J ULY—DECEMBER 1917. le 4 LONDON: TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET SOLD BY SIMPKIN, MARSHALL, HAMILTON, KENT, AND CO., LD. SMITH AND #0N, GLASGOW ;— HODGES, FIGGIS, AND CO., DUBLIN ;— AND VEUVE J. BOYVEAU, PARIS, “‘Meditationis est perscrutari occulta; contemplationis est admuirari perspicua .... Admiratio generat questionem, queestio investigationem, _investigatio inventionem.”—Hugo de S. Victore. ——“ Cur spirent venti, cur terra dehiscat, , Cur mare turgescat, pelago cur tantus-ansaror, Cur caput obscura Phoebus ferrugine condat, Quid toties diros cogat flagrare cometas, Quid pariat nubes, veniant cur fulmina ceelo, Quo micet igne Iris, superos quis conciat orbes Tam vario motu.” J. B. Pinelli ad Mazonium. JONTENTS OF VOL. XXXIV. (SIXTH SERIES). NUMBER CXCOIX.—JULY 1917. Prof. R. A. Millikan on a New Determination of e, N, and ee LISI ATILS 6/5 Poet hiss) 8llo//al oye) wos cutie 9 shoes rhe in me Dr. 8S. A. Shorter on the Theory of Osmotic Equilibrium .. The Earl of Berkeley’s Note on the above................ Mr. HE. M. Wellisch on the Motion of Ions and Electrons LDPE TS SS ST ce MRA A ge ee A a gen Prof. W. M.- Thornton on the Nature of Chemical Affinity in the Combustion of Organic Compounds.............. Prof. W. M. Thornton on the Curves of the Periodic | aw .. Prof. A. Anderson on the Focometry of Lens-Combinations. . NUMBER CC.—AUGUST. Sir Oliver Lodge on Astronomical Consequences of the Miccmes Picory of Mather 2.2... se ee ee oe we ee Lord Rayleigh on the Pressure developed in a Liquid during the Collapse of a Spherical Cavity ...............-..-- Dr. H. E. Ives on Hue Difference and Flicker Photometer Mr. Harold Jeffreys on Periodic Convection Currents in the PERM UIACTIO 21) os... «MENT IES steno! sy a) ay 2 eile oie ele’ al eh2" ale Prof. C. V. Raman and Mr. Ashutosh Dey on the Main- tenance of Vibrations by a Periodic Field of Force. TELAIPS LIS eI RR 5.0 2 Prof. Ganesh Prasad on the Failure of Poisson’s Equation inn certain Volume Distributions <2. eee ee: Page 81 94 99 1V CONTENTS OF VOL. XXXIV.—SIXTH SERIES. Page Mr. E. A. Biedermann on the Energy in the Electromagnetic UCLA eo ea Gh Gea wae ie oe o's = or 142 Dr. S. Chapman on the Partial Separation by Thermal Diffusion of Gases of Equal Molecular Weight.......... 146 — Notices respecting New Books :— Dr. W. H. Bragg and Mr. L. W. Bragg’s X-Rays and Crystal Structure... jets... sos ee Pie 151 Proceedings of the Geological Society :— The President’s Anniversary Address............-.. 151 NUMBER CCL—SEPTEMBER. Sir E. Rutherford on the Penetrating Power of the X Ra- diation froma Coolidge Tube: ~~... ... 22ers 153 Prof. A. S. Eddington on Astronomical Consequences of the Electrical Theory of Matter. Note on Sir Oliver Lodge’s Dugeestions..:. 1. cthe ce. er 163 Mr. W. J. Walker oa Thermodynamic Cycles with Variable Specific Hleat of Working Substance. ...... 2222. eee 168 Prof. A. Anderson on some Properties of the Nul Point of Thin Axial Pencils of Light directly refracted through a Symmetrical Optical System ........ 2a. eee 174 Prof. R. W. Wood and Mr. 8. Okano on the Ionizing Potential of Sodium Vapour’: :......:2eQ0. 09. Le Dr. Balth. van der Pol on the Relation of the Audibility Factor of a Shunted Telephone to the Antenna Current as - used in the Reception of Wireless Signals.............. 184 Dr. J. R. Airey on the Numerical Calculation of the Roots of the Bessel Function J,,(x) and its first derivate J,(x) .. 189 Mr. H. H. Poole on the Temperature Variation of the Hlectrical Conductivity, et Mica ...... ee eee eee 195 Dr. L. Isserlis on the Variation of the Multiple Correlation Coefficient in Samples drawn from an Infinite Population with Normal Distribution ........'.. (30eee eeee 205 NUMBER CCIL—OCTOBER. Mr. G. A. Hemsalech on the Origin of the Line Spectrum emitted by lron Vapour in the Explosion Region of the Air-Coal Gas Flame. (Plates IT. & IIL.) .............. 221 Mr. G. A. Hemsalech on the Production of Coloured Flames of High Luminosity for Demonstration and Experimental PUP POSES 0s. oi. vtec vie Gly eels itieosces «4 > or 243 CONTENTS OF VOL. XXXIV.—SIXTH SERIES. v Page Prof. E. H. Barton and Miss H. M. Browning on Vibrations : under Variable Couplings Quantitatively Hlucidated by Simple Experiments. (Plates IV—VIJ.).....5.......... 246 Prof. C. G. Barkla and Miss M. P. White on the Absorption and Scattering of X-Rays, and the Characteristic Radiations Reece ana os iho L-cce At Gycttah Lda eth Bad bod toad ole 270 Prof. O. W. Richardson and Lieut. C. B. Bazzoni on the Limiting Frequency in the Spectra of Helium, Hydrogen, and Mercury in the Extreme Ultra-Violet.............. 285 Miss N. Thomas and Dr. Allan Ferguson on the Evaporation meomma Circular Water SUBIACE.. .. 66.5 sc ends oud codes 308 Prof. A. Eddington on Astronomical Consequences of the Electrical Theory of Matter. Note on Sir Oliver Lodge’s Pape stone. LT... fr ptevayelem eso 'a/5 «tela: alles Halsieie's oon oe 321 Prof. Frank Horton on High Potential Batteries for supplying PeMEGUIECODIES: . .. <1 cute meamasS).)s dia efwre bya ice od ade w REL 327 Mr. J. Prescott on the Motion of a Spinning Projectile .... 332 Dr. Eva Bruins on the Application of van der Waals’ Equation of State. Remarks on Miss Bruins’ Communication by ee Fee ASI WON fone es ss ccs si we cid dls hw ee 380, 381 Proceedings of the Geological Society :— Mr. F. Dixey and Dr. T. F. Sibly on the Carboniferous Limestone Series on the South-Eastern Margin of the pow VW ales Coaltteld 2.05. fed ule Ge ee dese cee 382 NUMBER CCITIL—NOVEMBER. Mr. G. H. Livens on the Flux of Energy in the Electro- MRE ARO EMUCLC.. 5). 2 st ReMi ches icle svah chef eee eiaNe chee! wid 385 Dr. H. Bateman on some Fundamental Concepts of Electrical POE al sit, oy shay AUG eo As heel wlan eter en ies aes SUR ore) eels eve 405 Lord Rayleigh on the Colours diffusely reflected from some Collodion Films spread on Metal Surfaces............... 423 Mr. S. Ratner on the Distribution of the Active Deposit aebagmmim an KleehererBield: oe. bul esha ew ele wld. 429 Dr. Harold Jeffreys on Periodic Convection Currents in the Atmosphere. (Second Paper.) The Ear] of Berkeley on the Theory of Osmotic Equilibrium. 459 Proceedings of the Geological Society :— Mr. T. Harris Burton on the Microscopic Material of the Bunter Pebble-Beds of Nottinghamshire and its Pro- Ma pletsource Ob arama 6 lee nels sc vena 460 vl CONTENTS OF VOL. XXXIV.—-SIXTH SERIES. Page NUMBER CCIV.—DECEMBER. Prof. Frank Horton on the Application of Thermionic Currents to the Study of Ionization by Collision ................ 461 Dr. H. 8. Allen on Atomic Frequency and Atomic Number.— Frequency Formulz with Empirical Constants .......... 478 Dr. H.'s. Allen on Electronic Frequency and Atomic Number 488 Mr. T'. Chaundy on a Method of Line-Coordinates for Investi- gating the Aberrations of a Symmetrical Optical System.. 496 Prof. H. C. Plummer on the Action of Coupled Circuits’ and Mechanical, Analogies 02)... . 2. a eee 510 Sir Oliver Lodge on Astronomical ows safec cc: of the Elec- trical, Theory ot Matters (0. 45..)..... oe) ieee 517 Dr. 8. A. Shorter and Dr. F. Tinker on the Kinetic Theory ofthe Ideal: Dilute Solution 2... 0... 0. ane 521 Proceedings of the Geological Society :—— Dr. E. J. Garwood and Miss E. Goodyear on the Geology of the Old Radnor District, with special reference to an Aloal Development in the Woolhope Limestone .... 528 PLATES. I. Iustrative of Prof. C. V. Raman and Mr. Ashutosh Dey’s Paper on the Maintenance of Vibrations by a Periodic Field of Force. II. & IT. IMlustrative of Mr. G. A. Hemsalech’s Paper on the Origin of the Line Spectrum emitted by Iron Vapour in the Explosion Region of the Air-Coal Gas Flame. IV.-VI. Lllustrative of Prof. EK. H. Barton and Miss H. M. Browning’s Paper on Vibrations under Variable Couplings Quantitatively Elucidated by Simple Experiments. ERRATA, Page 78, line 22 (the last line but two of the third paragraph), for the errors are each equal to half a millimetre, read the errors are each equal to half a centimetre, Page 80, line 4, for OP,, or y,, read OP, or yi. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. | as ah 'T OFFICE Kf I. A new Determination of e, N, and Related Constants: By. A. MintiKay * 1. Introductory. ee only preceding determination of e and N for which a high degree of precision has been claimed was. completed in 1912 and published in full in 1913 7. This. determination yielded, for the electronic charge and the number of molecules in a gram-molecule, the following values :— e = 4:°774 x 10-+-0095 electrostatic unit, N = 6:062 x 10” +:012. Although these values, as well as the method by which they were obtained, received quite general recognition, it soon became imperative to reopen the problem and to attempt to make a new and, if possible, more convincing determination. For, first, Professor Ehrenhaft and his pupils began publishing in 1914¢ a series of results which, though obtained by a modification of my method, were wholly irreconcilable with the results which I hadfound. I wished, therefore, to see whether I could check their conclusions and find conditions under which my method failed. Secondly, there developed a tendency, especially among * Communicated by the Author. + Physical Review, 11. pp. 109-143 (1913). t Ann. der Ph ysik, xliv. p. 657 (1914), xlvi. p. 261 (1915) ; also Phys. Zeit. xvi. p. 10 (1915). Phil. Mag. 8. 6. Vol. 34. No. 199. July 1917. B 2 Prof. R. A. Millikan on a new British physicists, to regard the value of e given above as ‘somewhat too high, a value being commonly adopted which’ was about 2 per cent. lower. As this was much greater than the necessary error in the method, I was anxious to see by entirely new work whether a numerical error could have crept into the former determination. Finally, the constant e has recently taken on added importance, since not only does it now carry with it, as formerly, the knowledge of the most important molecular, atomic, and radioactive magnitudes (such as the exact number of moiecules in a given weight of any substance, the absolute weight of any atom or molecule, etc.), but all of the most significant of the radiation constants as well (such as Planck’s h, the Stefan-Boltzmann constant o, the Wien constant C,, all the X-ray constants, 2. e., the wave-lengths of characteristic X-rays) have recently been found to depend for their most reliable evaluation * upon the value of e. I urther, if electricity exists in nature only in exact multiples of e, then e is in a more complete sense than any other physical quantity a natural unit, having none of the arbitrariness about it which inheres in so-called absolute units like the centimetre, the gram, and the second. In a word, e is increasingly coming to be regarded both as the most fundamental of physical or chemical constants, and also as the one of most supreme importance for the solution of the practical numerical problems of modern physics. It seemed worth while, therefore, to drive the present method for its evaluation—a method which is certainly exceedingly exact 1f its validity is granted—to the utmost limits of its possible precision. Accordingly, early in 1914, the work herewith reported was begun. 2. The Method. For the sake of completeness, it may be stated again that the method consists in capturing electrons f on an oil-drop * Physical Review, Millikan, vii. pp. 353-388 (1916), and Webster, vii. p. 607 (1916). + The word electron is used with the meaning originally given to it by Dr. G. Johnstone Stoney, viz., “the natural unit of electricity.” This use has been consistently followed by the most authoritative writers, like Sir J. J. Thomson, Sir E. Rutherford, O. W. Richardson, N. Campbell, etc., all of whom speak in recent books or articles of positive as well as negative electrons, though the mass associated with the former is never less than that of the hydrogen atom. When an electron is found associated with a mass but ;¢75 of that of the hydrogen atom it may be called “a free negative electron,” or following Sir J. J. Thomson “a corpuscle.” Determination of e, N, and Related Constants. y situated between the plates of an air condenser between which a constant electrical field may be thrown on parallel to gravity. If v, is the constant speed of descent under gravity, and v the speed of ascent under the influence of the constant field I’, then it is found by experiment that when the charge on the drop is changed through the capture of electrons, or their loss through the direct incidence of X-rays or ultra-violet light, the series of speeds imparted to it by the fieid—namely, the series of values of w+, constitutes an exact arithmetical progression the greatest common divisor (vjy—vy)o of which is the value of the electron measured in terms of a velocity. This is reduced to apparent electrical units by means of the equation derived from Stokes’s law, Jn a L ) ee ey as 4h || eee I aR Nr ee at) oe OD in which 7 is the coefficient of viscosity of air, o the density of the particle, and p that of the air. The radius, a, of the drop is then found to a sufficiently close approximation by inserting an approximate value of ¢ in the equation 3 4 ie ; Lag Le (2) Amy (o—p) (%1+ v2)o this equation being derived from v mg mg == a Os Ci Spaqe UV ae Fa (eit Pa BS. eee and m = $1a*(o—p). Such observations are made on a considerable number of drops at various pressures or on the same drop at different pressures, and, for the sake of obtaining a linear relation, the values of e,3 are obtained and then plotted against the 4 iy i corresponding values of ——. This procedure amounts to pa adding a first-order correction term to Stokes’s law, and writing it in the form 2 ga’ bN ee 7 (o—p) (1 oh eb emchcus ais (3) pa. ' The relation between e, and e then takes the form a(1+") = at. WE yAd eed cd) pa B2 4 Prof. R. A. Millikan on a new From this equation * it is seen at once that the intercept of the ae line on the é;3 axis is the value of e:, and that the slope of this line divided by the above-mentioned intercept is the constant 6, the significance of which has been pointed out before ¢t and will be more fully discussed in a following paper. 3. Lhe Apparatus. The apparatus is new throughout, and every constant involved in this method of determing e has been re-evaluated with the aid of improved and refined methods. The old air- condenser { had consisted of ribbed brass plates, tested merely mechanically and found flat to about ‘01mm. They were held apart by ebonite posts 16 mm. long. ‘These posts were found to change in length slightly through the absorption of oil. The new plates M and N (fig. 1) were made optically flat by polishing and then testing them with the aid of mercury fringes against a standard optical test-plate. They were nowhere in error by more than two wave-lengths of green mercury light. They were 22 cm. in diameter and were separated by three small pieces of echelon plates about 1 cm. square and 14°9174 mm. thick, placed at points 60° apart about the circumference. These echelon plates, of course, had optically perfect plane-parallel surfaces. The dimensions of the condenser, therefore, no longer introduced an error of more than about 1 part in 10,000 instead of about 1 part in 1000 as in the previous work. The oil droplets from the atomizer A, blown by a puff of air through r (fig. 1) entered the condenser MN through five minute holes + mm. in diameter in the middle of the upper plate, and were observed as in the former work by means of © light from the are, a, filtered through a trough of water, w, and one of cupric chloride, d, for the removal of heat rays. The temperature was held constant to within one or two hundredths of a degree and very close to 23° C. by the oil- bath,G. The charge on the drop, p, was changed by X-rays from the bulb X passing through the window, g. The * Equations (3) and (4) are perhaps more easily visualizable if the correction term (1 eo is written in the form (I+A -); in which / is the mean free path of the gas molecule. Since the exact value of / is uncertain, I have for simplicity chosen to compute it uniformly from n="3502nmel, This gives at 23° 76 cm., /,..=-000009417 cm. + Millikan, Phys. Rev. xxxii. p. 381 (1911). t Millikan, Phys. Rev. i. p. 122 (1918). Determination of e, N, and Related Constants. 5 pressure, heldj absolutely constant during an experiment, was varied from 13 cm. to 76 cm. and was measured to a tenth mm. by the manometer m. The atomizer, A, fed from s with the highest grade of watch-oil, density at 23°C. =to pump cif Fig. 1. \ POOTLILIT ATLAS redetermined as *9199, was blown with carefully dried and cleaned air, let in through the cock r, the bulbs below A being to catch excess oil. The observing optical system was a specially constructed telescope of 30 mm. objective and a magnification of 25 diameters. The objective was about 6 Prof. R. A. Millikan on a new 25 cm. distant from the drop, which was brought into focus by advancing or withdrawing the whole telescope system by means of a nut and screw. ‘The distance through which the drops were timed was 1:0220 cm. It was mea- sured precisely as in the 1912 work, and could be duplicated in successive readings to one part in a thousand. The mean value obtained from ten readings varied from 1°0218 to 1:0223—a maximum difference of one part in two thousand. This factor in the determination of e shares with the coefficient of viscosity of air in introducing the largest uncertainty into the final result. The velocities of the drops both under gravity and in the field were measured with a most convenient and reliable printing chronograph made by William Gaertner & Co., of 5545 Lake Avenue, Chicago, and kindly loaned to the Laboratory for this determination. This instrument is controlled by a standard astronomical clock, and prints on a tape the hour, minute, second, and hundredth of a second at which the key is pressed as the drop crosses the cross-hair in the eyepiece—the maximum error, so far as the recording mechanism is concerned, being never more than a hundredth of a second. Some slight errors were found in the calibration of the Hipp chronoscope used in the previous determination ; but with the Gaertner Printing Chronograph the uncertainty in the time-mea- surements was reduced to a wholly inappreciable amount. The electrical field strengths were determined for each drop with the aid of a 750-volt Weston Laboratory Standard voltmeter, and contain no uncertainty larger than 1 part in 3000. For this voltmeter was repeatedly calibrated in the midst of the observations against three different standard Weston cells, with results which never differed by as much as the limit indicated. The volts are then actually mea- sured in terms of a Standard Weston cell, the above limit being merely the limit of certainty in reading the pointer in the part of the scale used. All the other elements of the problem were looked to with a care which was the outgrowth of six years of experience with measurements of this kind. This work was concluded in August 1916, and occupied the better part of two and a half years of time. 4. The Validity of the Method. That portion of the investigation which has had to do with the testing of the general validity of the method and the endeavour to discover the causes of the disagreement Determination of e, N, and Related Constants. ig between the results of Professor Ehrenhaft and his pupils. and those obtained at the Ryerson Laboratory has been reported in detail elsewhere*. It is sufficient here to say that, although we have worked extensively with droplets. of substances other than oil, particularly with mercury,. upon which the irregularities are chiefly found in the Vienna work, and with particles of the same order of magnitude as those there used, we have found no indi-- cations whatever that the method, when properly used, even: remotely suggests the existence of charges which are not. equal to or multiples of the electron. We have studied. thousands of drops of many different substances in a: number of different gases, and have never found one whose: charge did not fit into an arithmetic series whose greatest: common divisor was the electron. We have definitely dis-- proved Khrenhaft’s contention that this greatest common divisor is a function of the radius of the drop. Further evidence of this independence is given herewith. We have also discovered what seem to us wholly adequate reasons for the irregularities observed by Professor Ehrenhaft. 5. The Precision of the Method. The consistency of the results on different drops is: sufficient guarantee of the precision of the method, pro-- vided no constant error inheres in the measurement of the: dimensions of the condenser, the volts, the time, or the viscosity of air, and provided the speed with which a given. drop moves through the gas is strictly proportioned to the force acting upon it, as it is assumed to be. This last point was very carefully studied in the 1912 work, and: considerable time has been given to studying it again. Incidentally, since for convenience most of the preceding work was done on drops charged with electrons of one sign only, and since it was thought conceivable that the: electron of opposite sign might have a slightly different. value and thus account for the discrepancies between dif-. ferent methods of getting e, exact demonstration has here been made that the sign of the charge is wholly immaterial.. / This means simply that an un-ionized gaseous molecule: possesses no residual charge of either sign which is at all. comparable with the electron. Some evidence upon these: points is contained in Table I. (p. 8), which records experi- ments in which a given drop was alternately loaded with: positive and negative electrons. It will be seen, not only that. *® Phys. Rev. Dec. 1916, “The Existence of a Subelectron ?” ne Prof. R. A. Millikan on a new Tase I. Sign of drop. t sec. tp Bec. nN. é. 63:118 \ 63:030 ie Se ane | = 637332 41-590 62°328 \ é,=6°713 62°728 25°740 62-926 25°798 11 62-900 25°510 63:214 25°806 y) Mean =62:976 63°538 22694 12 ) 63-244 22830 | 63:114 25870 63°242 25-876 11 63°362 25484 fe 63-136 10-830 ) b e, =6°692 63:226 10-682 | | 63°764 10-756 | 99 63-280 10-778 (ee 63°530 10°672 | 63:268 10-646 } ) Mean=63'325 63°642 \ 63°020 el g | 62°820 71-248 63514 | 52-668 | : =i 63°312 52800 7p e, =6°702 63776 | 52:496 63°300 52°860 63°156 71:708 6 63126 | i Mean=63°407 63:228 | 42-006 \ 63:294 | 41-920 8 | 63:184 | 42108 | = 63260 | 53-210 | 63;478 | +52°922 7 63074 53°034 \ e, =6'686 63306 53438 | | 63°414 | 12°888 63'450- | - 12-812 19 | 63446 | 12748 63556 | 12824 ) Mean=63°'335 Duration of experiment 1 hr. 40 min. Mean e, + =6°697 Initial volts =1723°5. e,—=6700 Final volts =1702°1. Pressure = 53°48 cm. Determination of e, N, and Related Constants. 9 the mean values of the positive and negative units are the same within 3 parts in 6700, but that the time ¢, under gravity is quite the same when the drop contains 22 electrons as when it contains but 6 (see column n) ; further, that the value of e, computed by (1) from the speeds when the drop carries from 11 te 22 electrons is the same as that found when it carries 6 or 7 electrons. These numbers show, first, that the speed produced by a given field is an exact measure of the charge ; second, that the speed under gravity and hence the apparent resistance of the medium to the motion of the drop through it is independent of the charge; and, third, that the un-ionized atom is strictly neutral, or that the positive and negative electrons are alike in charge. 6. The Coefficient of Viscosity of Arr. The only remaining element of uncertainty is in the coefi- ‘cient of viscosity of air. In 1913, in view of measurements made in this laboratory upon this constant, in addition to measurements made elsewhere, I published * as the most probable value at 23° C. n=:0001824, and estimated that this could not contain an uncertainty of more than ‘1 per cent. The correctness of this estimate was questioned by Vogel f and Gille t, who, while reducing the value obtained by the Halle observers by one per cent., still retained a value which was half a per cent. higher than that which I had adopted. It is to be pointed out that if this result were correct, my value of e instead of coming down would go up by three-fourths per cent., but in any case it was obviously necessary to institute further tests as to the correct value. ‘These tests were carried out most thoroughly by Dr. E. L. Harrington, who, using the constant deflexion apparatus designed by Dr. Gilchrist and the author, succeeded, by the introduction of improvements in condition and perfections in detail, in inaking a determination of 7 which is, I think, altogether unique in its reliability and precision. I give to it alone greater weight than to all the other work of the past fifty years in this field taken together. For the individual determinations, though made with different suspensions and in such a way as to eliminate all constant sources of error ‘save the dimensions of the cylinders, never differ among themselves by as much as ‘1 per cent. and the error in the final mean can scarcely be more than one part in 2000. Indeed, the work has since been repeated by another * Annalen der Physik, xli. p. 759 (1918). t Ibid. xliii. p. 1235 (1914). { Ibid. xlviii. p. 799 (1915). 10 Prof. R. A. Millikan on a new observer, Mr. Stacy, and the result found dependable to: within that limit of uncertainty. Dr. Harrington’s* value 1S 493=='00018227. This value is within less than a tenth per cent. of my 1913 value. The constants of the sus-- pensions were determined by taking the periods in vacuo, and it is interesting that they differed from the period in air by as much as ‘2 per cent., because of the moment. of inertia of the air which is dragged along with the rotating cylinder. 7. The Observations. The results of the final series of observations on 25 con-- secutive drops are given in Table II. and fig. 2. The numbers. at the top of the sheet in the figure represent the approximate times of fall under gravity of the drops opposite which they stand. They are inserted to show the reader at a glance that. the value of the slope and of the intercept on the e,3 axis,. that is, the value of the electron, is not in any way a function of the radius of the drop. One can get this slope by com-. paring only relatively large drops at different pressures (for example, drops falling in from 14 to 20 seconds), or quite small ones (such as those falling in from 44 to 57 seconds), or by comparing drops of different sizes at the same pressure.. ‘The starred drops were those taken when the conditions of observation were considered as perfect as possible. No attempt was made to take observations on drops which fell through the fixed cross-hair distance of 1:0220 cm. in less than 14 seconds, since it was desired to keep the- timing errors negligible. The value of e?—namely, the. intercept on the e,3 axis—was taken from the graph, as. was also the slope divided by the intercept, which is the value of 6 in equation (4). The values thus found were GO lalla yee (Om = b = 000618, p being measured in centimetres of mercury at 23° C. and ain centimetres. The value of A (see foregoing footnote). corresponding to this value of 6 is "864, instead of °874 as found in 1913. The difference is due, I think, to small errors which were then made in the calibration of both the- Hipp chronoscope and the voltmeter, which, however, com- pensated each other in their effect on e, though not in that on A. The numbers given in the last column of Table II. are the values of e# obtained algebraically from (4), and the * Phys. Rev. Dec. 1916. Lah Determination of e, N, and Related Constants. iGtalOae = sf UBS T SL-19 PLLL F6-19 G6-GL t@ 9 G9-GL 46-09 Ou-¢L 16-09 Fe FL, 66:19 Leh 0¢-19 r6-€L 36-19 09-14 F6-09 C8.0L LG 19 88-69 68-19 16-89 [1-19 06-89 96-19 FI-L9 L0-£9 GI-L9 10-19 OL-99 91-19 12e-c¢ G0-19 61-9 IT-19 61-99 0G-19 10-99 61-19 0-9 16-19 69-79 16-09 LE-D9 91-19 ¥¢-€9 60-19 F0G-69 60-19 eee 16-69 7 : I 801 X giz? | 201 X gig ? GO?90.- F8P90.- GOG90- 60°F0- GIIFO: ITTFO: 6-S8P G-886 9-086 G-IL6 L-6¢6 POPE P- 165 F186 F-G9G 8-LEG L-O0G P-S0G 9-491 LO9T 9-0GT $-LOL L-60T 6-901 F-GOL 6-10 9-06 L-98 0-89 ¢.1G GP-LG pad I 08-61 GL-ST FL-0G LE-0G G9-E1 GL-0G FEET FS-06 OL-6G 89-06 CE-1¢ 08-96 16:08 19-96 96-66 19-G), FO-GL OF-GL 86-FL LL-GL 00-¢L LG-¢) GF FL CO-¢ 6P-FL "(SPT ‘ut0) d OF-€6 LG-9T ¢9-61 69-41 LT &1 LV-06 00-F1 00-86 GG LT GI &I IL-6 66-41 OG-E1 96°16 80-L1 GLE TS-GL LU-GI 0G-G1 cO-§1 c0-6 t8-F1 66-1 FE1G C686 "UD QT XB N ae Sumit i oD pam or | ro | BP © bh HOD ODN 4 OD © 109 CN A OIG 19 19) 1) © OD OD S> SO we) Le Te bo | S Teele ee Ni, Ve ala as N i! OD 4 19 | . > > SECU CK AR 618S0- 10é90- 6L160- 6GEG0- ecco. CO09GO- 6FGLO- 6180: ¥6cC0- GLGO- 62160: GFIGO- GEFCO- GOGE0: G9T90- 6LL10- 8SGLTO- E2810: C86T0- SR6T0: 16GG0- GOLZO- O81¢0- 66090- F6190: ‘oas/‘uuo ‘a 91-96 ¢9-61 06-97 88-SP 08-81 16-66 OL-FL C9-96 G8-9P Pe-81 GL-GE G9-LP 18-81 81-66 89-91 OF-LG FI-6¢ 90-9¢ 8F-1¢ 69-1G 60-00 68-LE 6-61 91-91 08-91 —_. (-oas) ”y FrSE GPSS TREE OLES OSES G89F 6SEE 169F 699F €6EE STOP Loor L89r 1L99P E99P LLOF G809 1999 6664 ILL? 199F GSIP S0EG OOTY 0G99 "(S}]0A) ‘dd GTL-€% 60:-€¢6 E1-€S 00:86 PI-&G FL-€% OL-SG GIES GL-&G CT-66 OL-S6G GL-€G 86-66 LI-&6 €1-E% OI-&% 00-86 10-86 80-83% GL-SG 90-86 80-8% CO-€6 00-86 LO-€% ‘O 9 dway, see eee seen ee eee tee ‘ON 12 o€ 19 O21 OG 09 Osi 00S O22 Ove 02” O6€ OSE OFF eantt? OBy OSe Prof. R. A. Millikan on a new £9 69 >< Gato N N NI W O N 6Z let + 4.7% 26 44 eo) - 27% % ‘SL Determination of e, N, and Related Constants. 13: above value of } taken in connexion with each individual set.. Sian | ; : of values of the observed quantities oe and e3. It will be seen that the final mean value of e3 obtained by this method of analysis—a method which vields the most reliable value- of e2 obtainable from the data at hand—is és = bI26 x 10" *- There is but one drop-in the table which yields a value of e differing from this by as much as one-third of one per cent., and’ the probable error of the mean computed by least squares is one part in 4000. No such precision, however, is claimed for this deter-- mination of e?. It is ‘07 per cent. higher than the value. (61:086) which I published in 1913, both values being computed in terms of :0001824, as the value of the co-. efficient of viscosity of air. Dr. Harrington’s new value of this constant, viz. 00018227, is, however, more reliable. than the old one and is °07 per cent. lower than that ; so that the new value of e and N computed solely from the new data obtained in this redetermination is exactly the same as the- value published in 1918. The maximum uncertainty in this. value was then estimated as one part in five hundred. This work has reduced it so that that it is no more, I think, than one part in a thousand ; for it now contains but two factors. which are uncertain by as much as one part in two thousand,. namely, the coefficient of viscosity of air and the cross-hair distance. The exactness of the agreement to four places with the 1913 value is, of course, accidental. This is,. however, the third time that with independent deter-- minations (one unpublished) I have come out well within one-tenth per cent. of the foregoing result. The result of this investigation may, then, be stated as. follows :— e= 4°71 14510, +005; N= 6062641072, +006, the Jast number being obtained from N x 4:774x 1071 = 9649-4 x 2°9990 x 102°. Since this is an attempt at a precise determination, and by far the most carefully carried out of any work which I have thus far done upon the evaluation of the electron, it. is perhaps worth while to give more of the original data than would otherwise be justified, in order that others may better form their own estimates of its probable reliability.. a4 Prof. R. A. Millikan on a new Accordingly, the actual records of the observations on all of the 18 starred drops which were chiefly considered in ‘determining the line in fig. 2 are given below. The other seven are omitted merely to save space. As the graph ‘shows, they do not modify in any way the result. With their inclusion the graph becomes the record of 25 con- secutive observations without any discards, so that the result ‘is entirely free from the exercise of choice. 8. The Values of some other Related Constants. I have already recorded in the 1913 paper the values of six fundamental but related constants which are at once known as soon as e is found, and with the same precision as that attained in its evaluation. These six are: (1) the electron, ¢; (2) the Avogadro number, N ; (3) the number -of molecules, n, in an ideal gas at 0° C. 76 em.; (4) the kinetic energy, Eo, of molecular agitation at 0° C.; (5) the constant change, ,in molecular energy per degree ; (6) the -entropy constant, £, or the gas constant applied to a single molecule. A seventh constant which should have been included at that time is the mass m of a hydrogen atom, given by : BE ees LO O13, au Bot ive BG a7) —24 im — arrow 9-000 x 10 = ]°062 X)105 >" era This list may now be extended as follows :— The constant of the Balmer series of hydrogen is known ‘with the great precision attained in all wave-length deter- minations, and has the value 3°290x10. From Bohr’s ‘theory it is given by 2rre*m 2Qtre 7B or. =e ee) I have shown that h may be determined photoelectrically* ae an error in the case of sodium of no more than 4 per cent., the value given by my work on sodium being e: 56 x 107 21. The value found by Webster T by the X-ray method discovered by Duane and Hunt{ is 6:53x 10. Taking the mean of these two results, viz. 6°545 x 10777, -obtained by wholly dissimilar methods, and substituting * Phys. Rev. vii. p. 374 (1916). + Ibid. vii. p. 599 (1916). t Ibid. vi. p. 166 (1915). Determination of e, N, and Related Constants. ee ° .J . . é . in (1), after introducing Bucherer’s value of —, viz. 70 1:767 x 10’, we obtain for the Rydberg constant, 3°294 x 10”, which agrees within one-tenth per cent. with the observed value. This agreement constitutes most extraordinary justification of Bohr’s equation, and warrants the use of spectroscopic data, combined with the foregoing data on e, for a most exact evaluation of h. The value of 4 computed thus from (5) with the aid of my value of e and the fore- . Z . . . e e going value of =a, which is now known with a precision of one-tenth per cent., is h=6540% 15-7? + -O1T. It will be seen that the uncertainty is just 2 the un- certainty in e, since e appears in (5) in a power 8 that of h, Se 3 navi i has while — affects h by an amount which is negligible in com- m parison. The foregoing value of h may be considered the most reliable thus fur obtainable, its uncertainty being one part in six hundred. It will be seen, too, that it agrees within just one part in five hundred with the value obtained for my sodium curves, which I estimated correct to only one part in two hundred. Having thus fixed the value of h to one part in six hundred, we may obtain from Planck’s equation the Wien constant, Cy, with the same precision, for it will be re- ‘called * that he 6°547x Lg x 2°399 Se Low et 1:03 (peas A Ue = 1:4312+:0030 cm. degrees. The estimated error set down above is obtained from the assumption of an uncertainty of one part in six hundred for h and one part in one thousand for &. The latest experimental result on C, given out by the Reichsanstalt f is C,=14300. Coblentz t gives as the result of his direct experiments C,=1:4369, while his combination of total radiation experi- ment and theory lead him to C,=1°4322. Os = * Phys. Rey. 1. p. 142 (1913). + Ann. Phys. x\viii. p. 480 (1915). t Phys. Rev. vii. p. 694 (1916). 16 Prof. R. A. Millikan on a new Again, from Planck’s equation, cee (“z=)" a) we can compute the Stefan-Boltzmann constant of total radiation a [or 25 | and obtain a result which is uncertain: by but six-tenths per cent. The resuit is a= 9712x105 7-034 watt Cm9? demas This is exactly the value found by Coblentz from his most recent and most thorough experimental work * ono. The. exceedingly close agreement between all these values of h, Cy, and o, computed on the one hand from the work on e,. and directly observed on the other, is an indication of the exactness of the work on e. The grating spacing in calcite computedt from the. foregoing value of e is 3°030 iN A summary of the most important constants the values of which are fixed by this determination of e is given below,, with the uncertainty attaching to each :— Mie electron. sa) maar rseeneteciens lars) 6's) é = 4774 +:000 x 10gE: The Avogadro constant ~..:...:.... N = 6:062 —=2-006>aI0-= BE ee molecules per c. ¢. a n = 2705 +:003x 10. Kinetic energy of translation of 4 ee bite molecwlerath (an OHytmnmesrtern ae chee By = 5621 + :006X10-™.. Change of translational molecular | ¢ = 9-058 42-000 oe energy per degree ©. ...24-...-. 0s ( a ei Mass of an atom of hydrogen in grams. m = 1662 +:002X 10-4, Planck’s element of action .......... h = 6547 -OlLD ate Wien’s constant of spectral radiation.. ¢, = 1481240030. Stefan-Boltzmann constant of ae oa 57D shoo TACTAUI OM, assole eee eM ao tehio! aS cs =a I have to express my hearty thanks to Dr. Yoshio Ishida. for invaluable aid both in observing and in computing the- accompanying data. Ryerson Physical Laboratory, University of Chicago, January 12th, 1917. * Phys. Rev. vii. p. 694 (1916). + Webster, Phys. Rev. vii. p. 607 (1916). ga Se a Determination of e, N, and Related Constants. Ee APPENDIX.—Observational Data. Ragan vil ee tne tp. (Ee) ee reese (ese Teste g F n ih = =): m | = ( i, =F a Drop No. 1. 16:56 | 97:96 8 | -008852 | V,=6658 volts. 16°46 97:99 V = 6647 volts. | ) 1 | -008831 | 16°46 | 52°35 t=23° 07 C. 9) -008849 16-48 | 52:70 p=74-49 om. 1 | 008816 | a 16°54 | 97°84 8) 008854 | »,=-061940 = 16°58 a='0002340 cm. 16-50 | _ pgs pa 1645 | 51:26 | 1 ‘009136 16°46 96:48 8 | -008872 e° — 63:21 16°46 oes | 1 ‘008948 9 008880 16°46 51:30 | Fe ='04111 1650 | 51-68 | 2 | -008856 16-44 | 624-39 7 | 008886 009016 1655 | 94:18 8 | -008903 5 | -008951 16:52 | 18-01 13 | -008921 1644 | 1810 | 5 | -008930 1663 | 93-27 8 | -008916 1 | -008924 16-44 | 50-90 9| -008917 16°46 Mean | 16°50 008934 "008895 e2/3—61:03 Dror No. 2. 1675 | 32:45 | V,=6107 volts. 11) 008225 16-72 | 32:49 | V,=6091 volts, 3 | -008211 | | | Phil. Mag. 8. 6. Vol. 34. No. 199. July 1917. C Mean 18 Prof. R. A. Millikan on a new Medinle yan ER N/M eel fhe tp. Pid pee (ipa NNA DS LE a aay i: iN Tt g ‘ wv! is = m nN iB ne a) Paste, al Drop No. 2 (cont.). 16°74 | 162:22 ay | 8]. -008231 . | 4=23°-00 c. 16°77 69°16 3 aes p=75:00 em. 1677 | 69:54 | », = 061022 2 1 008163. sec. 16°66 44°35 10! -008222 a='0002322 cm. 16°72 | 163:51 8 | -008225 ries 2 | 008218 Bye 16°76 69:93 9 | -008220 if ‘008184 16-74 | 162-59 8} -oogz29 | g— 041154 1 ‘008194 16°78 69°71 9 | -008225 2/3 — 63-904 “008249 16:76 | 44:26 10 | -008223 16°74 | 44:26 1 008234 16°75 69:65 9 | -008227 16°92 | 69:63 16°756 008218 ‘008227 e7/> 61:03 Dror No. 3. | 19°70 13:06 V,=5319 volts. 19°65 15°12 " : V ,=5303 volts. ‘007800 19-77 15:08 t=23°-05 C. 19:72 36°80 p=74:49 cm. 19°70 | 36:53 v,='051799 10 | -007799 BEC. 19°66 36:54 a=:0002134 em. 1 19°74 | 36°59 — =63-0 2 007814 Le 19°72 | 85:45 ; 8] -007801 = ='04509 19°77 85:11 is 21 -007831 e? =63°54 19°70 | 51:07 Mean Determination of e, N, and Related Constants. Wate 2 hae t typ fod BB BARE ae eel sie g | B AS =) S 7 7 tp EE eS ——a soe == | Dror No. 3 (cont.). 1969 | 51-01 | 9 | 007812 19°72 | 50-82 19°75 | 50-92 1 | -007807 19°73 | 84-52 19°72 | 83:53 8 | 007827 1962 | 83:39 2 | -007816 19°7 36°38 1977 | 36:02 | 10 | -007835 19°79 | 36:16 | 19°69 | 36:05 3 | -007819 19°70 | 233-47 7 | 007852 1 | -007845 19°82 | 82-45 8 | -007851 2 | -007872 19°76 | 35:88 10 -007856 1975 | 28-09 11 | -007844 19°76 | 228-15 7 | -007866 19°78 19°78 ‘007829 ‘007831 e7/3—6 116 Drop No. 5. 39:95 | 28°77 | | V,=4665 volts, 6 | 009957 39:99 | 28-70 V=4659 volts. 2 | -009890 | 39°87 | 66-43 | t=23°-06 C. 4 010015 40:06 | 66-24 p=75'00 em. 40:14 | 65°78 v= 025213 — 1 | .-010056 sec. 39:93 | 39:59 a='0001484 em. Cx Mean Mean 20 51°48 ty. 30°55 21°86 50°72 148°63 147-46 50°29 50°25 50°39 49°70 146-41 ' “(2 ~| nm. | —{| — — — n' tp tp] Prof. R. A. Millikan on a new 1 n it 1 (heh Drop No. 5 (cont.). 6 "010026 ‘010094 4 ‘010067 5 010083 ‘010148 4 ‘010120 "009995 3 010142 ‘010037 ‘010056 Drop Nose 4 ‘013040 5 013034 "0138015 3 ‘013047 012961 2 013092 013138 3 ‘013120 °013150 2 ‘013128 ‘013066 013084 5 010037 e612 V ,=5306 volts. | V p= 5295 volts. 1=230-08 C. | p=74:98 cm. v= 019852 — a='0001305 cm. |b ea94 pa Y . Bi = 07329 c= 65°07 e7/3 61:20 Mean Mean 55°69 59°90 56°18 55°91 56°12 55°90 56°29 56°26 56°33 56°06 58-92 58°92 59°78 59:04 58°69 58°36 59°50 59°34 58°85 59-95 59°14 Peete Determination of e, N, and Related Constants. 2] 1 | ; pNP d We ead tp. | sR aa Se peg (ese? (EE cu 7 ‘ ae A: if 2 S a = Dror No. 8. 29:97 V ;=6669 volts. _ 29°83 | 3 017094 V (f= 6657 volts. 29-69 t=23°-01 C. 1 | -017251° | 60:89 121 -017131 p=75'40 cm. | 29976 | 1 | -ol7179 | v, = 018230 %: sec. 29°49 a='0001250 em. 29:51 3 | oe | =1063 1 ‘017072 oe 59-49 Sorin | 2| ove | @~ 07608 A4 ghana 59 e °=6513 ‘017167 ‘017191 e737 —61-11 Drop No. 9. 21:23 | 2 | -015934 | 4 -016002 | V,=6091 volts. 65:52 2 | 016071 Vp=6077 volts. 65°75 | t—23°-00 ©. | 1 | 016222 | | 831-85 | p=75:04 em. 3 | 016121 2 31-72 | »,=-01728 1 | -016202 | sec. 20-98 4 016002 a—-0001217 em. | 2| -016074 | | 1 | 64:45 9 | “016211 | =—108-7 be | 2 | Olean pa g+ 1 | -016351 180240 1 016134 ] 20-90 4 | -016188 — = 07850 | PR =6519 ‘016107 | 016151 e/? —61-05 29 Prof. R. A. Millikan on a new eaten a Mave all irae ty. tp. N. a= — =) n. rs l au a Dror No. 10. 57°35 | 28:10 5 | ‘010598 nes 2 "010714 V,=4091 volts. 57:24 | 70-6 . 3 ‘010567 V -=4071 volts. 5692 | 69-2 ‘010731 t=23°°10 C. 5718 | 27:84 p=75°67 em. 57°20 | 27-97 a 5 | -010624 | v,=-017787 — 57°35 28:49 SEC: REY ERS a=:0001234 cm. 57-26 | 28-00 | 1 _ 1073 2 ‘010543 ae 57-48 | 68:35 ; 3 | -010678 — =:07680 57°30 | 68°53 4 57°76 68:20 7/2 — 65:21 5 | -010661 : 57°84 14-72 8 ‘010669 5718 | 21:32 6 | -010781 57°38 | 20:97 57°74 | 27-67 : 5 | +010709 2 | -010557 57°78 | 66:55 3 | -010810 57°80 Mean | 57-46 | 010661 010670 7/3 —61°16 Dror No. 11. 16°43 74:69 V,=4668 volts. 1650 | 74:31 | 11 | -006701 | V,=4660 volts. ISREOL flee ae ¢=23°°13 C. Rot 006869 1651 49:19 | * 12 | -006721 p=29:26 em. 1 ‘006630 | a 1652 | 72:98 | v= "061651 —— Mean Determination of e, N, and Related Constants. 23 Raa i? os bya eed ripe tp. Pa | ee ees 2 | lagen! (cena Oh Te ae g = ‘ ee = % e Tr =: Dror No. 11 (cont.). 16°54 | 72:55 | | me | @=-0002272 cm. 1 ‘0 16-56 | 72:35 + _ 1596 pa 1658 | 71:85 F 1 | 006625 ~ ~1078 16°59 | 137-12 10. 006762 d 1 | -006865 | | 16:55 | 70:58 11 | -006772 |e7?—66-70 1 ‘006898 | ; 1658 | 47°52 | 2 12 006782 1659 | 47-43 | 2 | -006734 | 10 006792 16°54 | 131-68 | 1 006945 1663 | 68-78 11 006811 1 | -006810 16-7 46°84 12) -006806 1 | -006723 | 1658 | 6837 / 11! -006813 | 1 | 006799) "| | 1665 | 127°77 (10 -006815 2 -Otestann | | | 16-75 | 46°60 | 12| 006815 | 1 | -096692 1665 | 67°72 11 006826 16-59 | | Lt ee Ie | fw 16-577 | 006783 006781 | =61-01 Dror No. 12. 29:00 | 3459 | | V,=4668 volts. 29:30 | 3465 | V-=4660 volts. 2899 | 3459 | | | | ¢=23°-11 ©. | 7 | . 009037 29:15 | 34:49 | p=36'61 em. 2918 | 34:50 |v, = 035023 —— sec. 99:12 | 34:38 | | | a=:0001708 em. | 2 | -909020 | | 29:19 | 90:53 | igo 5 | 009075 |? om Prof. R. A. Millikan on a new t t n' “(ze a2 ~(- +>) g a : I tp tp ; ; 2 ty tp ; Dror No. 12 (cont.). 29°35 | 89°61 ] 2 | -009071 aS oaG 29:32 | 3414 | a vf ‘009081 S13 ee 29:14 | 34-12 el =67 12 2} 009043 5 | -009097 99:11 | 89-17 1 ‘009184 6 | -009108 29°32 | 4907 1 009086 5 | -009113 29°23 | 88:55 29-27 Mean | 29:18 009081 009085 e738 —61-07 Dror No. 13. 1862 5182 | | -V,=4708 volts. 1867 | 51:52 V=4079 volts. 10 | -007263 | 1871 | 5083 | t=227-98 ©. | 1879 | 51:23 p=80°27 cm. | Rot 007209 | 1868 | 81-28 | 0, = "05432 — sec, 18:62 80°34 | - @="0002126 cm.| | 9 | -007291 18-81 | 80-29 14 | — =155°6 18°82 79:07 1 | 2 007353 bi 1S27ly emseber ent | 11 | 007319) ease | : 18:84 | 49-90 | e7/3 67-14 10 | :007325 |? , 18:82 | 49-66 : 2 007325 18:87 | 182-28 | 8 | 007330 18°87 1 | 007427 18:80 | 77-44 | | 9 | -007347 Mean Determination of e, N, and Related Constunts. 25 aE ae : ' y F: WW. a= ). nN. | 3 (7 +2): Ww ty tp a'\tp ai ip | | | Drop No. 13 (cont.). 1895 | 76-76 | 1 007187 18:24 | 49-47 | 18°86 49-23 | 10 007345 1881 | 49-15 | 1 ‘007212 1885 | 76:14 9 | -007364 | 1 | 007451 18:89 | 175-98 8 | -007355 1 | , 007333 18:83 | 76:89 | 9 -007365 1 ‘007468 | 18°87. | 48:82 10 | -007352 18:96 18813 | 007825 | -007331 e7/9— 61-26 Dror No. 14. 47-84 es | | -V,=4660 volts. 47°63 pal | V = 4447 voits. 47-78 | 66-79 | 13 | -o11986 | #=23°12 0. | 1 | -012150 | 47-40 | 354-00 2 ‘011905 p=36'80 em. | | 1 | -012350 | ra 47-57. «| «65°91 | 3 012058 |v, = 021448 | 1 | -012053 sec. 47°70 | 36°70 | | a=-0001320 em. | | 4 012053 | 4 47°38 | 86°75 | — =906-4 1 | -O12010 [28 4790 | 65°72 1 | | 3 012051 Fena 47-79 4+} -011959 47:90 15:86 7 012006 |e” ?=68-90 | 3 ‘011903 ; 47-55 | 86°57 | 4 ‘012083 1 ‘012181 47-66 66-20 3 ‘012050 Mean Mean 26 Prof. R. A. Millikan on a new aah i iain \ t.. tp. o ae CEN (aca Yas =) I . is w'\¢ pw =) n\i, ie ; Drop No. 14 (cont.). 47°81 65-70 | 47-70 | 47-65 012050 012040 3861-11 Dror No. 15. 32°50 | 34-73 V = 4656 volts. 32°57 | 34:36 6 009928 Ve= 4645 volts. 32°75 34:57 ¢=23°-10 C. 32-71 34-49 1 "009859 p=d1'35 cm. 32°44 | 52°25 me vy, = 08129 — 32°78 51°85 sec, 5 009994 a—-0001592 cm. 32:6) 5151 32°78 | 51-47 + _200-7 pa 32°88 | 51:27 y 1 ‘O10111° — =-1487 32°84 | 33°71 6 ‘(010032 ie7/3 — 68-97 32:97 | 33-91 2 ‘010011 32°50 | 104-92 4 010042 32°94 104-40 | 32-72, | ‘009994 009999 e/3 — 61-39 Dror No. 16. 18:20 | 29:66 | V ,=3403 volts. 18:23 | 29-76 | V ¢=3389 volts.’ ; 18:14 29°58 '16 | -005521 | $= 23°15, 18:29 29°55 p=20°58 cm. Mean Determination of e, N, and Related Constants. 27 ' at =) tp. m. |—{— — —), 0 aw an 1 /l 1 pes (- + =). nN ty tp Drop No. 16 (cont.). 18°28 | 29:33 wey 2 ‘005592 vy, ='05572 — 18°30 43-55 sec, a='0002111 em. 18°42 | 42-98 14 | -005552 i — =227°8 18°38 42°95 Pe 18:32 | 42°90 1 Tickers 2 | -005452 Pais 18°26 | 80-60 3 20. 1848 | 80-74 e7!° = 69'88 12 | :005586 18°46 80-42 18°38 | 79°55 2 -005600 18:29 | 42:07 | 14. | -005587 1848 | 42:35 | | 2 005541 18:36 79-80 12 | -005583 18°32 | 79-32 3 | -005556 "18-27 34-16 | | 15 | ‘005684 18-48 34-26 | | 18°36 | 18°45 18°34 ‘005557 005569 e7/3 61-97 Dror No. 17. 46°60 | 63:00 3 °012439 V,=4670 volts. 46°87 62°42 1 ‘012461 V ,=4669 volts. 46-79 35:11 4 012463 1 012314 t=23°-12 C. 46°60 | 61°85 p=29°10 cm. Mean 28 Prof. R. A. Millikan on a new ' “(2 ~ “(5 =) ZS eel ee Ng al ee I a'\tp tp n\t, tp Drop No. 17 (cont.). 46°81 | 62:04 3 | -012513 - : 46:31 | 61-65 % = "022940 oo 1 ‘012559 a=-0001312 cm. 46°85 | 273-08 2 | 012516 7 1 "012540 — =962°4 4697 | 61-76 pa 2 | -012524 46°72 | 61°68 1 | -012402 5 188 4699 | 34-94 e=70°85 46°95 | 35:13 ; 4 | -012494 46:88 | 34:83 46°84 | 34:92 46°82 012475 -012492 e7/> — 60-94 Dror No. 19. 1401 | 89:54 | V,=3342 volts. 16 | -005136 1407 | 88-32 V -=3827 volts. 1 | -005218 14:06 | 60-46 — t=23°°10 C. 17| -005146 1411 | 60°37 | p=13-24 cm. | 2 | -005146 ae 1411 | 31-27 | , ey. 0, = "072488 — | 0 ‘00! - 1416 | 3115 I a=-0002300 em. | 2 | 0052 1410 | 60-90 | = — 321-4 | 14:09 | 59-90 ) 17.| -005151 ees 1416 | 59-78 14:11 59:79 P= 13 34 2 | -004996 14:11 | 146°35 15 | 005184 14-09 14:14 | 86-20 16 ‘005157 Mean 1411 14-12 1408 13°94 14:05 14°18 14°10 43°71 43°99 44°11 44°22 44:27 43°75 44:02 44-19 43°95 43°68 43°98 43°75 43°56 43°83 43°71 a Determination of e, N, and Related Constants. 29 60:01 nN. 5) | 2 bo Eye ale ip ) — g Drop No. 19 (cont.). 005127. | 19 | -005153 005153 17| -v05153 005161 005154 Drop No. 22. 009512 009571 4 | -009563 009428 5 009533 009493 4 | -009543 009537 5 | -009542 009563 4 | -009515 009593 3 | -009484 009436 5 | -009465 009566 4 | -009451 009458 jie! 1 ala: = ). | | e7/3—61-20 V ,=3384 volts. Vi= 3369 volts. t=238°:-00 C. p=20°47 cm. v, ="02329 sec. a='0001317 cm. Mean Mean 30 A new Determination of e, N, and Related Constants. rae = 1 Cae hey pe al tp)” Ane) 7A tp. if (cae aa g x n (7 i Ba my — Yo Drop No. 22 (cont.). 43-61 | 29-43 | } | 6 | 099461 43°55 43°88 009516 009510 e7/? — 60-97 Drop No. 25. 26°74 V ,=8347 volts. 26:59 | 31:09 ; V p=8342 volts. 26°60 9 | -007733 $=23°°15 C. 26:84 | 31:06 p=13°80 cm. 26:66 | 30°94 Ne 3 007706 v, = 038191 26:85 | 108°66 sec. 6 | -007768 a=-0001657 em. 26:70 | 107:86 l 1 007829 — =438:3 26°74 5848 Be 26-65 } 7 | 00778 || =veler 26°70 | 58:37 27-02 | 58-42 Pi 1 007766 26:92 | 106-87 6 | -007792 96:96 | 106-32 1 007694 2665 | 58:47 7 ‘007782 fi ‘007785 96°80 | 107°32 3 ; 96°74 | 107-21 6 | :007777 26°82 | 108-48 1 -007668 96:80 | 59:22 7 007782 26°70 26°76 007742 ae) ae II. On the Theory of Osmotic Equiltbrium. To the Editors of the Philosophical Magazine. GENTLEMEN, — N the March number of the Philosophical Magazine appears a paper by the Harl of Berkeley relating to osmotic equilibrium in binary mixtures. The object of the paper is to obtain relations connecting the conditions of the various conceivable cases of equilibrium. The method adopted in the paper is, however, fundamentally erroneous. The fundamental error is contained in the following proposition * :— “Tt is possible to change the pressures on the solution and its mixed vapours (separated from one another by a membrane permeable to both components) in such a manner as to keep osmotic equilibrium between them without any change in concentration taking place.” This proposition is easily seen to be false. The change contemplated involves the constancy of three variables (temperature and two concentrations) and the variation of two (the two pressures). Now these five variables are con- nected by two relations (the conditions of osmotic equilibrium of the two components) so that only three are independent. Hence the change contemplated is impossible. We may prove the same thing in a slightly different manner. In the absence of the osmotic membrane the system would, by the Phase Rule, be bivariant. The membrane merely increases by one the number of degrees of freedom, by removing the condition that the pressure must be the same in both phases. The system is therefore trivariant. Hence ete. It is instructive to examine more closely the fallacy of this proposition. Suppose a mixture of two liquids A and B to be in equilibrium with the mixed vapours through a membrane permeable to A only, under conditions of pressure and concentration, such that they would also be in equili- brium, if placed in communication through a membrane permeable to B. Let pand wp be the pressures of the liquid and vapour respectively. Let us increase p to p+6p and wy to w+é, adjusting the increments so that there is no disturbance of the equilibrium. Let s, and o, denote the “apparent specific volumes” of A in the liquid and vapour respectively. Now the increment of pressure dp will increase the chemical potential of A in the liquid by an amount s,6p, Pa 26%, az On the Theory of Osmotic Equilibrium. while the increment dy will increase that in the vapour by g,0. Hence we must have Sop —o,01. |. el Suppose now that the liquid and vapour are put into communication through a membrane permeable to B. There will not in general be equilibrium, since the incre- ments of pressure which augment equally the values of the chemical potential of A in the two portions of the system, do not necessarily augment equally those of B. Let s, and o, denote the “apparent specific volumes” of B in the liquid and vapour respectively. The respective increments of the values of the chemical potential of B, will be sxdp and o,oy, and the relation ss6p=o,0 9: 21 eee will not in general be verified. Only when by accident the relation * fu _ a (3) holds, will it happen that increments of the two pressures which do not disturb equilibrium through a membrane permeable to A, also do not disturb it through a membrane permeable to B. If we assume the truth of the proposition, the simultaneous fulfilment of equations (1) and (2) at once follows, as does the universal validity of equation (3). This is essentially the method adopted by the Harl of Berkeley in establishing this relation. Hquations (3) and (4) of his paper f corre- spond respectively to equations (1) and (2) above, and lead at once to equation (3) above f. 3 In conclusion it may be pointed out that the fallacy in the Earl of Berkeley’s attempt to establish the proposition, lies in applying his “ Equivalence Theorem” § to a system containing membranes which differ with regard to the components to which they are permeable. am, Yours faithfully, S. A. SHORTER. Sp.) Op The University, Leeds, March 8th, 1917. * Tf this relation were generally true, it would lead to the rule that the densities of liquids were approximately proportional to their mole- cular weights! ; + P. 269. { Equation (6) on p. 270. § P. 265. The Motion of Ions and Electrons through Gases. 33 To the Editors of the Philosophical Magazine. Foxcombe, near Oxford. GENTLEMEN ,— By courtesy of the Editors I have been permitted to see Mr. Shorter’s letter. The limitation to which proposition (0) is subject is ex- plicitly stated in the first two paragraphs of p. 268 of my paper. I am disposed to think that had Mr. Shorter realised the importance of these paragraphs, the irrelevance of his criticisms would have been apparent. I am, Yours faithfully, April 11th, 1917. BERKELEY. III. The Moiion of Ions and Electrons through Gases. By H. M. Wetuiscu, Lecturer in Applied Mathematics at the University of Sydney *. 1. INTRODUCTION. HE experiments described in the present paper were carried out in the Sloane Laboratory of Yale University, and are a continuation of those which have already been described in the American Journal of Science for May 1915. In determining the mobility (4) of the ion as a function of the pressure (p) of the gas, previous investigators had found that the product pk showed an abnormal increase as the pressure of the gas was reduced. This result had been inter- preted as indicating a diminution in the size and mass of the ion at relatively low pressures; for the negative ion in air this diminution appeared to set in at pressures below 10 cm., while for the positive ion it did not occur till the pressure was reduced below 1 mm. The investigation to which reference has already been made provided experimental and theoretical indications which were entirely different from the foregoing. For the posi- tive ion in air no anomalous results were found; the law pk=const. held good to the lowest pressure employed (05 mm.). The negative carriers were {ound to consist of two distinct kinds, electrons and ions, the former coming more and more into evidence as the pressure of the * Communicated by the Author. Phil. Mag. S. 6. Vol. 34. No. 199. July 1917. D 34 Mr. EH. M. Wellisch on the Motion of as was reduced. When once this separation had been effected all the preceding anomalies disappeared ; the law pk=const. was verified for the negative ion in air from 1 atmosphere down to *15 mm., indicating that the ion remains unaltered in character over this range of pressures. The electrons appeared to travel freely through the gas without attaching themselves to molecules. No indication was found of any intermediate stage in the nature of the negative carrier, the separation between the ions and the electrons remaining throughout clearly marked. In the present experiments these results have been extended to other gases: in accordance with expectation, the abnormal mobility values found by previous investigators for the negative ions in hydrvgen and carbon dioxide were shown to be capable of a similar explanation, all anomalies dis- appearing as soon as the resolution of the carriers into ions and electrons was effected. A brief study has been made of the motion of free electrons through carbon dioxide at relatively high pressures; in addition, the motion of ions through a number of vapours has been investigated. A few discussions bearing upon the pbysical interpretation of the results have been included: in particular, certain outstanding problems of ionic theory have been specially considered. 9. EXPERIMENTAL MretHop AND ARRANGEMENT. A description of the experimental method and apparatus has already been published ; on this account it seems ad- visable to repeat here only the essential features, reference being made to the previous paper for further details. More- over, advantage will be taken here to enter into greater detail in connexion with certain features of the method to which only a brief allusion was previously made. The method employed in the determination of the mobi- lities was that devised by Franck and Pohl*. The ionization vessel (v. fig. 2) consisted of a brass cylinder divided into two compartments by a brass partition containing a circular aperture. In the upper compartment was a copper plug on which a layer of polonium had been deposited ; great care was taken that the radiation from the polonium was con- fined to the upper compartment. A circular electrode A was situated about 3 cm. above the aperture and was in metallic * Franck and Pohl, Verh. Deutsch. Phys. Ges. ix. p. 69 (1907). Ions and Electrons through Gases. 35 communication with the case of the vessel. The lower com- partment contained a gauze electrode insulated by a thin ebonite ring from the partition. Two centimetres below the gauze was the electrode e connected to the electrometer : this electrode was surrounded by a guard-screen (W) connected to earth by means of a guard tube. Fig. 1, +¥ -V epee Mil Fig. 1 illustrates the method employed to effect the com- mutation of potential. The commutating disks were of brass with a number of fibre segments of equal width placed at regular intervals along the periphery. The two potentials V, and —V, were connected across the terminals of a large metal resistance R in series with the commutator ; it was not in general convenient to alter the potential V, except in steps of 40 volts each, and on this account the potentiometer device (v, 7, p) was employed to effect finer gradations of potential. When the commutator is in action the potential of K (fig. 1) should alternate between # and —V,, where vp+r(Vi+ Vo) Bye Ri pe leah Wh (1) Owing, however, to the time involved in the establishment of potential, this formula will be sufficiently valid only if care be taken to maintain a satisfactory relation between the frequency of commutation and the resistance R. This was effected by an experimental method described later. We shall assume here that the potential of the gauze is given by a and —V, alternately, the former potential lasting for a D2 t= —V,+R 36 Mr. BE. M. Wellisch on the Motion of fraction 7 of the total time; this fraction can be determined experimentally. Under these conditions the mobility & of 2n the ion under consideration is given by hao where n is the number of complete alternations per second, d is the distance between the gauze and the electrode e, and Vis the critical potential, 2. e. the value of w which is just sufficient to enable the ions to reach the electrode e before the field is reversed. ANDINA & Pew eA Wd } y ab Ue aE v | it The diagram of connexions is exhibited in fig. 2. As in the previous experiments, two commutating disks were em- ployed: one of these had 20 fibre segments while the cireum- ference of the other was half fibre and half metal. The motor was worked generally on 110 volts which afforded approximately 42 revolutions per second. The double-pole double-throw switch 8, when thrown to the right, completed the connexions as exhibited graphically Ions and Electrons through Gases. 37 in fig. 1. When thrown to the left, connexion was made with a subsidiary potentiometer system (6); in this position the quadrants of the electrometer could be commutated in po- tential between zero and any convenient potential read off on the potentiometer. The use of this device in testing the contact at the brushes, in estimating the value of f, the fractional duration of contact, and in adjusting the position of the electrometer-needle for observations, has been de- scribed in the previous paper. For large current values readings were taken with the capacities B and © added to the electrometer system ; the capacity of the system was then increased 174 times. 3. EXPERIMENTAL PROCEDURE. For convenience in manipulation a table was prepared of the potentials assumed by the gauze for different values of p, V,, V2, and R. This was effected by means of formula (1), which for the purpose was put in the following form:— g=Vy+ [2 — (Vi+V.4 ee) \ oe CASA Ye oMEAMAG isk (oe a cal eva ow Dag Me) vw was always chosen equal to 40 volts and 7 was always 15,000 ohms. The calculated values of c for various values of pand V,+V,2 were then tabulated, and the value of # under any desired conditions could be quickly obtained. Establishment of Potential. It was important to ascertain that the experimental con- ditions admitted of an effectively instantaneous establishment of the withdrawing potential —V, through the resistance R; in other words, the ions must commence to retire as soon as the commutator-brushes make contact with the fibre seg- ments. This point was tested experimentally in the following manner: the commutator and resistance R were put in series with a battery V (fig. 3) of which one terminal was earthed ; K represents a Kelvin multicellular electrostatic voltmeter which was included in the manner shown in the diagram. The commutator was set in motion at its highest speed, and readings were taken on the voltmeter corresponding to different values of R. If the values of R were excessively large there would not be sufficient time during an alternation to admit of the earth-connexion with the gauze being fully 38 Mr. E. M. Wellisch on the Motion of established, and in consequence the steady reading of the voltmeter would be too large. It was found that when the small-frequency commutator was employed this steady reading remained constant for values of R up to 1,000,000 ohms; for the high-frequency commutator the value R=200,000 afforded a reading greater than the normal by less than 2 per cent. Inasmuch as the potential (V2) iH ca R DA PP LDA LIN TO GAUZE was always chosen considerably greater numerically than the advancing potential («), the value R=200,000 was sufficient to ensure the realization of the desired conditions. In the present series of experiments this value of R was chosen in preference to a smaller value because in the deter- mination of electron velocities V, is often small, and it is advisable to have c in formula (2) small compared with Yj. Manipulation of Switches. In general, when the gauze is raised to any potential, the electrode e is raised by induction to a potential which has to be taken into consideration when the electrie field is estimated. It was found possible, however, by a suitable manipulation of the switches 8, f, and g to arrange that the electrode e was practically at zero potential when the potential (x) had been established on the gauze, so that no correction for induction was necessary. ‘The series of operations involved in taking a single reading was as follows :— (i.) Potentiometer (5) fixed at a convenient value so that the electrometer-needle should have a suitable range of Ions and Electrons through Gases. 39 deflexion: & closed: S closed on 6 side: f and g both closed : earth-key K open: motor and commutator running but not Operating on account of the short-circuit at g: capacities B and ( included in the system. (ii.) & opened: S swiiched to the right. (iii.) g opened, if it is desired to work with added Sai. Or Qi.) Capacities Band C cut out and g then opened, if it is desired to work without added capacity. It will be seen from the foregoing that the effect of induction was to alter only the reversed field whose value did not need to be known at all accurately. The electrometer- needle always experienced a small kick when the switch g was opened, but this quickly subsided and the current was measured with the needle in steady motion, the midpoint of the range of deflexions being so chosen as to coincide with the zero of the instrument. 4, EXPERIMENTAL RESULTS. (A) Electrons in Gases. In fig. 4 of the previous paper typical curves were given showing the relation between the current due either to positive or negative ions and the potential (#) for various pressures; from such curves the critical potential Vy could be deduced and the ionic mobility determined. In figs. 5 and 7 of the same paper there were given the curves corre- sponding to the negative carriers in air at relatively low pressures ; the characteristic feature of these curves is their compound nature resulting from the independent passage through the gas of electrons and ions. It is convenient to designate as I curves the former type which is due solely to ions, while the latter type may be referred to as HI curves ; moreover, those curves or parts of curves which arise solely from the motion of electrons will be called E curves. On resuming the experiment, an investigation was made of the gases CO, and H;. The CO, was prepared in a Kipp’s apparatus by means of the action of diiute HCl on marble and was passed through NaHCO; Aq. in order to remove acid fumes: the H, was obtained by the action of dilute HCl on zine, and was passed through KOH Aq. In each ease the gas was passed through a series of tubes of CaCl, and P,O; in order to remove traces of moisture. A series of I and EL curves was obtained for these gases under various conditions, 40 Mr. EH. M. Wellisch on the Motion of ai few examples of the latter type being given in fig. 4. The free electrons were more numerous in each of these gases than in air at the corresponding pressure: this point is brought out by the fact that with the same frequency of commutation the electrons appeared at much higher pressures than in air, e. g. it was just possible to detect electrons in air at 8 cm. pressure, whereas in CO, they appeared in large numbers at a pressure of 14 cm., and in H, they were readily Fig. 4. ee eee So. ————— ¢ Oa iS | ots e ey ~H x Deg cy ig pity 3 ie CY BN ¢ fo eae iS tah ——- dR fo se samen enn cult i Vee Wea ie / ee [oN ve eameet | i it fi i a) 0) | 40 _ 60 50 {00 led observable at atmospheric pressure (v. curve A, fig. 4, which was obtained with a frequency of only 42-6; also curve in fiv. 5). This result was to be expected from the conclusions of previous experimenters who had found that the abnormal increase in the ionic mobility set in for these gases at higher pressures than for air. It should be remembered that we cannot form any definite inference as to the relative number of electrons by comparing the ionization currents in the E curves for different gases at the same pressure because these currents are due to the electrons which have passed through the meshes of the gauze electrode, and the fraction of electrons which accomplish this depends upon the gas concerned. When the pressure of the CO, or H, was relatively high, the free electrons appeared to be extremely sensitive to the presence of impurities in the gas under consideration ; the number of free electrons was greatly decreased if the gas 2 Ions and Electrons through Gases. 41 were allowed to stand undisturbed for a few hours in the measuring vessel, which was presumably air-tight. This effect is illustrated in the curves of fig. 5: curve A refers to CO, at 79 mm. pressure, the readings being taken quickly after the introduction of the gas; curve B exhibits the values after the gas had been allowed to remain 24 hours in the closed vessel. For lower pressures of the gas this effect practically vanishes; with CO, ata pressure of 44 mm. the HI curve obtained after the gas had remained undisturbed in the vessel for two days was identical with that obtained immediately after the introduction of the gas. It is probable that the above effect arose from a very slow leak of oxygen into the vessel from the outside atmosphere ; actual experiments were performed to test this point, and it was found that traces of air added to CO, or H, at relatively high pressures resulted in a marked decrease of the number of free electrons, whereas when these gases were at low pressures the number of electrons was not appreciably affected by the admixture. It should, however, be mentioned that a similar though much more intense effect was found in experimenting with the free electrons in the vapour of petroleum ether (v. sec. 4D); in this instance the diminution in the number of electrons was very rapid, and could not reasonably be ascribed to a small leak of air into the apparatus. All the indications pointed to the appearance in the vapour of a constituent capable of absorbing electrons at ordinary temperatures. It is con- venient to refer to nuclei, whether molecules or aggregations, which possess this property, as “electron sinks”; the electrons cannot remain in the free state during their motion through a gas which contains these sinks other than in excessively small quantity. Ali the experimental evidence indicates that the molecules of oxygen do not belong to this class of impurities, and that the larger electron velocities attendant upon the act of ionization are necessary for the formation of negative oxygen ions. It is of course possible that the decay of the electrons in CO, and H, does not arise from an air leak but is due to an ageing effect similar to that in petroleum ether. In this connexion several unsuccessful attempts were made to remove possible nuclei from CO, which had been allowed to remain for several hours at a pressure of 81 mm. in the measuring vessel. In one experiment the gauze electrode was main- tained for several hours at a potential of —160 volts in the hope that the electrons which were being continually produced 42 Mr. E. M. Wellisch on the Motion of would ultimately remove the nuclei from the gas; however, the current measurements failed to indicate any tendency to restore the original condition of the gas under which permanently free electrons were in evidence. The same gas was subsequently passed several times through P.O; by means of a mercury-reservoir attachment in order to remove any trace of water vapour which might have arisen from the metal walls; the free electrons, however, did not reappear, and the possibility of the existence of nuclei con- sisting of molecules of water vapour was thus excluded. A few experiments were made to ascertain whether free electrons are present in carbon monoxide. This gas was liberated by the action of concentrated sulphuric acid on potassium ferrocyanide and was passed through solid caustic potash, calcium chloride, and phosphorus pentoxide before admission into the measuring vessel. A typical HI curve was obtained for CO at a pressure of 13 mm., demonstrating thus the existence of free electrons ; these were, however, not nearly so numerous as in air at the same pressure, and, as the manipulation with this gas presented difficulties, it was not considered expedient to extend the investigation. It seems fitting to refer here to an apparent difficulty in connexion with the existence of free electrons in gases. The electrons were shown to appear in measurable amount in dry air at pressures as highas 8 cm., and yet it has been mentioned in this section that a trace of oxygen is sufficient to cause them to disappear from CO, or Hy, at relatively high pressures. Reference is made later (sec. 5) to this apparent discrepancy; the difficulty is in large measure removed by a consideration of the experimental fact that the sensitivity to oxygen decreases rapidly as the pressure of the original gas. is reduced. To take actual figures, it was found that a trace of air would rob H, at 1 atmosphere of its free electrons, and yet in a mixture of H, at 825 mm. and air at 25 mm. the electrons appeared in considerable numbers. (B) Motion of Free Electrons. A number of experiments were undertaken to determine the velocity with which the free electrons moved in an electric field through CO, and Hy. Mobility values have already been assigned by Franck * for the electrons in argon, helium, and nitrogen at atmospheric pressure; the values given were respectively 209, ca. 500, and 120 cm. per sec. per volt per cm. The mobility values were found to be * Franck, Verh. Deutsch. Phys. Ges. xii. pp. 291, 613 (1910). Ions and Electrons through Gases. 43. extremely sensitive to the presence of impurities in the gas under consideration, the slightest trace of oxygen, for example, causing a considerable reduction in the value. Recently Haines * has investigated the motion of free elec- trons in pure nitrogen at atmospheric pressure, and has obtained a mean value of 367 for the mobility. Carbon dioxide appeared especially suitable for expe- riments in this connexion because the electrons were rela- tively numerous in it, and at the same time the density of the gas was sufficiently great to justify the belief that the velocities would not be inordinately large and thus incapable of measurement with the apparatus at disposal. Even with the high frequency of 800 alternations per second and at the highest practicable pressures of the CQ., it was found that the values of the critical potential (Vo) were considerably less than 10 volts, so that the observation error in the deter- mination of the electron mobility was of necessity considerable. Moreover, there was also the difficulty connected with the presence of the ageing effect which, as mentioned above, occurs at the higher pressures ; it was of course not feasible to attempt determinations at the lower pressures where this effect is absent, because the electron velocities become excessively large. It was in every instance found that the effect of age (2. e. of allowing the CO, to remain for any length of time in the apparatus) was to reduce considerably the velocity of the electrons. On this account great care was taken to exclude impurities, the gas being in all cases swept several times through the measuring vessel, and the observations quickly made after the final introduction. In figs. 5 and 6 there are given a few typical H curves which were obtained in the determination of Vy for the free electrons ; the ions do not make their appearance until much higher potentials are employed. Reference will be made later to the fact that the experimental results rendered doubtful the assumption that the velocity of the electron is proportional to the applied field, so that the use of the term “mobility”? is not certainly justified; however, it was thought useful to make the calculations on the assumption that there exists a distinct mobility for the electron just as for the ion. In the following table there are given the results of the mobility determinations for freshly prepared CO, together with some of the results for CO, in various degrees of impurity; the symbol K denotes the mobility * Haines, Phil. Mag. vol. xxx. p. 503 (1915). 44. Mr. HE. M. Wellisch on the Motion of reduced to atmospheric pressure on the assumption of the validity of the law pk=const. Freq Ff nat We k K Remarks. 8079 | 545 | 35:5 | 14 | 4982 | 1976 | Fresh. 890°6 | -518 54 9-45 | 9587 | 183°8 04-5 | “527 87 98 | 2933 | 9556 7968 | 530 | 137 45 | 1336 | 241-0 8333 | -540 79 30 | 2056 | 2140 8301 | -545 BA 8 "461-7 | 54:1 | 2 days old. 834-2 | -550 93 22 | 2757 83:4 do. 833°3 | -540 "9 46 | 1341 | W894 oe iares onele ; 79 mm. CO 898-2 | -540 81 Gan | ead) { Ce In fig. 5 there is given an E curve for freshly prepared hydrogen at atmospheric pressure; the value of K deduced from this curve was 1700 cm. per sec. Fig. 5. Cuirrent., . The values of K for freshly prepared CO, are scarcely in sufficient agreement to justify the assignment of an average value. Itis evident from the table that the electron velocities are very sensitive to the presence of impurities ; the highest value of K obtained was 255 cm. per sec.; but even this a Ions and Electrons through Gases. 45 cannot be regarded as a maximum, as a greater degree of purification would probably result in still higher values. Fig. 6. Ss pee | ae ! i Current. Volts. There is some evidence of an indirect character that the electron does not move with a velocity which is strictly pro- portional to the applied field, but traverses with an accelerated motion distances comparable with the distance between the electrodes. The close approach for small potentials of the E curves in fig. 6, which refer to CO, at the same pressure (137 mm.) but with different alternation frequencies, sug- gests very large values for the velocities of the electrons; this js more readily understood if we apply the formula 2 = eg — where 7’ is the current for potential V when the alternating field is employed and 7 is the current for potential V directly applied. If we take the velocity calculated from vi at the higher frequency, viz. k=1336 at 137 mm., the above formula (with d=2) gives i,'/i,,=4'1 tor V=5-78 volts, where the suffixes 1 and 2 refer to the low and high frequency respectively; the value obtained from the experi- mental curves is only 1:3. Similarly for V=6-96 volts we cbtain a calculated ratio of 2°6, whereas that obtained experimentally is 1-2. These considerations would seem to imply that the value £=1336 is too small, and that the critical potential is really smaller than the value (V)>=4'5) apparently obtained. In this connexion it is a significant fact that several of the A6 Mr. H. M. Wellisch on the Motion of H curves, especially those obtained at the highest pressures, showed a distinct curvature in the neighbourhood of the potential axis, the tendency being to shift the point of inter- section towards the origin. ‘This shape of the current- potential curves in the vicinity of the origin suggests acce- lerated motion of the electron or a slow acquisition of a terminal velocity. Further experimental data are of course necessary before the nature of the motion of the electron is definitely ascer- tained; the suggestion here given is that the electron may traverse a considerable distance with accelerated motion before its terminal velocity is acquired. It should be re- membered that Franck and Hertz* have already shown that the collisions of electrons with the molecules of the inert gases are practically perfectly elastic, so that the drift motion of the electron would under these circumstances be accelerated. ‘The experiments with regard to the effect of impurities upon the number of free electrons in CO, or H, strongly suggest that the collisions of electrons with the molecules of these gases have a high degree of elasticity, although naturally not so high as with the inert gases. The effect of this high but imperfect elasticity would be to cause the electrons when moving under an electric field in CO, or H, to move with an accelerated motion until their terminal velocity is acquired. On this view the effect of traces of impurities in the gas in diminishing the velocity of the electron is readily explained ; the impact of the electron with the molecule of the impurity is in all probability either inelastic or considerably less elastic than the collision with the gas molecule, and, in consequence, the electron is unable to acquire as great a velocity as in the pure gas. (C) Electrons in Vapours. The demonstration of the existence of free electrons in air, CO,, and H, at relatively high pressures rendered it fairly obvious that all permanent gases were able to contain electrons in the free state. Franck’s experiments had shown previously that the inert gases were especially conspicuous in this respect, the negative carriers appearing to consist entirely of free electrons. It became of interest to extend the investigation to the case of vapours, especially as these are liable to occur as impurities in gases. It was thought extremely improbable that the electrons, if they were present in the free state, would occur in large numbers except at * Franck and Hertz, Verh. Deutsch. Phys. Ges. xv. pp. 373, 613 (1918). Tons and Electrons through Gases. 47 very low pressures; preliminary trials with a few vapours justified this conclusion. With the high-frequency com- mutator there was not the slightest indication of the presence of free electrons either in dry SO, at a pressure of 7 mm. or in CH,I at a pressure of 28mm. It was, however, quite possible that lower pressures would bring the electrons into evidence; but as the apparatus did not readily lend itself to securing low vapour pressures, the investigation was resumed in a slightly different manner. A small quantity of the vapour under consideration was mixed with a permanent gas, and experiments were made to ascertain whether free electrons cbuld continue to exist in this mixture; if the vapour molecules behaved as electron sinks and were present in appreciable amount, then the number of collisions and subsequent : attachments between electrons and vapour mole- cules would be sufficiently great to prevent the existence of a Th PORE he Coy Coa ee OSes Avene anaes PY lear? Ter er weet | err LCL Perera ert eee PE er EH oo oS 6/77 CurFent, Volts. free electrons. ‘This information was of importance in view of the experimental results with regard to the effect of im- purities on the number of free electrons ina gus. Three vapours were tried in this connexion, viz.: ether, alcohol, and water ; these were chosen because they were deemed to be the most probable absorbers of electrons. In each of these instances hydrogen at a reduced pressure was chosen as the gas with which the vapour was mixed because of the copious supply of free electrons which it affords. An EI curve was first obtained for dry hydrogen at a pressure of 36 mm. (fig. 7); ether vapour was then Jase) SRuGeP ar Zon Pou AG. |, 166 go 100 120 140 160 | 48 Mr. E. M. Wellisch on the Motion of admitted until the pressure of the mixture was 38 mm., and the readings were again taken. It was found that even in the presence of 2 mm. of ether vapour a considerable number of free electrons were able to traverse the distance between the electrodes. The number was less than in the pure hydrogen, but the EI curve for the mixture (fig. 7) was sufficiently definite to justify the conclusion that the molecules of ether vapour do not behave as electron sinks. The experiments with alcohol vapour were conducted in a similar manner; an EI curve was obtained for a mixture consisting of hydrogen at 35 mm. and alcohol at a pressure slightly less than 1 mm. The number of electrons was again distinctly smaller than in the pure gas, but was suffi- ciently great to make it evident that the molecules of alcohol were unable to absorb the free electrons. In order to experiment with traces of water vapour present in the gas, the tubes containing the drying agents were removed so that the hydrogen passed into the measuring vessel directly after generation in the Kipp’s apparatus. The moist hydrogen was introduced at a pressure of 37 mm., and a current-potential curve (v. fig. 7) was obtained in the usual manner; the presence of the moisture caused a reduction in the number of free electrons, but these were in sufficient evidence to show that the water molecules do not behave as electron sinks. It is of course quite possible that in all these instances a loose attachment may occasionally exist between the electron and the vapour molecule; the experimental results indicate, however, that such an attachment, if it occur at all, persists only for a time which is smallin comparison with that during which the electron remains free. Vapour of Petroleum Ether and the ageing effect. The previous experiments with CO, and H, suggested that the atoms of carbon and hydrogen were in great measure responsible for the relatively large number of free electrons in these gases as compared with air. It became of interest to make aspecial study of some member of the paraffin series whose molecules contain only atoms of carbon and hydrogen, or indeed of any vapour which does not contain electro- negative atoms such as those of oxygen or iodine. It was originally proposed to make the experiment with pentane, but as this was not immediately available the vapour of petroleum ether was employed instead. Petroleum ether (sp. gr. ca. 67) consists of a mixture of pentane (C;H,,) and hexane (O,H,,) ; its molecules contain, therefore, only Ions and Electrons through Gases. 49 atoms of carbon and hydrogen. A number of determinations of ionic mobilities were also made for this vapour; reference is, however, made to these only as far as they concern the motion of electrons, the actual values obtained for the mobi- lities of the ions being deferred to a later section (4 D). The first experiments with this vapour, which was in- troduced at a pressure of 95 mm., gave a normal value (K=°41) for the mobility of the positive ion; the current- potential curve for the negative carriers (fig. 8) was distinctly abnormal, as it afforded evidence of two types of carriers : Seo i Riri Fi OS ae es as eka ied CukFé@At, nN) aS 7auuner uae Baer be 40 ‘60 BO 100 {20 140 Vo/lts. in addition to the normal negative ion (K=-44) there appeared a carrier (a, fig. 8) for which K had the value 1°692, which is about four times as great as one might have reasonably expected for the negative ion, and, on the other hand, considerably less than the value corresponding to a free electron. On attempting to repeat the experiment after the vapour had been allowed to remain for about two hours in the vessel, only the normal value was obtained for the mobility of the negative ion. In the next experiment, after a preliminary evacuation of the vessel, streams of vapour were swept through repeatedly in the hope of removing traces of impurities; the vapour was finally admitted at a pressure of 76 mm. and the readings quickly taken. The curve obtained is given in fio. 8; the direction of the arrow signifies that the current measurements were made in descending order of potential. It will be seen that this curve shows the presence both of Phil. Mag. 8. 6. Vol. 34. No. 199. July 1917. E 50 Mr. E. M. Wellisch on the Motion of ions and of free electrons; the ions enter at 80 volts and possess a normal mobility, viz. K=:430. The curve marked 1 was obtained only a few minutes before that marked 2, and it will be observed that the electrons have decayed appreciably during this short interval of time. The third curve was obtained 24 hours after the introduction of the vapour; there is now only the slightest indication of free electrons, while the negative ion has still a normal mobility (K=:428). Subsequent experiments were made with freshly intro- duced vapour at a pressure of 20 mm., and gave evidence of a very large percentage of electrons ; however, the ageing effect was very pronounced, the free electrons decreasing in number so rapidly that no regular curve was obtained. The general indications seem to be that in the pure vapour of petroleum ether a large fraction of the negative carriers are free electrons, the negative ions if presentatall appearing only in small numbers; the free electrons are, however, extremely sensitive to the presence of some constituent which arises gradually in the vapour, with the result that at the expiration of a few hours the electrons have disappeared and the current of negative electricity is due entirely to ions. The nature of the constituent which occasions the ageing effect in the vapour can at present only be conjectured : systematic experiments are necessary before a definite con- clusion can be reached. There is distinct evidence, however, that we are dealing here with a true electron sink; in other words, this constituent, whatever be its nature, is capable of absorbing an electron during its drift motion through the vapour, and in this respect must be carefully distinguished from impurities such as oxygen, which seem to require for the absorption of electrons velocities considerably higher than those which are afforded by thermal agitation at ordinary temperatures. The effect of the latter type of impurity is to reduce the number of free electrons ina gas and at the same time to diminish appreciably the velocity of the electron through the gas ; this diminution in velocity has been ascribed (v. sec. 4B) to the comparatively inelastic impact between the electron and the molecule of the impurity. In the experiments with the vapour of petroleum ether the effect of the impurity is to cause likewise a reduction in the number of free electrons and a diminution in the electron velocity; the electron, however, appears now to be capable of acquiring all velocities intermediate between that of a free electron in the pure vapour and that of a negative ion. We seem, therefore, to be dealing with a carrier which changes continuously and progressively from a free electron fons and Electrons through Gases. 51 to a negative ion; the most feasible hypothesis is that the electron as it drifts through the vapour is for part of the time in the free state, and for the remainder in attachment with the molecule of the impurity. It is highly probable that this attachment, occurring as it does as a result of ordinary thermal motion, is of a very loose nature and is liable to be broken at molecular encounters; we would thus expect continual alternations of the electron between the free and combined states. With regard to the nature of the sink which is gradually formed in the vapour of petroleum ether, nothing at all definite can be said. We may imagine that polymers or small aggregates of pentane or hexane are formed gradually under the influence of the radiation from the polonium ; such systems would probably be able to form stable negative ions tor large electron velocities and unstable ions for smal velocities. Initially, when the vapour is pure, the negative carriers are for the most part electrons; as the sinks appear, the velocity of the electrons would be reduced throngh the formation of unstable ions. The fact that ultimately the carriers consist entirely of negative ions may be explained by ascribing to a polymer the property of being able to effect occasionally a union between an electron and a molecule of the vapour. (D) Lons in Gases and Vapours. Gases.—The law pk=const. was verified for both the positive and the negative ions in dry air over a wide range of pressures. Some of the values obtained experimentally for K at the lower pressures have been given in the previous paper, and should be sufficient to illustrate the unchanging mature of the negative ion. A set of values obtained for the mobility of the positive ion in air at low pressures is given below. The first table refers to values obtained by means of Press. V , mm. alts | I Freq. hy. K,. 8-31 530 | “548 8643 1195 1°31 1:66 115 540 834-2 537-5 1:17 1-64 10:2 540 850:9 6180 1:33* 1:01 60 | ‘574 851-7 989-0 1:31 563 30 | ear 8365 1944. 1:44 -416 2:5 584 862°1 9363 1:29 321 1°75 “585 8333 3254 1:37 Mean value of K, : 1°32 * a=8 volts (v. fig. 2); for the other determinations a-=20. K 2 52 Mr. E. M. Wellisch on the Motion of determinations of the critical potential Vo, while the values given in the second table were determined by means of the an 1 formula k= vi a Press, V : - ate Spe 1. w. be Freq. Ten K,. 1°66 14:3 418 52 540 829-2 557-4 22 1:64 14:3 2°84 34 540 851:1 568°2| 1:23* | 1:01 9:03 2:20 40 ‘5T4 851°7 $65 1:28 563 5:00 1:73 34 57 836°5 1781 32 ‘416 6:26 1°506 46 “584 862°1 1975 1:08 B21 4:17 1°84 60 585 8333 3090 1:31 151 1:13 1:05 "155 “565 856°0 7265 144. | 052 55 298 068 565 842°4 | 18250 1:25* | 052 1°15 "333 137 =| *565 842°4 | 191380 1:31* Mean value of K, : 1:27 * a=8 volts (wv. fig. 2); for the other determinations a=20. The mean value obtained for K, at the higher pressures was 1°23, which is in sufficient agreement with the above values to justify the conclusion that the nature of the positive ion is independent of the pressure. The mobilities of both the positive and the negative ions in CO, and H, were determined over a wide range of pressures: there was no evidence in either gas of any systematic alter- ation in the value of pk, or pk, as the pressure was reduced. It was deemed unnecessary to extend the determination for these gases down to the low pressures employed in the case of air inasmuch as it was apparent that the processes were. entirely similar ; for this reason the law pk=const. was only verified down to a pressure of 4 mm.in CO, and 12 mm. in H,. The mean values of K for air, CO., and H, were estimated from the results obtained at the higher pressures, where the observation error is relatively small, although the results at the lower pressures showed good agreement. The values thus obtained are as follows :-— Ke K,. Kaj Mies he 1:28 1:93 1-57 COL eae 73 1:07 1-47 Ply ores 511 9-67 1:89 The values of K refer as usual to the dry gas at atmospheric: pressures, and are expressed in cm./sec. per volt/em. Ions and Electrons through Gases. 53 For each gas the value of K, is less and that of K, is greater than the value usually assigned ; for the sake of comparison the values of the mobilities obtained by Zeleny* are given below:— Ki | K,,. Keak Wai ec: 1:36 1:87 1:375 BOM 76 81 1-07 a et 6°70 7:95 1:19 Zeleny’s values for air and CO, are in fair agreement with those obtained by the writer in former experiments and by several other observers. The cause of the discrepancy is not apparent: it is scarcely probable that any defect either in the method employed in the present experiments or in the determination of any of the constants involved would cause the ascertained values of K, and K, to vary in different directions. It should be mentioned that the present values for CO, and H, are in fair agreement with those obtained by Blane +, who used Franck and Pohl’s methed with X-rays as the ionizing agent. Blanc’s values are: K,. | K,. K,/K,. Ser... | 83 1-027 1-24 a eae | 5°33 10:00 1°88 Haines} has recently obtained the value 5°4 for the mobility of the positive ion in pure hydrogen; the negative ions did not appear till a trace of impurity was present, and under these conditions their mobility was about 8. Effect of Water Vapour. A few experiments were performed to ascertain the effect on the ionic mobilities of saturating with water vapour the gas under consideration. In these experiments the water vapour was introduced by ebullition when the gas in the vessel was at a low pressure; after this operation the gas was admitted till the desired pressure was attained. Asa result of condensation of the vapour, the insulation was extremely defective and the currents could not be determined by observation of the rate of deflexion of the electrometer- needle. However, it was observed that the spot of light * Zeleny, Phil. Trans. A. excy. p. 193 (1900). Tt Blanc, Journ. de Phys. vii. p. 825 (1908). 1 Haines, Joe. cit., also Phil. Mag. xxxi. p. 3389 (1916). by ie Mr. E. M. Wellisch on the Motion of assumed for each value of the potential a definite position on the scale: in this position the ionization current is balanced by the current due to the leak through the condensed vapour, and is thus proportional to the steady reading of the electro- meter. The conductivity of the condensed vapour remained constant over a sufficient interval to enable the critical potential to be determined in this manner. Typical curves obtained for saturated H, are given in fig. 7*. The results of the mobility determinations for saturated H, and CO, together with the calculated values for the dry gases at the same pressure are given below :— mm. | 1 io He dayne ee ae g09 | «4:80 9-09 SauUmatedesessecmecccees 809 | 5:03 6:105 COME y Oa I eae 79) ae ‘72 1-06 saturated .........6. 769.) Ri "88 | In addition, other experiments were performed in which small quantities of water vapour and ether vapour were mixed with hydrogen: in these experiments the mobilities were determined in the ordinary manner. The following results were obtained :— EM kG, K,. mim. ry ELS ry Ny s/t aiscie osete 517 T51 14°21 H,, with water vapour... 517 7:43 10:27 H, with ether vapour ... 766 4°13 7:25 The presence of water vapour appears thus to be without effect on the mobility of the positive ion, but occasions a marked diminution in that of the negative ion. This is in accordance with the results obtained by previous investigators. The diminution in the mobility of the negative ion is too great to be accounted for by the extra resistance to the motion of the ion which arises when the vapour is mixed with the gas. The diminution may be explained in part by assuming that the effect of the water molecules is to cause inelastic impacts with the negative ions, and thus prevent them from acquiring the larger terminal velocities which they attain in the dry gas. * The current scale for these curves is different from that for the other curves in fig. 7. fons and Electrons through Gases. d5 It seems probable, however, when the gas is saturated with water vapour, that condensation occurs round the negative ion and that the diminution in mobility is to a large extent due to this process. We should thus have the negative ion constituted by a cluster of water molecules round a charged nucleus : it should be carefully noticed that the existence of such a cluster in a moist gas affords no evidence as to the nature of the ion ina dry gas. Inalater paper experimental evidence will be given which indicates that the water molecules are not held together i in the eluster by the electrostatic forces due to the charge on the ion, the function of the charge being merely to determine the act of condensation. Vapours. A number of measurements were made of the mobilities of the positive and negative ions in a few vapours; this was of interest as affording : a comparison with the results obtained by the different method employed in a previous investigation”. The mobilities were determined in the usual manner; the average values estimated from a number of determinations in good agreement are recorded below, together with the corresponding values taken from the previous research. The figures in the second column give the minimum and maximum pressures employed; the mobilities given correspond as usual to a pressure of 1 atmosphere. Pressure 1915. 1909. Vapour. range. mim. KG. KS. EEA OM al ie Ethyl ether ............ 67—126| -27 "346 ‘29 31 Ethyl alcohol »......... 23— 39| 39 ‘412 34 “27 do. (saturatedt) ...)38— 42] -365 392 Petroleum ether ...... 74—115|} -370 "440 "36t 35t Sulphur dioxide ......|73— 94| °415 “414 “44 “41 Methyl iodide ......... 63— 65} :24 "233 "21 "22 The agreement in the case of the positive ion is as good as could reasonably be expected in view of the difficulties attendant upon experimenting with vapours; we can say with a high degree of certainty that to each vapour there corresponds a definite value of the mobility of the positive ion. The mobilities of the negative ions in alcohol and petroleum ether are, however, in greater disagreement than * Phil. Trans. ser. A. vol. ccix. p. 249 (1909). + Measured by the method employed with saturated water vapour. t Pentane. 56 Mr. E. M. Wellisch on the Motion of can be accounted for by experimental error. We have seen (sec. 4 (") that in the pure vapour of petroleum ether the negative carriers are practically all electrons, and that the negative ions come into evidence only when the vapour is allowed to remain for some time in a closed vessel. We are therefore constrained to associate the negative ions in this vapour with impurities ; and it is of course not improbable that there are other vapours in which the existence of negative ions is conditioned by the presence of some impurity. Refined experiments on ionic mobilities in vapours are necessary before the nature of the negative carriers can be determined. 5. Discussion oF RESULTS. It is proposed to discuss briefly in this section the signi- ficance of the results of the present experiments in connexion with the theory of electric conduction in gases. Several of the points brought forward have already received attention in the previous paper; reference is made here to these only for the sake of continuity. The experiments with air showed that the mobility (£) of the positive ion varied inversely as the pressure (p) of the gas down to the lowest pressure which it was convenient to employ (‘05 mm.). It was not thought necessary to proceed to very low pressures in the case of carbon dioxide and hydrogen, but all the indications were that the law pk=const. would continue to be valid. The validity of this law over a wide pressure range signifies that the nature of the positive jon remains unchanged throughout this range. The same law was found to be valid for the negative gas ion, but only after care had been taken to separate the negative carriers into the two components, electrons and ions. It was found that the apparently anomalous increase at reduced pressures of the mobility of the negative ion to which many observers had previously drawn attention was occasioned by this dual nature of the negative carrier ; when the ions were considered apart from electrons all the anomalies disappeared, the velocity being expressible in the form v=k)—. It is instructive in this connexion to consider the difference between the present and the older point of view. It has long been known that in air at very low pressures the current of negative electricity is due practically entirely to free electrons; at the higher pressures, however, the current is due to the motion of negative ions. What is the nature of the negative carrier at intermediate pressures ? ‘The answer hitherto given Ions and Electrons through Gases. 57 to this question was that the carrier altered in nature during its motion between the electrodes, but in such a manner that for a given pressure it possessed an “average” mass. If, for instance, we regard the jon as being constituted at high pressures by a cluster of molecules, then we should have to assume that as the pressure was reduced the average number of molecules in the cluster decreased; as the pressure was still further reduced, any individual negative carrier would be for part of the time in the ionic state (say now as a single molecule), and for the remainder would exist as a free electron; at this pressure we should have at any given instant a number of free electrons and a certain number of ions, but if we were to follow one electron throughout its motion we should find it associated on the average with a mass intermediate between that of an electron and that of a molecule Ultimately at very low pressures the carriers would be all free electrons. Prof. Townsend’s* point of view differed only slightly from this in that he regarded the average nature of the carrier to be determined by electric force as well as gas pressure. The answer afforded by the present experiments is funda- mentally different. Wenowregard the electrons and ions as passing independently through the gas, each kind of carrier remaining constant in nature throughout. The transition from the ionic conduction at high pressures to the electronic conduction at low pressures is effected by means of an increase in the number of free electrons relative to the number of negative ions without any alteration in the nature of either kind of carrier. The appearance of the phenomenon of ionization by collision would further affect the relative numbers of carriers, but would not influence the nature of the conduction. Looked atfrom this point of view it seems clear that, as far as the so-called permanent gases are concerned, we must regard the free electrons as occurring theoretically at all pressures. These gases differ, of course, considerably in the relative number of free electrons and ions for any given pres- sure, and, practically speaking, there is for each gas a pressure at which the number of free electrons is negligibly small, but the general rule is in no way invalidated on this account. In the above illustration we considered the electric current passing through air. It was shown, however, by Franck t that for certain gases, viz. the inert gases and nitrogen (which behaves often as an inert gas), the negative carriers * Townsend, ‘ Electricity in Gases,’ Oxford (1915), Chap. IV., VIII. Cf. also Pidduck, ‘ Electricity, Cambridge (1916), Arts. 214-215. t Franck, loc. cit. 08 Mr. E. M. Wellisch on the Motion of consist entirely of electrons even when the gas under con- sideration is at atmospheric pressure. Chattock and Tyndall* gave good reasons for believing that hydrogen possessed similar characteristics; more recently Hainesft bas shown independently that the negative carriers in hydrogen consist. practically entirely of electrons. In all these instances a slight trace of impurity (especially oxygen) was sufficient to convert the carriers into ions. The older point of view was to regard these gases as possessing, by virtue of their inert character or otherwise, the exceptional property of being able at high pressures to contain electrons in the free state: on this account they had to be clearly distinguished from gases, such as oxygen, chlorine, &c., which were regarded as being unable to contain free electrons except at very low pressures. The present. experiments indicate that the difference is merely one of degree inasmuch as the electrons are capable of existing in the free state even in air at considerable pressure. We may now regard at any rate the so-called permanent gases as being able to contain both negative ions and free electrons, each kind of carrier maintaining its identity throughout its motion. The inert gases and hydrogen are now regarded as being exceptional, no’ in their power of containing free electrons, but rather by reason of their great reluctance to form negative ions, 7. e. by reason of the exceptionally large proportion of electrons to ions. It was shown in sec. 4© that the vapour of petroleum ether is able to afford a copious supply of electrons and to maintain them in the free state provided we reduce the con- tamination toa minimum. As the molecules of this vapour contain only atoms of carbon and hydrogen, this result suggests strongly that the negative ions in air, OO, COs, &e., are due almost entirely to the presence of the atoms of oxygen. Franck t has arranged gases in the following order of increasing electron affinity: helium, argon, nitrogen, hydrogen, oxygen, nitric oxide, chlorine. This list was obtained by considering the relative power of the different oases, when present as impurities, to deprive helium of its free electrons. If, in accordance with the views embodied in this section, we regard this series of gases as affording a relative idea of the proportion of electrons and ions which results from the process of ionization, i would seem probable that in order to supply an appreciable number of negative * Chattock and Tyndall, Phil. Mae. xxi. p. 585 (1911), + Haines, loc. cit. t Franck, loc. eit. Tons and Electrons through Gases. oo ions, the molecules of a gas must contain atoms either of oxygen or chlorine; we may by analogy include other electro-negative atoms such as bromine, iodine, &c. This. statement is to be regarded merely as a suggestion for further experiments; a study of the ionization in pure ammonia might prove of interest in this connexion. Tt is known that the presence of a trace of oxygen in an inert gas or in hydrogen at atmospheric pressure will reduce considerably the number of free electrons. The present expe- riments showed that in hydrogen the sensitivity of the free electrons to traces of oxygen was greatly decreased if the gas pressure was reduced so that, for instance, a considerable number of free electrons was obtained in a mixture of hydrogen at 824 mm. pressure and air at 24mm. In a revious communication * a definite theory in explanation of these results has been given: the underly: ing idea is that an electron cannot effect a permanent union with an un- charged molecule to form a negative ion unless the relative velocity at collision exceed a critical value characteristic of the molecule concerned. We have seen that in a large number of gases the electrons persist in the free state, so that it would appear that the negative ions in these gases must in general t be formed immediately after the act of ionization. We may regard the electron as being expelled with a certain velocity from an uncharged molecule, but owing to the positive charge acquired by the molecule the velocity of the electron will decrease as it recedes; in accordance with the above view we may imagine a sphere drawn round the parent molecule of such a radius that the electron will be effective in forming a negative ion only for impacts within this sphere. It is probable that the circumstances of an encounter as well as the relative velocity will determine the effectiveness of a collision, so that only a fraction of those impacts will result in the formation of ions: outside the sphere, however, the electron must continue in the free state. It is easy to see that on this view the relative number of electrons will increase with decreasing pressure. The potential required for the formation of a ‘negative ion must of course be less than that required to ionize a molecule, inasmuch as in the latter case a fresh pair of ions originates. We should expect that for those gases which have a high ionization potential the proportion “of negative ions to elec- trons would in general be small. This is borne out by the * Wellisch, Phil. Mag. xxxi. p. 186 (1916). + When the applied field is sufficiently great to generate the critical velocity in the electron, negative ions will again commence to be formed. 60 Mr. E. M. Wellisch on the Motion o) results for the inert gases and hydrogen, although the value assigned by Franck and Hertz * for the ionization potential in nitrogen (viz. 7°5 volts) would not indicate on this view a very large percentage of free electrons. It does not seem advantageous to discuss in great detail the question as to the nature of the gas ion; all that is proposed is to indicate here the leading features of this outstanding problem. Itshould be remembered that the notion of the ion as consisting at moderately high pressures of a cluster of molecules grouped round a charged nucleus was first intro- duced in order to account for the observed mobility and diffusion values, which were found to be considerably smaller than the values which were to be expected from theoretical considerations if we regard the ion as consisting of a single molecule. It was shown, however, by the author?+ that the observed values were consistent with the view that the ion was a single molecule, provided we took into account the extra resistance to the motion of the ion resulting from the attraction between the charge on the ion and the charges induced on neighbouring moleculest. A definite decision in favour of the cluster theory appeared to be given by the results of the series of experiments, which indicated a departure from the law pk=const. even when the gas was at a pressure of several cm.; the abnormally high mobility values were naturally interpreted as corresponding to the disintegration of the ionic cluster. The fallacy of this series of results has already been discussed in the present paper; it is sufficient here to repeat that no indication has been obtained of any change in the nature of either the positive or the negative ion as the pressure of the gas changes over a wide range. We would certainly expect at least a partial disintegration of anionic cluster when the electric field and the gas pressure were such that the ion acquired energy comparable with that required to ionize a neutral molecule. According to Townsend ionization by collision in air commences to be appreciable when X/p=60 (X being measured in volts/em. and p in mm.Hg). X/p is proportional to the energy acquired by an ion after traversing a distance equal to its * Franck and Hertz, Verh. Deutsch. Phys. Ges. xv. p. 34 (1918). + Phil. Trans. loc. cit. p. 272. + It was the author’s idea that this extra resistance was due entirely to increased frequency of collision between the ion and the molecules. Sutherland later maintained that the increased frequency was responsible only for part of the extra resistance and that 1t was necessary to introduce another type of electric viscosity. These points were discussed further in two communications (v. Phil. Mag. xix. pp. 201, 817, 1910). Ions and Electrons through Cases. 61 mean free path. In the present experiments the positive ions are shown to have a normal mobility for values of X/p as great as ]1, even if we take for X only the small values of the critical field from which the mobility was estimated. The normal character of the complete curves which were obtained in the process of determining the critical potentials indicates that neither the positive nor the negative ion is appreciably altered in nature for much greater values of X/p. It should be remembered, however, that at the lowest pres- sures employed we are nearing the conditions for which the mobility law would be no longer valid even for an unchanging ion. A simple calculation gives that in air for p—0p' mime and with 1 volt fall of potential, the positive ion makes about 330 collisions in traversing the distance of 2 em. between the electrodes. It is surprising that the mobility law should be so nearly valid at this stage: the explanation is probably that the velocity (3°3 x 10* cm./sec.) acquired by the ion after describing freely a distance equal to the mean free path is still smaller than the mean velocity of thermal agitation of the molecules (4°6 x 10*). If we suppose that the mobility law is valid as long as the mean velocity of agitation predominates, we find by calculation that in air the law should hold for values of X/p up to 20. Loeb* has recently made a series of determinations of the mobilities of the ions in air under high electric fields, and has shown that the mobilities remain normal at atmospheric pressure for field strengths up to 12,450 volts per em.; the law pk =const. was verified for values of X/p up to about 20. The preceding considerations indicate that the notion of an ion as a cluster 1s unnecessary; the cluster theory must depend for its continued existence on arguments essentially different from those which have hitherto been advanced. Moreover, it should be stated that evidence of a more direct nature in favour of the single molecule theory has of late years been forthcoming. Chattock and Tyndallf in their experiments on the point-discharge showed that the absorption of positive ions of hydrogen by a metal corresponded to a withdrawal of two atoms of hydrogen from the gas. Hriksonf, * Loeb, Proc. Nat. Ac. Se. vol. ii. No. 7, p. 345 (1916); also Phys. Rey. vol. viii. p. 683 (1916). Loeb has misunderstood me when he states that I verified the law pk=const. for the ions in air up to values of X/p as high as 34:5; as a matter of fact I maintained merely that the negative ions were still 7 evidence for this value, whereas Townsend’s theory would necessitate their complete disappearance fer a value of X/p- equal to0°2. (EH. M. W.) + Chattock and Tyndall, Phil. Mag. xvi. p. 24 (1908); also Joe. cit. p- 60. t Erikson, Phys. Rev. vol. vi. p. 345 (1915), 62 Mr. KE. M. Wellisch on the Motion of in experimenting with regard to the variation of ionic mobility with changes in temperature, concluded that his results were not explicable by the notion of clusters. In the present experiments it has been shown that an electron passes un- encumbered through ordinary gases at considerable pressures notwithstanding the strong electric field which is associated with it; it is hard to reconcile this fact with the basic idea of the cluster theory, viz., that the cluster of molecules is held together by the electric field assogiated with the ion. Leaving the question as to the nature of the gas ion, we may now with advantage consider another outstanding problem of ionic theory, viz., the explanation of the difference in the experimental values obtained for the mobilities of the positive and negative ions ina gas. The greater mobility ot the negative ion in most gases has usually been regarded as indicating that. this ion is constituted by a smaller cluster of molecules. On another view* we could explain the greater mobility of the negative ion by supposing that the electron ‘is able occasionally to leave the ion, so that the increased velocity would arise during the free motion of the electron. The present experiments show, however, that this view is untenable as an explanation : it was shown that the electrons pass through the gas independently of the negative ions, and still the latter have a mobility greater than that of the positive ions. If we regard the ion as consisting of a single charged molecule, it seems evident that the difference in the mobilities of the two kinds of ions must be ascribed to a difference in the attractive forces between each kind of ion and the uncharged molecules. In the Bakerian Lecture of 1890T Schuster remarked that “if the law of impact is different between the molecules of the gas and the positive and negative ions respectively, it follows that the rate of diffusion of the two sets of ions will in general be different.” Franck and Hertz f were the first to bring out clearly the possibility of great differences existing in the nature of the collisions between an electron and the molecules of different gases. On their view the electrons are regarded as possessing different degrees of elasticity when in collision with the molecules of different gases, the collisions being extremely elastic in the case of the inert gases, but only partially elastic or even almost inelastic for most other gases. * Cf. J. J. Thomson, ‘Conduction of Electricity through Gases,’ 2nd edit. pp. 28, 29. + Schuster, Proc. Roy. Soc. vol. xlvii. p. 553 (1890). + Franck and Hertz, loc. cit. Ions and Electrons through Gases. 63 Itseems to the writer to be perfectly natural and logical to extend this conception so as to apply to the collisions between the zons and the neutral molecules. The difference in the mobilities of the two kinds of ions is thus regarded as being due to the different degrees of elasticity between the neutral molecules and the positive and negative ions respectively. Tf we regard the collisions between neutral gas molecules as being moderately elastic, we would expect that collisions between an ion and a gas molecule should have a smaller degree of elasticity on account of the attractive forces resulting from the charge on the ion. These forces would result in a small fraction of the translational energy at collision being transformed into energy inside the ion or molecule. A very high degree of elasticity would imply (v. sec. 4B) either an accelerated drift for the ion or a slow acquisition of terminal velocity: experiment shows that in general the ion quickly acquires a terminal velocity so that its collisions with gas molecules must he imperfectly elastic. The experimental fact that the negative ion has the greater mobility would imply that at collisions between the neutral molecules and the positive and negative ions respectively the latter have the higher degree of elasticity. It is of interest to inquire whether we know any properties of the negative ion which would suggest that it should be associated at collision with a higher degree of elasticity; moreover, we have to explain the experimental fact that the difference in mobilities is especially marked in the case of the light gases (€. g. impure hydrogen and impure helium) and practically vanishes for the heavier gases and vapours. The general effects can be accounted for by two consi- deraiions : firstly, the discrete nature of the electronic charge, and secondly, the assumption that the positive and negative charges are differently distributed in the respective ions. If we consider a negative ion which is about to collide with a neutral molecule, the discrete nature of the electronic charges both in the ion and the molecule will be mani- fested by an intense force of repulsion when the distance is very small. This field will be superposed upon the attraction due to induction, and will resist any mutual penetration at collision. The ion and the molecule are at close approach resolved as it were into constituent charges, and the simpler the structure the more effective the resolution. The eftect of the forces due to polarization will be to decrease the elasticity of the collision while the repulsive forces will act in the opposite manner. We would thus expect the collision in the case of the negative ion to 64 Mr. HE. M. Wellisch on the Motion of be fairly elastic, this elasticity being especially marked with light gases, such as hydrogen or helium, which have a simple structure and a small coefficient of polarization. In the case of the positive ion the charge is either more centrally situated than is the electron in the negative ion or, what is effectively the same, the positive charge is not discrete but distributed : the ionic charge will thus act so as to produce a collision of small elasticity; the ion will probably penetrate an appreciable distance into the molecule, and the mobility will in consequence be diminished. For the heavier gases and vapours we would expect the negative charge in the ion to be situated more centrally than for the light gases; in any case the discrete nature of the electronic charge would not be so readily manifested with these complex molecules which would approximate more closely to metallic conductors. The forces due to the approach of a positive or negative ion would be more nearly equal and, in consequence, there would be no great dif- ference in the values of the two mobilities. In pure hydrogen at atmospheric pressure the negative carriers consist practically entirely of electrons ; a trace of an impurity such as oxygen is sufficient to convert the carriers into ions. An interesting question arises as to the nature of the negative ion in slightly impure hydrogen : is it con- stituted by the hydrogen or by the molecules of the impurity ? Haines * has recently niade an investigation with regard to the negative carriers in hydrogen, commencing with the gas in a very pure state and allowing impurities to accumulate. In this manner he has brought into evidence three distinct types of negative ions, the normal ion being the slowest of the three. His conclusion is that these ions are composed of clusters of hydrogen molecules, each type of ion com- prising a definite number of molecules. The possibility that these ions are composed of the molecules of the impurity present is not discussed in the paper, nor indeed does the part played by the impurity receive consideration. No evidence was obtained in the present experiments of the intermediate types of ions described by Haines: this was possibly due to an excess of impurity in the hydrogen employed by the writer, although it should be observed that it was sufficiently pure to yield a copious supply of electrons at atmospheric pressure, whereas in some of the curves given by Haines the intermediate 1ons are in evidence when free electrons are practically absent, With regard to the question as to the nature of the negative ion in impure * Haines, loc. ctt. Ions and Electrons through Gases. 65 hydrogen, the suggestion is here made that the molecule of the impurity may act as a catalyst, enabling the electron te enter the hydrogen molecule; in the pure gas the electron will, however, remain in the free state. In the present paper the motion of the free electrons through a gas at relatively high pressures has been con- sidered. It appears that in general an electron is able to effect a permanent union with an uncharged molecule so as to form a negative ion only if the encounter take place quickly after the act of ionization, when the electron still retains a considerable part of its velocity of projection ; if it fails to combine initially, it would seem that it can remain in the free state even in the presence of electro-negative molecules such as oxygen. However, there may arise occasionally certain systems (electron sinks) which possess the property of being able to absorb electrons which drift through the gas ; the union appears in these cases to be of a loose nature, and is liable to be broken by molecular encounters. In a recent communication* Sir J. J. Thomson has ex- pressed the view that the electron is able to unite with a molecule during its drift motien so as to form a negative ion ; before such an attachment occurs the electron in general traverses distances through the gas which are large compared with its free path. The distinct separation between the ions and the free electrons which is shown in a whole series of EI curves lends, however, strong support to the view that the electron traverses the whole distance between the elec- trodes without effecting any permanent union with a gas molecule ; a well-defined bend in the experimental curve could not have been obtained if any considerable fraction of the electrons had become attached to molecules during their passage through the gas. 6. SUMMARY. 1. The separation previously effected between the electrons and the negative ions in dry air at the lower pressures has in the present investigation been extended to other gases, notably CO, and H,; for these two gases the electrons are relatively more numerous than in air at the corresponding pressure. 2. A trace of impurity is especially effective in reducing the number of free electrons when the gas is at a relatively high pressure : at low pressures the effect of the impurity is often inconsiderable. In most cases a velocity greater than that arising from * J. J. Thomson, Phil. Mag. xxx. p. 321 (1915). Phil. Mag. S. 6. Vol. 34. No. 199. July 1917. F ee eee 66 Prof. W. M. Thornton on the Nature of Chemical thermal agitation at ordinary temperatures appears to be necessary to enable the electron to effect a permanent union with an uncharged molecule of the gas or impurity. 3. For the vapour of petroleum ether, whose molecules contain only atoms of carbon and hydrogen, the negative carriers appear to consist practically entirely of free electrons; a trace of impurity, however, is sufficient to effect the production of a considerable number of negative ions. 4, A brief investigation has been made of the motion of free electrons through CO, ; the results do not indicate that the velocity of the electron is proportional to the applied field, but suggest that the electron may traverse a consi- derable distance with accelerated motion before its terminal velocity is acquired. 5. In no instance was any evidence obtained of a change in the nature of either the positive or the negative zon as the pressure of the gas was reduced. 6. The present method was employed to determine the values of the ionic mobilities for a few vapours; the results have been compared with previous determinations. 7. A discussion is given with regard to the bearing of the results on certain outstanding problems of ionic theory. The University of Sydney, December, 1916. IV. The Nature of Chemical Affinity in the Combustion of Organic Compounds. By W.M. THornton, D.Sc., D.Eng., Armstrong College, Newcastle-upon- Tyne™. The HE final stage in the combustion of organic com- pounds is combination in a gaseous state. Affinity is then probably of the simplest kind, for molecules are far apart, and they are in contact for the briefest possible time. In the absence of any knowledge of how an electron, acting as a bond, is anchored into the atomic structures of combining atoms, the force that a bond sustains may be assumed as a first approximation to obey the usual laws of attraction and to be proportional to the products of the nuclear charges—that, is of the masses of the molecules. It is clear, however, that affinity is not a simple electrostatic attraction, and that between elements it is dependent upon atomic formation. 2. The most recent comparisons of affinity In gases are those of Prof. W. A. Bone and colleaguest made by ex- * Communicated by the Author. + “Gaseous Combustion at High Pressures,” by Prof. W. A. Bone and others, Phil. Trans. Roy. Soe. ser. A, vol. cexv. pp. 275-318. Affinity in the Combustion of Organic Compounds. 67 ploding mixtures at high initial pressures. They found that in a mixture CH,+0,+2H, the distribution of oxygen in the products of combustion was, under the best conditions, 97:1 per cent. combined with methane, 2°9 per cent. with hydrogen—a ratio of 33°4. The molecular weight of methane is L6, with which four atoms of oxygen of total weight 64 combine, the product of these being 1024. Hydrogen of molecular weight 2 combines with oxygen of weight 16, giving a product of 32. If the affinity of these gases for oxygen is proportional to the product of their combining masses, the affinity of methane for oxygen relative to that of hydrogen should be 1024/32=32. The coincidence between. this and Bone’s observed value, the only direct determination available, is remarkably close. The case of carbon monoxide is complicated by oxygen already in com- bination, and by the uncertain role of steam in its explosion _ with hydrogen present. 3. In an explosive mixture before ignition molecules collide without liberation of heat; but when activated in the wave front they combine with increased kinetie energy of which the heat set free isa measure. This old conception of the source of heat in combustion has been criticised as inadequate, but it has the merit of giving a clear picture of the commencement of the process, and the only way in which it can be extended is by further consideration of the velocities of combination. It takes no account of changes of heats of formation. Let two masses, m,, mz, of combustible gas and oxygen respectively, combine. Their kinetic energies at collision are 4mv2=Jh, and 4m,v.?=JSh., where hy, hy are the heat equivalents of the translational energies of each and 1, v, the components of velocity due to attraction. Since the force of combination is applied equally to both, h,=h,=A say. The total heat H contained in the products of combustion is 2h, or, if there is suppression of heat at collision, 2kh, where £ is less than unity. The product 9 2 2 MyM = Nh? (=) =H?( a) ) : 01029 U0 It has been shown * that H is proportional to ms, the mass of oxygen burnt, in the ratio H/m,=3°31. The heat of combustion H is also proportional to m,, the mass of the combustible molecule, and on an average for organic com- pounds H/m,=11°6. It follows that H?/m,m, and (k/vyv.)? * “The Relation of Oxygen to the Heat of Combustion of Organic Compounds,” Phil. Mag. vol. xxxiil. p. 196, Feb. 1917. 2 68 Prof. W. M. Thornton on the Nature of Chemical are both constant, and to the same degree of approximation. According to a gravitational law of attraction or power of acquiring and retaining oxygen, affinity, measured by the product mym,, should then be proportional to H?. Con- versely, when the ratio H?/m m, is constant, it is strong presumptive evidence that affinity follows this law, whatever the ultimate electromagnetic explanation may be ; but it is only when the products of combustion are of similar com- position in every case, as they are in the examples to be quoted, that heat of combustion can be used to find a law of affinity common to all the compounds examined. 4. Taking the heats of combustion given by Thomsen (‘Thermochemistry,’ Tables 35 to 44), the values of H?/m ym, for the paraffin series, though fairly constant, fall slightly to a steady number as the series rises (Table I.). (H—15)?/mym,z is more constant. TaBue I. ‘Compound.| Oxygen. | H (H— 15)? m, m, 1b. mI, mM, igdiere | aren 16 64 212 439 | 3BT9 Ethane ...... OSE ee 30 112 370 40°7 376 Propane...... C,H, 44 162 529 39°7 37°6 Butane ...... Cr 58 208 687 39°2 37°5 Pentane ...... C,H, 72 256 847 39°'1 37°6 Hexane ...... C,H, 86 304 992 38°7 376 Heptane......| 0,H,, | 100 352 1166 386 | 378 Decane ...... ©) 3TH 5 142 496 1645 38°6 37:8 | | Means...| 298 37°6 * Kilogram-calories per gram molecule. For the other compounds of which the heats of combustion are given by Thomsen the mean values of H?/m,m , are as in Table II. TaBue II. Compounds. H?/m,m,. Aromatic hydrocarbons .................. 345 Unsaturated hydrocarbons ............ 38°2 Halogen compomndsy |)... 0... -2esces 30°4 Ethers tandiacetalet Woe. 2./u. ee csceneae 36°4 PAT colOTS UE Ne A SUMDRE S42. oCdldl lot nen 358 Aldehydes and ketones,’ ...........-.-:0-. 34°6 ISLES 4 (icone eee cate ata! aria ate eee 34:7 Cyanogen, err itera niase. (sc. ne cgosseh ee 39°9 | Nitrocompounds .................-..+24++-- 380 Affinity in the Combustion of Organic Compounds. 69 The mean of these including the paraffins is 36°7; but, since this gives equal weight to each group, a fairer mean is obtained from the figure which contains all Thomsen’s compounds. From this H?/mym,=38'6. Affinity between O25 50 75 100 125 I50 175 200 225 250 275 30m MyM,. x 102. these typical compounds and oxygen is then very fairly proportional to the product of the masses of the combining molecules. 5. The fact that the heat of combustion of organic com- pounds is proportional to the number of oxygen atoms required for complete combustion is now seen to be that an affinity similar to the gravitational law holds over a wide range ot those compounds that are reduced to similar products of both steam and carbon dioxide. Hydrogen and carbon monoxide are not reduced to the same products, and their affinities are not comparable in terms of heat of combustion. For this reason it is impossible to compare the affinities of elements thermally, and heats of oxidation of metals differ widely. Prof. Bone’s results are of special interest in the ease of hydrogen, since they show that the affinity of this gas for oxygen follows the same mass law as organic com- pounds, and that heat of combustion is only under strict limits a reliable means of comparison. 70 Prof. W. M. Thornton on the That chemical affinity in its simplest form is proportional to the product of combining masses does not necessarily identify cohesion and gravitation, though it makes such a relation possible. The difficulty in establishing it in liquids or solids would be to find an expression for, or an experi- mental means of observing, the action of surrounding mass. By analogy with its known influence in electro-optical phenomena, it might be expected that this effect would be to dilute affinity without changing the ultimate law; but the coefficient of dilution would certainly not be constant. ————— V. The Curves of the Periodie Law. By Prot. W. M. THornton, D.Sc., D.Eng., Professor of Electrical Engi- neering in Armstrong College, Newcastle-on- Tyne. Ike ()* the disintegration theory of matter atomic structure is modified by the loss of electrons in a regular sequence, from positions of maximum atomic volume to those of maximum density. The periodic curves of density and atomic volume both have the inflexion characteristic of hysteresis. They can be built up on the assumption that the internal force by which atoms are held together passes through a simple periodic change, and that in the resultant change of atomic volume there is structural hysteresis. 2. If all aggregates of electrons forming atoms had the same mean density of concentration, atomic volumes would increase indefinitely in a straight line. The density p of the elements oscillates, however, between two limits, a lower line, IIT. fig. 1, of nearly constant density, and a straight line I. through the maxima which cuts the density axis at the same point m as the lower line. If these lines passed through zero so that p=kw=kVp, the corresponding atomic volumes V would be constant, and the maxima of the curve of atomic volumes (fig. 2) would have equal values. The minima would lie on a lower horizontal line. As it is, p=potkw for the limiting densities, and V=wu/(po+tkw). Here V=0 when w=0, and the curve is asymptotic to 1/h. ; * “The Dependence of Optical Phenomena on Physical Conditions,” by. Prof. T. H. Havelock, Roy. Soc. Proc. -Ixxxiv. pp. 492-523 (1911). + Communicated by the Author. § Curves of the Periodic Law. (a Fig. 1. ee ie Co een Bete ter eee php ae ae NERY PEAR HE is 40 ° 25 So res 100 125 150 i75 200 ATOMIC WEIGHT» ATOMIC VOLUME ATOMIC. WEIGHT 72 Prof. W. M. Thornton on the The equation of the upper line of density is py=0°62+°1146 wv, and the minima of the curves of atomic volume lie on the curve V,=w/(0;62 + 0:1146w), - 2 ae) asymptotic to a volume 8°72, osmium being at 85. The maxima lie on the curve Virr=w/(0°624+0°0083w), . . . (2) asymptotic to 120. The slope of the line of mean donnie is0°057, and for the corresponding volumes Vir=w/O'62-+0-05 Ta), 2 ee asymptotic to a value of 17°5. This line passes between thallium and lead, so that when the region of elements at present radioactive is reached, the mean atomic volume is rather below than above that of lead. 3. Superposed upon the lines of mean density of volume there is an oscillation which, since the change from maximum to minimum takes place ina regular periodic manner, and so that the rate of change is never discontinuous, may be assumed to be caused by a simple periodic variation of cohesive force. Using now the conception that there is structural hysteresis in taking up new atomic formations, we have the following interpretation of the curves, considering first atomic volumes. Fig. 3. i ATOMIC VOLUME CENTRAL FORCE O! OENSITY Assume for the moment that the mean connie force is constant, represented by the line AB (fig. 3), and that the Curves of the Periodic Law. 73 curve C is the change of cohesive force caused by the suc- cessive removal of nuclear electrons from the atomic structure. Ordinates Pa of the latter curve from the base-line are the effective forces of cohesion. The result of there being a Minimum atomic density is to move the curve of hysteresis to the right, clear of the vertical axis; a steady component of the force of cohesion moves the centre of the loop vertically downwards. The ordinates of the line of densities in the right-hand figure are obtained by projecting any points P, P’, on to the hysteresis curve, and setting up vertically lengths ap, ap', which, if there is lag, are the values corre- sponding to the force aP. The resultant curve resembles closely that of observed densities, fig. 1. The period decreases with loss of atomic weight. To a first approximation the maxima of cohesive force occur at the cubes of the first six natural numbers. Atomic Element. wciene N. n>, n?/A.W. |- EA ae 1 1 1 1:0 15) ae 11 2 8 0°72 72) i Pf Coll 4! QT 1:00 nila) 2k *, < 63°5 4 64 1:00 Tee 101°5 5 125 1°22 (I aera 191 6 216 113 Mean ratio 1:01 Superposing a series of such oscillations, of increasing amplitude, upon the curve of equation (3), with minima of force at 1°, 2°, 3°,...., we obtain a fair approximation to the full curve of fig. 2. The equation of the density curve if there were no hysteresis would be p=p)tkw(l+ae”? sin ¢), where ¢ is proportional to w. The change of atomic volume with atomic weight, gene- rally taken to illustrate the periodic law, can then be built _ up from two components:— (1) An asymptotic rise of mean volume to a maximum which coincides with the radioactive elements. (2) A periodic oscillation, giving a displacement modified by structural hysteresis, increasing in amplitude and period, there being six known minima which in the absence of abnormal forces correspond approximately with the cubes of the first six natural numbers. The last maximum of cohesive force appears to be ata weight of 6?=216, and it is to be remarked that although 74. Prof. W. M. Thornton on the there is a suboscillation which displaces the minimum the atomic weights of the last permanent elements are 207 and 208, lead and bismuth, whilst radium emanation (niton) has. 222-4 and radium 226-4. 4, It is plain from § 2 that the atomic formation oscillates between two limiting conditions, that in which the density is constant and a minimum, shown by the lower straight line of fig. 1, and that in which the volume is constant and a minimum, remembering that if it were possible to have zero atomic volume, the maxima and minima of fig. 2 would lie on parallel horizontal lines. There isa gradual adjustment between the conditions of minimum density and minimum volume. In the former the ratio of the mass to the volume of the atom is constant, so that the closeness of packing of the positive electrons within the atom is constant. In the latter the ratio of mass to density is constant; the system has. constant volume and is therefore elastic in the sense that at the higher densities more electrons are packed into the same space. As the internal cohesion is relieved by the successive loss of units carrying positive electrons the atomic weight falls, the volume expands to a maximum, declining again to a minimum under the elastic forces of the remaining matter, and any theory of atomic formation must be capable of explaining the change of volume in so great a ratio as 13°75 to 1. | 5. It is to be observed that in the curve of atomic volumes between cesium and osmium there is evidence of a sub- oscillation, fig. 2, reaching a maximum at an atomic weight of about 180, that of tantalum. Such an oscillation super- posed on a regular periodic curve has the effect of displacing the maximum of the wave on which it falls. The atomic volumes with cesium asa maximum are clearly spaced un- symmetrically, being more to the left than the earlier parts of the curve would lead one to expect. Ourve 1, fig. 2, is the harmonic force which gives rise to the curve of observed volumes; curve 2is that which would occur if there were no such harmonic. The minima of the latter are close to 125 and 216, the cubes of 5 and 6. | The origin of this harmonic, which plays so important a part in the genesis of the higher elements, is clearly shown in the curve of densities. The form of the part of the curve in fig. 1 to the right of cesium is the same as that of G, fig. 3, derived from a hysteresis loop symmetrical about the line of mean values. In other words, the variation of atomic volumes appears there to depend chiefly upon the periodic component. Curves of the Periodic Law. 75 As disintegration proceeds and atomic weight falls the steady component of cohesion begins to rise in importance, in those elements having an atomic weight below 180. The atoms are then changing from a condition of constant volume, but the steady force coming into play is sufficient to delay the change to the opposite condition of constant minimum density at n, fig. 2. There is in fact a return to the minimum of volume at p, from which the oscillation of force proceeds regularly. It is remarkable that this region of variation is that of the rare earths, and it is now sugg cested that the reason for their comparative fewness is that the forces under which atoms are formed are modified by the rise in importance of the steady central force, so that there is here in a sense in- stability of type of structural formation, and permanent elements are rare. 6. So far as it is permissible to draw inferences from the shape of the curves, it may be predicted that the atomic volumes of elements between 150 and 180, such as gadolinium and neoytterbium, will be found tu be low, probably between 9 and 10. ‘The corresponding densities are about 16 and 17. The atomic volumes of cerium (21) and samarium (19°3) are relatively higher than those of elements in corresponding positions in the other series, and elements having atomic weights between 155 and 170 should have relatively lower atomic volumes, those from 170 to 185 rather higher volumes than would be obtained from a smooth curve similar to those of the lower periods. 7. On the degradation theory of atomic formation some form of retarding action is necessary, otherwise matter would slip without resistance to its lowest elements. In the con- secutive loss of nuclear units from the outer ring the forces controlling the size of an atom rise and fall in a regular manner, and because of this periodic change retardation takes the form of cyclic hysteresis. Anatom about to lose a unit of mass must be supersaturated, which is a lag of state behind controlling force. Ls Aha VI. Note on the F ocometry of Lens- Combinations. By Prof. A. ANDERSON *. l AM much obliged to Mr. Robert E. Baynes, M.A., for directing attention to what is undoubtedly a defect in my paper which appeared in the Philosophical Magazine of January last—the exampleat the end. The combination con- sisted of two concave lenses. but I cannot now submit it to further measurement, as I kept no note of the particular lenses used or of their distance apart. The measurements were very rough and rapidly made, and the nodal slide I used is far from being an accurate instrument. It was made in the laboratory workshop for demonstration purposes. The principle of the method is the same as that of the method of Abbe, and whatever defects the one method pos- sesses the other has too. The distinctive characteristic of the method I proposed is the simple way in which the magni- fication is measured, which is suitable not only for converging combinations, but also for diverging combinations where an auxiliary convex lens must be used. I admit, however, that if its claim to recognition rested solely on the example given it would not have much to recommend it. I have measured with some care another diverging com- bination, but I must point out that the results are much more a test of the capability of my rude nodal slide than of the method. The combination consists of two concave lenses A and B. The focal length A is 20°34 cm., and that of B 30°91 em., and their distance apart is 12°45 em. By calculation it will be found that f=9°87 cm., BH,=6°04 cm., BH, =8°47 cm., and H,H,=2°43 em. The light falls first on A, and an auxiliary lens is placed to the left of B to form an image on ascreen. It is this image which is not displaced when the combination is rotated about O. The position of an object whose image is formed on the screen by the lens alone gives _ the position of the image formed by the combination. The measurements gave OP, or 2, =142, OP, or y,=9°4, OP,’ or t= 29°1, OP,! or y2=8°3, and d, the distance through which the combination was moved to the right, 113°8. We thus have 113-8 _ 142 _ 29-1 i 9-4 era giving f= 9°81 cm., which does not differ much from 9°87 cm., the calculated focal length. * Communicated by the Author. - On the Focometry of Lens- Combinations. 17 Mr. Baynes points out that the relation d (1- nbs) = a cat —(%,—Yo) Uywvy must hold between the five quantities measured. The left- hand side, when the numbers are substituted, gives 111°65 and the right-liand side 111°8. This is also satisfactory. The above formula may be also written A+ &—ay +1 Yo = Ay1Yyo2/2 122, which becomes and this is, apparently, less satisfactory. With the lens to the left of B, and with a diverging combination whose focal length is small and in which H,H;, is small, d is nearly equal to #,—4#, and y, and y, do not differ by very much. Thus the quantity d+.a,—.2,+7,—Y2 will be small, and small errors in the five quantities involved may have a considerable effect on the amount of the error in it compared with its correct value. In the example at the end of my paper the value of this quantity is —0'1, the right-hand side of the equation being 0°067. The right-hand side being always positive, it is impossible for the left-hand side to have a negative value. Mr. Baynes points out that the focal length may be obtained by getting only one centre of rotation, that is, only one value of the magnification, together with three other mea- surements. Denoting the quantity d x.—a,+y4,—y, by 5; the formule are faa/(e—$ us) =a|( oe /\yr 8 Si Wye | yas In the first formula the magnification is = and the other 1 measurements to be made are «,;—y,, t,—ye, d, and similarly for the second formula. But, as has been pointed out, s is liable to a large relative error, which may have a great effect on the value of f. Substituting the numerical values, the first formula gives f=10 cm., and the second f=8-95 cm. The utterly incorrect values of / obtained by Mr. Baynes from the data given in the example at the end of my paper are explained by the value —0-1 of s, which is quite wrong and could not possibly be negative. The value of H,H, obtained from 2, y,, and fis 3°38 cm., and that from ay, Yo and f 3°23 cm., 7 being taken to be 9°81 cm. These values, 78 Prof, A. Anderson on the it must be admitted, are not very satisfactory, as the value of H,H, calculated from the focal lengths and distance apart of the lenses is 2°43 cm. An adequate explanation of all these sents is furnished by a consideration of the effect of possible errors of measurement which are due (1) to the instrumental error, (2) to want of precision in determining whether there is any motion of the image, and (3) to the error in finding the position of P,. We may assume that the error in the measurement of d is negligible. The formula for the focal length is Tf «1, %, 81, 8, are the possible errors in 2, 2, 41, Yo, the possible error in fis 7? (a 4 Bits a) a ata d ae a yy Ys all the quantities 2, 4:, 8), 8, being taken with the positive sign. Let us suppose that these errors are each equal to a millimetre ; then, for the combination above described, the greatest possible error in f amounts to 0°19 cm., or nearly 2 millimetres. For the combination referred to in my paper, the error would be very nearly 3 millimetres. If instead of 1 millimetre the errors are each each equal to half a millimetre, the largest possible error in the focal length would amount to about 1°5 cm. Taking, now, one of the formul referred to by Mr. Baynes, et = Be Jo ONG 8 Gh where s=d+x.—2,+ 4, —¥Y2, we find the error in f to be filet Y1 — 1) (f -2- 4) | — (2 By Se nay; “ae Say Say doe Taking the most unfavourable case for the above com- bination, and supposing the error in each case to be one saflllinnesine. this becomes 0°44 cm., or more than 4 millimetres. For the combination in my paper the possible error, on the supposition of a small error of 1 millimetre in each of the Focometry of Lens- Combinations. 79 measurements, comes out to be over 500 cm. It has ceased to be a small quantity. This is due to the very small value of s. The value of H,H, is, in any case, given by the formula HyHi=7 (2-2-4) -a—y Y, aw ; Xv Y1 s Vy XL» = 21 —' -a(2 +4 -2)/(/2-%), te Yy vy YY. Ya Assuming that the error is 1 millimetre in the measure- ments of 21, 41, 2, yz, and taking the most unfavourable case, it may be shown that for the above combination the greatest possible error in H,H, is about 1:16 cm. This is a large possible error for a quantity whose actual value is 2°43 cm., and, to obtain a reasonably correct result, the errors in measurement must be much less than a millimetre. But this defect is not one which can be attributed only to the method I proposed in my paper; it is inherent in all methods in which the positions of H, and H, are deduced from the observed positions of an object and its image. As I have mentioned above, the special characteristic of the inethod is the way in which the magnification is measured. Taking, for instance, the case where the magnification is Oe 29a Vy a 142 = 020493, and supposing an error of 1 millimetre to be made in the measurement of both 2, and y,, and the most unfavourable cease to be taken, the error in the magnification will be 0°0015, or about 0°75 per cent. This accuracy could not be attained by using transparent scales, when, as is frequently the case, small images have to be dealt with. But the measurement of d and two magnifications are all that is required for the determination of the focal length. Additional Note. Since writing the above, I have been led to another method of using the nodal slide to obtain the constants of a lens- combination. It is clear that the method described in my former paper, and further considered above, although quite satisfactory as far as a determination of the focal length is concerned, fails to give an accurate value of H.H,, because small errors are multiplied by the numerical value of d the 80 On the Focometry of Lens-Combinations. distance through which the combination is moved, and this in the case given above is large, 113°8 cm. The method which I now propose is, first, to find O. This gives OP), or 2,, and OP,, or y,, and the magnification y;/2,. Also, as was shown in my first paper,,O divides H,H, externally in a ratio equal to the value of the magnification, or Yy mA OH, Vy OH, aes or enya a ma arma MTT aT ag Hi H, fe) H, H, ‘he nodal slide is now turned about O through two right angles, when H, and H, will oceupy the positions Hy’ and H,'. The combination must now be moved through the distance H,’H, so as to produce an image on the screen which is not displaced. let this distance, which can be easily measured, be d. Then, if H,H,=a, we have a (=a + 1) =a, or In the case above mentioned, 2, is 142 cm. and y, 9°4 em. A careful measurement of d gave, for these positions of the object and image, 2°5 cm. 132°6 Thus CS ‘TAL == "4,9 emmy Also, OH,= — = (0-75 com C= tial = 2-625 om. Hence HP, =142—2°625 =1389°375 cm. EEE = 9-4 — I (S = 9°925 cm. And 1d a f Gee, 189s or t = 9°87. The focal length and the distance H,H, have thus been deduced from the measurements of three lengths, and the actual positions of H, and H, determined. THE LONDON, EDINBURGH, an» DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. VIL. Astronomical Consequences of the Electrical Matter. By Sir OLIveR Loper*. HE inertia of an electric charge was predicted by Sir J. J. Thomson in 1881; the dependence of electric inertia on speed of motion through the ether was calculated by Oliver Heaviside in 1889, and confirmed by Thomson, who also isolated the unit charge in 1899; while the expe- rimental verification of inertia as a function of velocity, by Kaufmann, occurred in 1902. On the electrical theory of matter a body of any material moving at high speed through the ether acquires extra or spurious or apparent inertia; and this inertia is presumably | not subject to gravity, since, as in the case of a solid moving through a fluid, it probably is an effect of pressure reaction and not a real increment of mass. All this is quite independent of the theory of relativity. The inertia factor for small speedsf is (1—v?/c?)~®, or * Communicated by the Author. + The simple factor for inertia-increase 2/m,=(1—v2/c?)—2 is supposed to be an approximation, though a close one, to a more complicated expression, such as Bs ed. 2 46, —s_- 26 — sin 20 My a(: sin 29 sin so) where sin 9=v/c. . See my book on ‘ Electrons,’ pp. 183 & 225. This becomes 1 when 6 is small, slightly less than sec 6 when 6 is moderate, and approaches rs. 9 / 20 infinity in this form, ig (aps +1), as 6 approaches 37. For instance, if v came within a tenth of 1 per cent. of the velocity of light, so that sin @=‘999, the ratio m/m, would be 20. Phil. Mag. 8. 6. Vol. 34. No. 200. Aug. 1917. G $2 Sir Oliver Lodge on what is practically the same thing, 1+ v7/2c?; where ¢ is the characteristic velocity with which every known disturbance is propagated by free ether. It is just this uniformity of transmission which makes the connexion between ether and matter so elusive to experimental observation: measurement is foiled save when matter moves relatively to matter. That is the foundation, and I venture to think all the foundation, for the Theory of Relativity considered as a philosophical reality instead of only a more or less convenient summary of experimental results. An increase in inertia, without corresponding increase of gravitational control, cannot fail to have astronomical con- sequences, though they may be so small as to be barely observable. For inertia is a function of speed even when speed is but planetary, and if the force of gravitation is not correspondingly increased—a thing which we have no reason whatever to think likely, since zther is presumably the vehicle of gravitation and not subject to it until contorted into the singular points called electrons,—then some small but cumulative effects may be caused in the more rapidly moving bodies. Professor Hinstein’s genius enabled him in 1915 to deduce astronomical and optical consequences (some not yet verified) from the Principle of Relativity. See an interesting account by Professor Eddington in ‘ Nature,’ vol. xeviii. p- 328 (28 Dec. 1916). I wish to show that one of them at least can be deduced without reference to that principle. In so far as the deduction is incompletely in accord with quanti- tative observation, there is something further to be considered ; but it is unlikely that a result of approximately the right order of magnitude can be devoid of significance. Consider the amount of the perturbation caused by extra inertia in the case of Mercury, whose orbital speed is approxi- mately half as great again as that of the earth, or say 1°5 x 107~* times that of light. In one part of its orbit it ‘will be travelling parallel to and in the same sense as that component of the sun’s way which lies in the plane of the planet’s orbit, which we may call Mercury’s ecliptic. In all this half of the orbit therefore its inertia will be slightly greater than the average Historical Note.—Ueaviside’s expression for the coefficient of 3v? in the value of kinetic energy for a charged sphere was first given in Phil. Mag. April 1889. J. J. Thomson's expression for the coefficient of v in the value of momentum is contained in his ‘Recent Researches in Electricity and Magnetism,’ page 21, published in 1893. The above trigonometrical expression is intended to represent this. Astronomy and High-speed Inertia. 83 value; whereas in the opposite half of the orbit, where it is travelling against the sun’s way, the inertia will be less than the average. But in all cases the effective or apparent inertia of the planet will be slightly larger than the mass on which the force of gravity acts. Hence we may expect the orbit to revolve in its own plane, or, in other words, the apses must slowly progress; for the effect will be much the same as if gravity were proportionally diminished, the inertia remaining constant. The theory of the apsidal angle in nearly circular orbits, under a central force varying as any power of the distance, is given by Newton with unexampled genius in Principia, Book I, Section IX, and is thrown into orthodox analytic form in Tait & Steele § 248. The result is that for a central orbit subject to a law of force as the nth power of distance, the angle swept through by the radius vector between two consecutive apses, 2. e. between maximum and minimum radii, in orbits nearly circular, is T V(3+n) Newton himself remarks that for a direct distance law this angle will be a right angle; for a constant force, 7/V/3; and for the exact law of inverse square, precisely 7; thus giving a perfectly repeated orbit without any apsidal progression. But if the inverse square law were inexact, the apsidal angle would differ from 7 by a corresponding amount; and the direction of motion of the apses would be progressive if gravity diminished faster than the inverse square law, 7. e. if the index n exceeds — 2 numerically. To get the perturbed rate of progression for Mercury, Al or 43 seconds of are per century,—which value was reckoned by Newcomb as the outstanding discordance from theory,—we have only to remember that the planet makes 4 revolutions per annum, or 8 journeys from apse to apse, so that in a century the discrepancy m/4/(3+n) —7 has accu- mulated 800 or more accurately 830 times; so, to give the observed value, (OE RIE 20 NER s/ (3+ 7) ~ 830x180 x 3600" ‘Whence n= — 200000016. This value was reckoned by Professor Asaph Hall in Astr. J. vol. xiv. page 49, Boston 1894; but then there is no other reason for supposing that the index is not exactly —2. a : | | ; | 84 Sir Oliver Lodge on If, however, instead of a reduction in force, the mass were slightly greater than is proportional to weigkt, the result will be similar, and similar aspidal progression should occur. It is noteworthy that a discrepancy of electrical mass due to planetary speeds will be of the order 107°, and that that is. of the same order as the above imagined discrepancy in n. But to determine the amount of the extra or unexplained advance of perihelion properly, we must know the absolute speed of the planet through the ether. This involves a knowledge of the proper motion of the sun in our cosmic group of stars, and also an estimate of the unknown drift of that group itself as suggested by Kapteyn. Let the sun’s true way be inclined to the plane of the planet’s ecliptic by an angle » (latitude). Let the sun’s gross velocity through the ether be w, meaning its motion towards Vega (which alone would make A about 60°) com- bined with the unknown drift of the whole cosmic group of stars to which the sun belongs. Let the velocity of a planet in its orbit be v, and let @ be the angle which its direction makes with the projection on its orbit of the sun’s true way; then, taking account of the normal component w sind, as. well as of the component in the plane of the orbit, wcos dv, the resultant speed of the planet through the eether is n/ (ov? + 20w 00s.r COsi7);\) 2) nee so that its apparent mass at any point in its orbit is w” -+ v? + 2vw cos r cos 8 (14 Zhe Ae coe Noe In nearly circular orbits this aagle @ may be taken as also representing the angle between the radius vector and the normal to the above line of reference; 2. e. as the @ of ordinary polar coordinates. Now writing the differential polar equation for an inverse. square orbit as usual, with uw the reciprocal of the radius vector, and h twice the rate of description of areas, du 1 cae de ut w is the acceleration multiplied by the square of the distance, and is constant for constant mass. But mw contains the effective inertia in its denominator, and is therefore affected inversely by the mass factor just considered. So on the electrical theory of matter the equation becomes du jb w+? vwecosr ‘ a ale ee gga a), e (2 ) times the gravitational mass. Astronomy and High-speed Inertia. 85 which contains a slightly modified constant, and also a periodic term. It will suffice to take the case of orbits sufficiently circular to enable us to treat v as constant, so as to put a mean value in the small correction term instead of introducing av?=u(2au—1). In that case also we shall have no trouble about the precise meaning of 8. In so far as the angle between tangent and radius vector is not a right angle, 2. e. in so far as the velocity 6 and the position @ are not the same, the effect for small excentricities is to modify the solitary & in the denominator of equation (5) below into k(1—2e/3); but, as this is just the & which can be neglected, the slight complication will not be here attended to. Before solving (2), we may note that this equation has the form of the ordinary resonance equation, “+Ke+n?x= EH cos pt, with x equal to w minus the constant terms above, but with the damping coefficient « zero and with the frequencies n and p equal. So it is an equation which is liable to give infinite values; and even in practice it gives large accumu- lative amplitude when the damping is small. It is the equation on which all tuning or syntony in Wireless Tele- graphy is based. The ordinary particular integral for this case, C= ea, COS Pi n —p? ” gives an oscillation of infinite amplitude, the main infinite part of which, however, may be got rid of by combining it with the supplementary part of the complete solution with arbitrary constants, A cos nt +B sin xt. For putting p=n+<, and proceeding to the limit when z is zero, we get ae Eecos(n+z)t_ _ Heosnt , Ezisin nt n—(n-zy ) Qnz 2nz The first term is infinite, but when combined with the arbitrary term Acosnt it disappears, and the solution is left, Let sin ne, which exhibits an amplitude steadily increasing with time. 86 : Sir Oliver Lodge on So in like manner the astronomical solution for orbits of small excentricity, with v reasonably constant and both w and v small compared with ¢, is wu Cos A ne U= B(1— SS +605 (6-2) — 5 Osin 8), (3) instead of the usual u= oe cos (9—a)). Now @ being defined with reference to the sun’s way we cannot make a zero; infact « must be practically the angle between the major axis of the orbit and the line of reference; for, save for a minute correction term, it represents the value of @ at the perihelion apse. The extra constant term in (3) only matters in cases where w or v is beginning to be comparable with c; but the progressive term containing 0, an angle which steadily increases with the time, is important. For in a century the whole angle swept through ‘by Mercury’s radius vector, at the rate of 4 revolutions per annum, is 8007. The progressive term shows either that the constants of the orbit must change, or that the orbit must revolve in its own lane. To find its rate of revolution, consider the apses as places where du/d0@=0, and where O=a. Then in general i. Ele sin (@—«) + ao coe Spelt) 4 cos8) } . (4) So at an apse, writing & for Ae cos A/c’, e(sin @ cosa—cos Osin «)+k(sin 6+6 cos 0)=0, or (ecosa+k) sin @=(e sin «—k@) cos 0. So aes Mane Sim aa eU esin a kO mes ecosatk ecosatk ecosa’ kK being very small; so that if @) is the initial value of @ at an apse, and @, its value after n revolutions, Qank tan @)-- tan «, = — , €COS @ but dtana=sec? ada; so Qarnk Cos a Tnvw iy — ty, == = —___ 08 A cosa. (6) 6 te? Astronomy and High-speed Inertia. 87 The sign shows that for progress the sun’s effective motion must be in opposite sense to that of the planet at perihelion. Now cos @cosa is the cosine of the angle between thie sun’s and the planet’s motion, so cos a cosX is the cosine of the angle between the lines of the sun’s motion and of the planet’s motion at an apse; or say between the sun’s way and the minor axis of the orbit; call this ¢. So the apsidal progression during 7 revolutions is mnvw cos d oe ee YD and vwcos@ is a scalar, as it ought to be to compare with c?. But the change in & during a century (2. e. the known progress of the perihelion not accounted for. by orthodox gravitational perturbations) amounts to 40 or 43 seconds of are in the case of Mercury; so this gives us, for the unknown motion of the sun, da= ae eh ectda v4 ecxX 43 oe! = Gam 0) AeeO =x 400 ~180 x 3600 A3ec ec ~ 6x 180x36 904" The excentricity of Mercury is given in Galbraith & Haughton’s ‘ Astronomy’ as 20°56 per cent. ; 50 eee 2056 in 7, GOz een OSC eg Cay or two and a quarter times the orbital speed of the earth. I hardly think that astronomers will regard that large velocity as quite unreasonable. From (7) it appears that a nearly circular orbit can be made to revolve very easily, though the revolution would be un- important, while an excentric orbit would be stiff. Another curious result would seem to be that if the major axis of the ellipse points along the projected component of the sun’s way, this extra apsidal progession disappears; whereasif the minor axis lies along the sun’s true way, the kind of apsidal motion now under consideration reaches a maximum. Can this be true? In its favour we can say that the perturbation under con- sideration, though depending on velocity, is equivalent to a radial force varying sinuously with position, directed away from the sun during that half of the orbit where the motions. 88 Sir Oliver Lodge on _ compound additively, and towards the sun during the other half where they compound subtractively. If the sun’s motion concurs generally with that of the planet at aphelion, the virtual decrease in solar attraction near aphelion will cause that point to progress; while the virtual increase of solar attraction at and near perihelion will cause that point alse to progress. On the other hand, a concurrence of the motions at perihelion would cause both apses to regress. So the line of apses steadily revolves either forwards or backwards according to the sense of the sun’s proper motion in the direc- tion of the latus rectum. But if the sun’s way is directed along the major axis, the varying inertia of the planet is equivalent to a radial rorce acting oppositely on the approaching and receding halves of the orbit; so one apse progresses while the other recedes, and there is no cumulative effect on the line of apses. Nevertheless there should in this case be a change in the excentricity. If the sun’s proper motion concurs generally with the motion of the planet from aphelion to perihelion, this half of the orbit will be subject to a virtually diminished solar attraction, and so the excentricity will diminish. The other or receding half of the orbit will be subject toa virtually increased solar atraction, and that also will diminish the excentricity. The causes combine. A reversal of the sun’s motion, so that the component velocities are added from perihelion to aphelion and sub- tracted on the reverse journey, will have an opposite effect; and in that case, in both halves of the orbit, the excentricity will increase. Concerning any possible effect on the Moon:—astronomers appear satisfied that Dr. G. W. Hilland Professor H. W. Brown have settled the small residual discrepancy in the acceleration of the moon’s mean motion, but inasmuch as the frequency of revolution in the case of the moon is considerable, and its speed through the ether is compounded of many causes, it may seem worth while to examine whether its fluctuations of inertia do not call for residual attention. ‘True its monthly speed is almost insignificant, being only about a thirtieth of the earth’s orbital speed, but it shares in the motions of earth and sun, and so the wv to be compounded with it is con- siderable. Any cumulative effect, however, can only be a slight residual one, since its monthly orbit presents every aspect to the sun’s way in the course of a year ora decade. As to the constant.term in equation (3), it would seem that a modified w might affect the period of revolution, because T=27,/(a?/u); unless there werea compensating effect on a. Astronomy and High-speed Inertia. 89 And a very minute acceleration would become important with lapse of time. But the modified pu is no new thing, it has been there all the time, and so presumably it is only the fluctuations that we have to attend to. Moreover, the virtual force, whether variable or not, being always central, does not seem likely to affect h or T. In any case the action, being wholly in the plane of the orbit, has no effect upon the nodes. Another way. To check the calculation of (7), take the expression for the reciprocal of the semi latus rectum, (3) with (@—a)=90°, « being the angle between the latus rectum and a projection on to the orbit of the sun’s way, | ae w+ v wv COS Xr i a ate 92 +é€COS (O—a)— oes Osind), and see how « must change to keep it constant although @ increases by 2nzr. Initially the variable part equals ecos$a7—k @ sin 0, finally it equals e cos (47 —da) —k(2n7 + @) sin 8, so the difference, which is to be zero, gives é sin da=2nrrk cos «, or NTUW GOS NX COS a da= ce : which is the same result as before. Another check. Initially let d=a, so that the planet is at an apse and w is constant for an instant; then the terms in u ecos (9—a)—ké sin 0, differentiated, become e(sin 8 cos a—cos@ sin a) +k(sin 0+ cos 0) =0. After n revolutions @ has returned to its old value +27n, but « has changed slightly to 2’, such that e(sin @ cos «'—cos 6 sin «’) Ee ia 0 + (0+ 2n) cos 0) =0. So subtracting e sin O(cos «’—cos «) —e cos O(sin «' —sin «) +k2ancosO=0; 90 Sir Oliver Lodge on but still 6 and a are practically equal, so esin?ada+e cos? xda=27nk cos a, or 2ank COs a da= : é with K=vwcosr/2c?._ Again the same result as (7). Prof. Hinstein’s result, as quoted by Prof. Eddington in ‘Nature’ of Dec. 28, 1916, is sagt) ea radians per revolution me T(1—e) Pp . To make my result agree with this in appearance, we must eratuitously replace wcos@ (which is quite foreign to Prof. Hinstein’s ideas) by the following, e . or, what is much the same for nearly circular orbits, we can without reason write 6ev instead of wcosd. But the ditfer- ences, both in reasoning and in result, are fundamental. Problem for an orbit of greater excentricity. The equations for an inverse square orbit whose excen- tricity is not small are dazu vu? + w+ Qvw cos wee pa/(1- e *).| cos d=cos dX cos 6’, ia sin (0'—@)=esin (a— 0’), | v= 2uu— p/a, , ) where @' is the angle between the tangent and the projection of sun’s way on the orbit, @ is the angle between radius vector and the normal to. that projection, a is the angle between major axis and the same normal. But to get a cumulative result in a reasonable time the period of revolution should be not too great. Other Planets. Substituting, from (8), 2°27 x10~‘e for wcos@ in (7), w can get the apsidal progression for any other planet of call Astronomy and High-speed Inertia. of excentricity (though only on the assumption that the aspect of its orbit to the sun’s way does not differ numerically from that of Mercury). Its value for n revolutions of the planet is 000227 ©. Ra hich wero For Mars e is variable but is about 9°33 per cent., v=10-4c/1°22, and the period of its revolution is 687 days. So in a century of earth years its perihelion would progress Vat bed 365 x LO iar 67:3: x °09335 x22 =7 seconds of arc. I do not know what the actual outstanding discrepancy is in the case of Mars: the longitude of its perihelion differs from that of Mercury by about 100°. As stated above, in order to rotate the orbit forward, 2. e. to secure apsidal progress, the main solar motion which is to be compounded with that of the planet must have a component agreeing with the motion of the planet at aphelion: a component agreeing at perihelion would cause regress, and a component along the major axis would only modify the excentricity. I now point out, for what it is worth, that if the main solar-stellar-drift in plane of ecliptic were of magnitude 3 (meaning 3 x 10~‘*c) and were directed towards longitude 294°, it would have a component 2 in the direction of the aphelion motion of Mercury, : ” os 99 Mars, 4 29 29 39 99 Earth, ae ” 9 ” oe) Venus 5 thus suggesting that through a comparison of the outstanding discrepancies between theory and observation for different planets, if they were definite enough, it might be possible to get some indication of the direction as well as the magnitude of the sun’s true motion through the ether of space. Summary. The arguments are :— 1. That motion of matter through ether has a definite meaning, apart from relative motion with respect to other matter. 2. Thatan extra inertia due to this motion is to be expected at high speeds, in accordance with the FitzGerald- Lorentz contraction. 92 Sir Oliver Lodge on 3. That this extra or high-speed inertia is not part of the mass but is dependent on the ether and hence is not subject to gravity. 4, That from this reasonable hypothesis astronomical consequences follow which may be detected when cumulative. 5). That under certain specified conditions merely a small change in excentricity is to be expected as the chief result, in certain others an apsidal progress or regress is to be expected. 6. That the outstanding discrepancy in the theory of the perihelion of Mercury would be aceounted for by attri- buting a certain value to a component of the true solar motion through the ether in the direction of the planet’s aphelion path. 7. That using this value for the solar-plus-stellar drift, yiz. two or three times the earth’s orbital velocity, a result can be obtained for the perihelion of Mars, subject to a hypothesis about direction. 8. That by discussion of discordances in the elements of different planets an estimate may be formed of the magnitude and direction of the locomotion of the solar system in its invariable plane. ADDENDUM. I have inquired from Professor Eddington what is the outstanding discrepancy of Mars; and he replies eoa= + 0'"6440'"35. This value (without the probable error) happens to agree exactly with what is reckoned above in equation (10), since it gives an apsidal progress 6a of 7” per century. Note on the possible deflexion of Light. It becomes a question whether the gravitative deflexion of a ray of light, predicted by Hinstein, also follows from ether theory. A wave-front undoubtedly simulates some of the properties of matter. It conveys momentum, as Poynting has shown, and an advance wave-front presumably has to sustain and convey the light-pressure until» target 1s struck. The mechanical stress exists ultimately between source and receiver; but though one end acts on the source, all the time, the other end of the stress has to react on the advancing wave-front until the receiver is reached, unless Newton’s Astronomy and High-speed Inertia. 93. Third Law meets with some exception. It is, therefore, not unnaturai to associate a specific semi-material inertia with the travelling etherial disturbance which occurs in light. The nature of the disturbance is presumably quite different from that accompanying high-speed locomotion of matter : there was nothing likely to simulate general material pro- perties about that. So we are at liberty to ask, concerning the special kind of eetherial inertia associated with light-pro- pagation,—Is this momentum-inertia likely to be subject to gravity? In other words, how far does the temporary or travelling eether disturbance in a wave-front correspond to the permanent ether disturbance constituting an electron, which is so interlocked as not to need any locomotion for its existence? Is ita sort of temporary matter, not permanently constituted, but possessing material properties while it lasts ? Hxperiment must answer the question. An _ affirmative answer would be of the greatest interest. If it be assumed, as probable or possible, that gravitation is one of the properties of this imitation or temporary matter, though its gravitation constant may be g times the ordinary y (gq being either greater or less than 1), the problem of deter- mining the gravitative deflexion of a ray becomes the easy one of reckoning the angle between the asymptotes of a small comet flying in hyperbolic orbit near the sun with immense velocity c. The angle is poe Gin Ren eae: fv gamed Nes ale where 7pis its nearest approach to the central body of mass M. This expression for the deflexion 6 agrees with Hinstein’s predicted value if gis taken as 2; though the only reason I can see for that at present is the very speculative one that its gravitational entanglement may depend on the maximum amplitude squared, while its inertia depends on the mean. The observational determination of this quantity q is clearly an important one. It may be anything from 2 to 0, though to me the value 1 seems more probable than anything except 0. Taking q as 1 for arithmetical purposes, and writing yM as gR?, the value of the deflexion for grazing incidence is 2gR/c?; but ./(gR) is the velocity in a grazing circular orbit, which for the earth is 5 miles a second and for the sun 50 times as 94 Lord Rayleigh on the Pressure developed much. So the deflexion for a ray of starlight grazing the sun 1s PAR Oh ee 9 : Lobe = 6) | opuanne 2x (ssso00 dO x) LOR, woruGienees Longitudinal Possibility. The velocity of light issuing radially froma body might on this hypothesis also be affected, since it could be pulled back by a maximum amount represented by the free fall from infinity, 2. e. by /(2ggR); though the longitudinal g need not be the same as the transverse g concerned in deflexion. But this sort of action, if it can be imagined as likely to occur, and even if it caused a reduction in sunlight-velocity of 26 miles a secondin the neighbourhood of the earth, would not yield any Doppler effect ; the waves would still be received at their emitted frequency. VIII. On the Pressure developed in a Liquid during the Collapse of a Spherical Cavity. By Lord RayuLeEtes, OVO Ts Se QR/7 HEN reading O. Reynolds’s description of the sounds emitted by water in a kettle as it comes to the boil, and their explanation as due to the partial or complete collapse of bubbles as they rise through cooler water, I proposed to myself a further consideration of the problem thus presented; but I had not gone far when I learned from Sir C. Parsons that he also was interested in the same question in connexion with cavitation behind screw-pro- pellers, and that at his instigation Mr. 8. Cook, on the basis of an investigation by Besant, had calculated the pressure developed when the collapse is suddenly arrested by impact against a rigid concentric obstacle. During the collapse the fluid is regarded as incompressible. In the present note I have given a simpler derivation of Besant’s results, and have extended the calculation to find the pressure in the interior of the fluid during the collapse. It appears that before the cavity is closed these pressures may rise very high in the fluid near the inner boundary. * Communicated by the Author. during the Collapse of a Spherical Cavity. 95 As formulated by Besant *, the problem is— “An infinite mass of homogeneous incompressible fluid acted upon by no forces is at rest, and a sphericai portion of the fluid is suddenly annihilated; it is required to find the instantaneous alteration of pressure at any point of the mass, and the time in which the cavity will be filled up, the pressure at an infinite distance being supposed to remain constant.” Since the fluid is incompressible, the whole motion is deter- mined by that of the inner boundary. If U be the velocity and R the radius of the boundary at time ¢, and wu the simultaneous velocity at any distance r (greater than R) from the centre, then | | J Le ee a INC Gk, and if p be the density, the whole kinetic energy of the motion is 4 { ee Oo WERE: SER EVR aan 2) R Again, if P be the pressure at infinity and Ry the initial walue of R, the work done is AnP eee I ME 0G) When we equate (2) and (3) we get 2P (R,y° Doe R2 a: Ny (4) expressing the velocity of the boundary in terms of the radius. Also, since U=dR/dt, oer BaF He 3 a - p= Poy /(s5) | EVM EB if B=R/R,. The time of collapse to a given peaien ve the original radius is thus proportional to Ryp?P~?, a result which might have been anticipated by a consideration of “dimensions.” The time 7 of complete collapse is obtained by making 8B=0 in (5). Am equivalent expression is given by Besant, who refers to Cambridge Senate House Problems of 1847. * Besant’s ‘ Hydrostatics and Hydrodynamics,’ 1859, § 158. 96 Lord Rayleigh on the Pressure developed Writing 6?=z, we have 1 BePdB 1 ae eee te which may be expressed by means of [ functions. Thus fe ey anes | T=Ro4/ oT Wig 2) = 91468 Riv/ (oP). . (6) _ According to (4) U increases without limit as R diminishes. This indefinite increase may be obviated if we introduce, instead of an internal pressure zero or constant, one which increases with sufficient rapidity. We may suppose such a pressure due to a permanent gas obedient to Boyle’s law. Then, if the initial pressure be Q, the work of compression is 47QR,° log (R,/R), which is to be subtracted from (3). Hence 3 log _ 2P /R,? 2QR Us g 1) = logs. Te Re and U=0 when Pad —z)+Qlooez=0, (5) 2 ee (8) z denoting (as before) the ratio of volumes R?/R,°?. Whatever be the (positive) value of Q, U comes again to zero befure complete collapse, and if Q>P the first movement of the boundary is outwards. The boundary oscillates between two positions, of which one is the initial. The following values of P/Q are calculated from (8): z P/Q. 2. P/Q. mae 69147 1 arbitrary an 4:6517 2 0:6931 > 2-5584 4 0-4621 : 18484 10 0:2558 > 13868 100 0-0465 1 arbitrary 1000 0-0069 Reverting to the case where the pressure inside the cavity is zero, or at any rate constant, we may proceed to calculate during the Collapse of a Spherical Cavity. a7 the pressure at any internal point. The general equation of pressure is ldp _ Dug du y te (9) p dr 7 iy, ee ea ae u being a function of r and ¢, reckoned positive in the direction of increasing 7. As in ee w= UR?/7?, and du _ R di a qn ). Also 2} (on ee 2 ont Ro, dt dt and by (4) a Uae dt p 0 R*’ so that AWR) sore P Ro? LE 1a yaaa Thus, suitably determining the constant of integration, we get pa By a7 ee or Le 5 3rt re eae At the first moment after release, when R= Rp, we have DRG T ye sa eh ee CRD) When r= R, that is on the boundary, p=0, whatever R may be, in accordance with assumptions already made. Initially the maximum p is at infinity, but as the con- traction proceeds, this ceases to he true. If we introduce z as before to represent R,?/R®, (10) may be written p R R4 | Pa1=3 (e-4)-54(e-1), - - (12) and dp/P _ "i (42 —4) R? a Oe = (eA) p18) The maximum value of p occurs when y _ 4z—4 ; Rs ph? (14) and then Dey ee —4)5 pole tt pean (15) Phil. Mag. 8. 6. Vol. 34. No. 200. Aug. 1917. Jal 98 Pressurein a Liquid during Collapse of a Spherical Cawty. So long as z, which always exceeds 1, is less than 4, the greatest value of p, viz. P, occurs at infinity; but when z exceeds 4, the maximum p occursat a finite distance given by (14) and is greater than P. As the cavity fills up, z becomes great, and (15) approximates to ee ae | Pe AS BERR Ol (16) corresponding to r—BR=1587R. . 2) It appears from (16) that before complete collapse the pressure near the boundary becomes very great. For ex- ample, if R= Ro, p=1260P. | This pressure occurs at a relatively moderate distance outside the boundary. At the boundary itself the pressure is zero, so long as the motion is free. Mr. Cook considers the pressure here developed when the fluid strikes an abso- lutely rigid sphere of radius R. If the supposition of in- compressibility is still maintained, an infinite pressure momentarily results; but if at this stage we admit com- pressibility, the instantaneous pressure P’ is finite, and is given by the equation | re Py iio? 28! —2pl’=. Ts —1), (18) 8’ being the coefficient of compressibility. P, P’, 8’ may all be expressed in atmospheres. Taking (as for water) 8’ =20,000, P=1, and R=3,Ro, Cook finds P’=10300 atmospheres=68 tons per sq. inch, and it would seem that this conclusion is not greatly affected by the neglect of compressibility before impact. The subsequent course of events might be traced as in ‘Theory of Sound,’ § 279, but it would seem that for a satis- factory theory compressibility would have to be taken into account at an earlier stage. April 13, 1917. Disha a IX. Hue Difference and Flicker Photometer Speed. By Hersert i. Ives *. N the first of the writer’s papers on the flicker photo- _H meter some data were given on the speeds of operation of the instrument when the luminosity of the spectrum was measured against a carbon-lamp comparison standard. These speeds, which are critieal speeds for the position of intensity match, show a minimum near "58. In explanation of this minimum it was remarked f: “In order to compare lights of different colours it is necessary to attain such a speed that the colour flicker, due to difference in hue, dis- appears. It is therefore to be expected that at the ends of the spectrum where the hue is most different from the comparison lamp, a higher speed is necessary.” This explanation appeared adequate to the writer, and partly for this reason, partly because no quantitative theory was then available whose verification depended on fuller data, and partly because no flicker photometer then existed which was entirely free from purely mechanical flicker, or abrupt transitions which might in part behave as such, no further experiments were made on this line. Recently, however, Mr. L. T. Troland ¢ has published somewhat fuller data of the same kind, which he explains in a different manner. According to his view the wave-length-speed curve may be interpreted as the reciprocal of the luminosity curve, the minimum in the yellow-green indicating the greatest whiteness, which he considers as depending on the same underlying process as luminosity. This view is so antagonistic to the present writer’s ideas on the meaning of luminosity, and on the mechanism of intermittent vision as developed in recent theoretical papers §, that it appeared desirable to secure some addi- tional experimental data, using the new polarization flicker photometer ||. These data, which are given below, appear to substantiate the theory upon which the experiments were based. * Communicated by the Author. t “ Photometry of Lights of Different Colours,” Ives, Phil Mag. July 1912, p. 167. The italicizing is added in the quotation. t “Apparent Brightness, its Conditions and Properties,” Troland, Illuminating Engineering Society Convention, Sept. 1916. § “Theory of the Flicker Photometer,” Ives & Kingsbury, Phil. Mag. Noy. 1914, p. 708, and April 1916, p. 290, || “A Polarization Flicker Photometer, and some Data of Theoretical Bearing obtained with it,” Ives, Phil. Mac, Apr. 1917, p. 360. H 2 100 Mr. H. E. Ives on Hue Difference Theory. The greater part of the special theory necessary to handle this question is contained in the next preceding paper, on “A Polarization Flicker Photometer, &c.’’*, in the dis- cussion of the brightness and hue discrimination fractions. It is there shown (equation 14), that the critical speed at the equality setting of two different colours is given by : le nae preven |S a: where w,, is the critical speed of the mixture, w, and w,¢ are the critical speeds for the two colours (R and G) separately, 6, is the brightness discrimination fraction, and 6, the hue discrimination fraction. The latter is defined as the differ- ence in the quantity of one of the colours in the mixtures at the opposite phases, divided by the mean quantity, and it was pointed out that this fraction, unlike the brightness. discrimination fraction, varies with the size of the colour difference. ; For the purposes of the present paper it 1s convenient to consider this fraction in a slightly different hght. Thus, instead of identifying it with one colour of the mixture only, it may be identified with both by considering it to repre- sent a just distinguishable distance along a line of a colour- mixture diagram, divided by (half) the length of the line. It is thus twice the magnitude of the just distinguishable fraction that would be most naturally derived if the definition were developed solely from colour-mixture considerations. Now this just distinguishable distance along the colour- mixture line remains fixed, no matter how far in either direction the line is extended, but the value of the fraction decreases directly as the length of the line. Consequently, if we wish to learn the effect of increasing the colour difference between the lights compared (confining ourselves for the present to lights whose equal luminosity mixture is always the same), it is only necessary to consider the value of §, as varying inversely as this difference. If we call the distance apart of the two compared colours. * Tves, loc. cit. and Flicker Photometer Speed. 101 on a colour-mixture line c, in any convenient units, we then have the speed given by WG OT on a ee, ° ° . ° (2) which may be simplified by the combination of constants to M (log =? ) 3) oO = | og) Se ey diver tine lL iec cL ne ( ) where M is a function of the working intensity. From inspection of this equation it is seen that the speed becomes zero when the two compared colours are separated on the mixture diagram by only the just distinguishable difference, and that the speeds go up apparently without limit as the colour difference increases. Actually, as experiment shows, no hues exist sufficiently far apart to make the critical speed of an equal luminosity mixture ever more than a fraction of the speed necessary to eliminate flicker of the coloured light against darkness. In order to plot this equation on such a scale as to represent an actual case it is necessary to know the value of 6,. One method of obtaining this was developed in the preceding paper. Another, bearing more directly on the present problem, may be outlined. Suppose two coloured lights o£ equal intensity to illuminate the two glasses of the mixture photometer described in the previous paper. Let these be represented as the end points of a straight line of convenient length ec. Suppose the critical speed determined. Then let the two glasses be turned until each is illuminated by two parts of one light and one part of the other. The colour of each is then represented by a point one-third along the line from the end, and the colour-mixture distance of the two glasses is 1c. Suppose the critical speed deter- mined for this condition. We then have two equations from which 6, can be found. In fig. 1 is shown a plot of equation (3), in terms of a against ae re en curve was not undertaken, since the curve shown by Troland * for mixtures of red and white light is closely of * “ Apparent Brightness, &c.,’’ Ilum. Eng. Soc. Convention, Sept. 1916. Direct experimental verification of this 102 Mr. H. EK. Ives on Hue Difference this type, the deviations being no greater than can be ex- plained by the fact that he moved the hue of the mixture continuously from one end of the mixture line to the other, thus probably encountering differences in the value of the just noticeable distance along this line. Fig. 1. REALE IET: teh | eae en : a i | } 3000 ©4000 jan mscos Oy a 5) Lee, 2¢ =) and hue difference (=) , as calculated CH from theory. Relationship between Sr amanivse Gus tole) Pp a 7 ieee meus Ente re, Va Wiehe a oe au Q+AD ie ely u Serr “—exp(~ V9 2) } T+. +exp(— Fe4/ 2). bg ne shee ahs 5 Further, the equation of transference of heat across the bounding surface gives ott( Kaa/ 2 Jf 2) rAK,@ Ae K uF raulaly AO a Xo ae a aie /? ke v) Lx) KEG) Ne eee ee Ar aa ‘ (15) The value of ® thus found must be substituted in the formule for V,) and V.. 118 Mr. H. Jeffreys on Periodic If, then, V,' be the value of V, over the sea, we shall get the values of EF’, ®', and V,' simply by putting the sufhx 1 for 0 throughout in the above formule. All of the exponents are unaltered. We therefore see that V, can be put in the form Vaden + Be + Coxp(— tg /2), .. GEG) 74 2 where B and C have different values according as the point considered is over land or sea, but A is the same in both cases. Now it is obvious that if the temperature of the air were a function of the height and time alone, so that all the sur- faces of equal temperature were horizontal, there could be no horizontal variation of pressure and consequently no hori- zontal movement of air. Without altering the effect of temperature on horizontal movement we can therefore sub- tract from each of A, B, and C their mean values over the surface of the earth. Then, except near the coast, we can treat A as zero, while B and C have equal and opposite values over land and over. sea. Hven if we include the effect of the coast, it is evident that both B and C can be expanded over the whole plane in sines and cosines of multiples of wa/l. As henceforth we shall be considering only the atmo- sphere, the suffix 2 can now be omitted. TH The solution of on) =kV/7V that contains sin 7 on factor and vanishes when zis +© is as a e-™ sin mall, where m is the solution of aT mat + i that bas a positive real part. . (17) We shall consider the motion of the air that is produced when we have Veov=bey” sin 72/14 ce 2 Sim a7) 1 ee) where + and ¢ are constants. The solution for all other terms can be derived from those for these two. Convection Currents in the Atmosphere. 119 II. The Periodic Motion of the Atmosphere. The equations of motion of the atmosphere are assumed to be Du _y _ lop 2 1,08 | Di Seles oat Dv 1 Op 5 i 00 ee = Ye eyo eS a ee , Di ont Mets | (1) | Dw 1 Op 1, 06 | eee 2 ih a ZL — nae + kXV/?w Wt xk ae where Ou ov ow Bo ah cit dagen tee Aa Dean (tap) and & is the kinematic coefficient of pseudo-viscosity. It has been shown by G. I. Taylor to be equal to the ratio of eddy conductivity to heat capacity per unit volume in the case of an eddying fluid with no vertical motion apart from eddy motion. 1 assume this to be the case in general. The true kinematic viscosity is not equal to the quantity that Fourier denotes by K/CD, or the ratio of true conductivity to heat capacity per unit volume; the latter is indeed 7 times the former on the kinetic theory of gases, where f is about 2°5. Both are, however. very small in comparison with the eddy viscosity. For instance, Taylor gives™ for his 27/4¢, which is my k, values ranging from 0°57 x 10% to 3°4 x 10°, whereas the true kinematic viscosity for air is 0-017 at 0° C. The eddies therefore produce overwhelmingly the greater part of both the effective conductivity and the effective viscosity, and we are justified in putting & in the terms depending on both of these in the equations. The equation of continuity is 1 0p ) — EF a . . e . . . e But, by Charles’s law, p=po/iteV), . . . e . . (4) where pp is the undisturbed density and & the coefficient of expansion. Then neglecting V’, we have Sc Le ce nae ay * Phil. Trans. vol. cexy. A. p. 10 (1915). 120 Mr. H. Jeffreys on Periodic By hypothesis all the terms in the second equatteE are zero. Put 9/dt for D/Dt, and let p=jo+p', where po is the pressure in the undis- turbed state. jo is a function of z only. Then to the first order, p Ou Po Ox ar © +aV OPo Le 0 Lop _ il 0 0 0 ‘Rae : } . (7) 3 55 = —poLi= —gpo. (8) Hence 1 Op lhe yey Se oN =) ae ae (9) Further, X=0 and Z=—g. The equations of motion then reduce to ee £4 fa LD) Ot Po Ox 3 fs) i Ow _ ,_t Op" ah ee Mi) my O21 eo eee Ou Ow OV aes a a ee ° ° ° o (12) Differentiating the first with regard to 2, and the secon with regard to z, and adding, we find Oly, 4 ON o’7V Soe NY//2 ae 2 ~ Vip! = Toa ta ar Oe 4 9 7” pa LE =| ore bit) +9 ee ~ noe 2 | Ge-m gin ~~ ev" on | mee Convection Currents in the Atmosphere. 121 Hence 1 _ 4k — rl? i et SEE ag oe oi, 8 pvt Po& y* 7a ae “Ve l ei (gm+ ky )k comm? gin a ont uy l +- Le-v#! gin en", SSR hati: SE igs 1) where L is an arbitrary constant. To find the velocities we have now to solve ~~ _ hV7*y = = —— — kX7?u Rta oS Ou 1 me oh dp! a) ot Po Ox” 4) ( Ow 7h gaa rey oo | G7 == gts = CP oa, Ot Ma shy ae Po OZ Ae Ne where both the right sides are now known. The boundary conditions to be satisfied are :— (1) There shall be no motion when ¢ is very large. (2) wand w shall both vanish when z=0. This is the condition that there shall be no slipping at the boundary. The motion of the ocean is thus neglected. The solutions are therefore an = —gv+kiy(v?— rl?) + y? lL (P—a7/P?)tiy—k*—r’/l)} agnuk 7 Cz u=— b cos ev! (e—" — em) en MZ cos = eve uy Ll 2mk amt L ayt 72 /L —mz 9D wager —= 2g aN ant Keer. == gv + kiry(v? — 7/2?) +9? > TTX yt —pr —mMz OP = oe) 2) Ley — KP? aay oy ee ) mee sin wy—k(v?—7?/P?) agn*k CZ + = et! —e-™) = o-™ sin ert viyle Dmk © l amt LL Le Nee Sc’, 72 rie RD + i a / é (e é ) (16) 10) Mr. H. Jeffreys on Periodic It remains to determine L from the condition that Om Of olen On collecting the terms with the same exponent in this equation, we find that the terms involving e~”, e~7*/, and ze-™ vanish. The coefficient of e~” sin (7a/l)eY remaining gives fiy—k(P?—?/P?) (PP —e) PP? _ age ee eee 2mluy San 1 lury ma 7} =O. (17) Up to this point the solution is exact, whatever be the values of the quantities concerned. Our object being to apply the results to the variation of atmospheric mass distribution and pressure in the interior of the continents, we proceed to consider the order of magnitude of the quantities involved. For Asia / is about 6000 kilometres; for Australia it would be of the order of 2000 kilometres. Taking k=3 x 103 and y= 2/(1 year)=1°99 x 10 “/I sec., we have | 0°316 x 10~° em. so that the annual variations depending on the e~-”* term will extend upwards for several kilometres. Nevertheless, a/l may be taken as small in comparison with ,/ (y/k)- Again, it is stated that 2 of the insolation received by the earth is absorbed in the atmosphere. Thus 1/y must be of the order of the height of the atmosphere. Hence v and m are of the same order, and both are large compared with w/l. Applying these approximations to the last equation, it reduces to wybom(v—m) byl gre yrel TG (iy —kv?)v?lm 1 Pa 22 © ma - var lone Then the value of p’ at the surface is given by a 4 2 2 Une k Fenn dosee == eee ue aa yes (om anni Po l V by Jnighon (vm) 4 oak cma (ty—kv? lm vl © 2ley ma 4 2 2 kh 2 a ates LY ta ie Ce ie oCLS) v iy = mT Convection Currents in the Atmosphere. £23 All approximations hitherto made hold when the period is a year or shorter. Hvyen for a daily period we see at once that the only considerable terms are given by 1 mel ge ay P 2=0 = — pore” sin= {2 + (2 “- ye. - @9) m MIT We likewise want to know the variation in the mass of air per unit area of the surface. The total surface density is given by o=| paz, e 0 and therefore its variable part = ce Vide 0 Obie He PFie a ery iar: Fak P) e e = — poxe'v sin 7 45 “- <}. (20) Hence p'=go', provided we can neglect is in comparison 2 with g. for the yearly terms a is of the order of 6 x 10~°, while for the daily terms it is Piscat 1 cm. /2 sees)-4 0) lite assumption made by meteorologists that p'=go' is therefore justified in both cases. The ‘result. seems somewhat sur- prising at first sight, since the last term neglected has a factor almost equal to the radius of the earth in the numerator. As the result holds for the term in a involving sin > we see that it will also hold for cos — = | by a simple change ot origin ; further, by writing //(2r Lh and 1/2r respectively for 1, we see that, provided r is not very great, the result will hold for the terms in sin(2r+1)7e/l and cos 2r7a/l. Thus, for every term separately in the Fourier expansions of the quantities concerned, p’ is equal to go’. This relation therefore holds for the sum of the series. III. Case of Circular Symmetry. In this case the distribution of land and sea is supposed to be symmetrical about a vertical axis, but the circumstances are otherwise as in the last problem considered. Let 7 be the distance of any point from this axis, z its height above 124 Mr. H. Jeffreys on Periodic sea-level, and @ the azimuthal angle. Then it is obvious from symmetry that if the velocity at any point be resolved into three components, corresponding to 7, @, and z respec- tively, the @ component is always zero, while the other two, with the pressure and the temperature, are independent of od. Sow, as before, for a periodic variation the variable part of the radiation absorbed per unit heat capacity is of the form Qe~”, where Q is a function of 7 only multiplied by ev, Most functions of r can be expressed in the form 70) Toran “0 for all values of ; if Qe-% is of this type it will be suf- ficient at present to treat only the case of Q=BeYJ, (Az), where B isa constant. Then the temperature satisfies the equation ioe Tt hN/e We == Berd (Ar) Cnn s . e (1) A particular solution is V=be''J)(Ar)e~”, where bf iy —k(v?—A2)) = B. oo ae The complementary functions must have no singularity when r=0, and vanish when ¢ is infinite. Further, when zis zero, V must reduce to the temperature at the surface of the earth. Let this be (b+c)eY%J (Ar). Then the complete solution is Ve (be + ce ™ \Jo(nev = es where m™ is the root with a positive real part of m?=)? + wy/k. By making 6 and ¢ functions of » and integrating with regard to X from 0 to «, it will be possible to obtain the most general solution. The equations of motion are as before Ow) lL Op Pik Ov s. 1 Op oe 08 Ot 8 96 a ie tem oo 1 é | i ha Convection Currents in the Atmosphere. IDS From these we can deduce mee av ov Ov ee Pig + + ght AE Ae = | —(be9+ 5 7B ae 37 Je I, > = (emg+ gvele™ b Jo(Arjer". Hence a bvg + SupB+ gy a eee ES See = e—™ ( Jo(Arje” ser IMO ME i Toy 00 B) where L is an arbitrary constant. As the horizontal motion is all radial, we can put u=Ucosd and v=Using, where U is independent of @. Then the first two equations of motion give IME wy —kV2U + 5U= “aH _ hau), Ldsittew ut stato ODE and the last gives uyw —kV7?w = gv (E — : kavyV | PTY ted aD. ZN Dp while Bey = a (= ea + uk bem Po 2 t vy —)? ey comm LY o(Ar) ev? uy 5 + ale7*Jy (Ar) nal 126 Mr. H. Jeffreys on Periodic Then a typical value of U is given by Slee) yk (pine a ey ey b(e~” —e-™) U=arf Y a~ py MZ V orc LTiOwer alin — . J(Arjev'(e** — e- m2), mma Ve 5)) age f (eS uyky (Vian) —Vz —mz\ aan (Pj Toy — ba} OM ae) Mge ~ Syl ryt Tees ii Jo(Ar)er a ; +. a Jo(Arjev(e-* —e-™), a (9) oy L must now be found from the condition that or 4 8 OP b= anyV. 80 hai) Now OU 00 TNO | ay +. 3y = F Or ( [Gye ° ° . . (11) We find then the equation bod? (v—m) . buy (my —X°*) i Ge (v?—NX?) {oy—h(v?— 27) } v—)r 2muy —wye+ Zm—a)=0. aa (li!) ~.Lhis is converted into equation (17) of the case of infinite parallel strips by simply writing 7/l for 2. Case 1. Large island or continent. > is small compared with v and m. In the case of a daily period, this splits up at once into two sub-cases, according as the radius is large or small compared with 1000 kilometres. If it be small, as Convection Currents in the Atmosphere. 127 it usually is, then in (12) only the terms involving g need be considered. With sufficient accuracy, then, U=—anrq \ aR (e7"? —e7™*) + a JiArjev ee \ ap sat (e-*— 0-2) J ,(arjor, (13) p =—ag \<— + a ce-me Jo (Ar) e'%# Po ¥ y Bi {| ab Fah dol arent Ce aly In each case the ratio of the second part to the first is of order X/v, so that the second part may be omitted. Then = = | ee fe SHO g = = ryt JL = any Uv(ay a kv?) (e a é ) — Duy“ Jy(Ar)ev : (15) ! b j p=—4p9 art Sem gl OA a RIAD cord a AUR og a8 110) ree ce o( AT ee yw ble fly oye, oo CLT) It } were zero, the pressure would reach its maximum an eighth of a period after the temperature minimum, and the maximum outward velocity at a small height would be a quarter of a period after the temperature minimum, and an eighth of a period after the pressure maximum. The phases would be the same at all points. If the radius be a large multiple of 1000 km., the co- efficients are all small. Hence the daily motion does not penetrate far into the interior of a continent. In the case of an annual period, these approximations are always justifiable even for the largest continents. In both cases the relation p'=go’ holds. An amplitude of 1° C. in the temperature variation corresponds to one of about 0:42 mm. in the pressure variation. 128 Convection Currents in the .Atmosphere. Case 2. Small island. » large compared with v and (ey/k)?. | My iy ed a Akg Ue 2k (6 oy) + WM Bliss vg , 2 Ppa (eal ue ; fF a{(Zadas)iens (tbo ae (19) 24 Seen fe b(e~"%—e7**) + ee Jar”, (20) Ve (be 4 co Jo Are ea It is no longer true that p’=go'; in other words, in problems dealing with small islands the neglect of the vertical velocity is not justifiable. The largest term in p' is —2pya(g/r)be~*J o(Ar)ey". Thus, the maximum pressure coincides with the minimum temperature. Summary. 1. In land-masses of continental dimensions the customary neglect of the velocity in the equation of vertical motion is justifiable for harmonic disturbances with a period of a year. If the diameter of an island be between about 10 km. and 1000 km. the same will hold good for a daily period. In such a case, all heating being supposed done by eddy currents, the pressure would reach its maximum an eighth of a period after the temperatnre maximum, and the maxi- mum outward wind-velocity at a small height would occur a quarter of a period after the temperature minimum. In the case of an annual period an amplitude of 1° C. in the tempe- rature variation would give rise to one of about 0°42 mm. in the pressure variation ; this agrees with the facts of obser- vation within the limits of variation of the eddy viscosity. 2. Daily motion cannot penetrate far into the interior of a large continent. 3. In the case of a small island it is no longer justifiable to neglect the vertical velocity. When it is very small the pressure maximum and the maximum outward velocity occur at the same time as the temperature minimum when the heat is supposed supplied from the earth’s surface. In order that this may hold it is, however, necessary that the island should not be more than 0:1 km. in diameter for a daily period, and 2 km. for an annual one. _ eh sor: | XI. The Maintenance of Vibrations by a Periodic Field of Force. By C. V. Raman, M.A., and AsnutosH Dey *. (Plate I.] HE effect of a periodic field of force on the motion of a body subject to its influence has been discussed by one of us in a previous publication in this Journal +, one of the results of outstanding importance noticed being the series of special relations between the frequency of the field and that of the steady vibration possible under its action. It was shown that the motion is capable of being maintained when its frequency is either equal to, or is $ or 4,4,4, or ¢ of the frequency of the field, that is, any submultiple of it, but not when the frequency has any intermediate value. The experimental work and theory published in that paper related to the motion of a system with only one degree of freedom, the period of free vibration of which is determined entirely by the field ft. Recently, when experimenting with the electrically maintained vibrations of wires, we have noticed certain interesting effects which may he classed with the phenomena referred to above, but which merit separate discussion in view of the fact that the system in this case has a series of free periods of its own, quite independently of the field. These will now be described. EHaperimental Method and Results. The present investigation relates to the vibrations of a steel wire about 2 metres in length, stretched vertically under an adjustable tension, and subject to the transverse periodic force exerted by an electromagnet placed near some selected point on it. The electromagnet is excited by an intermittent current from a fork-interrupter of frequency which in our experiments is generally 60 per second. The forced vibrations (having the same frequency as the inter- mittent current), which are usually excited in the first instance, when the tension of the wire is adjusted for * Communicated by the Authors. + “On Motion in a Periodic Field of Force,” Phil. Mag. January 1915, by C.V. Raman, M.A. Also ‘ Science Abstracts, 1914, p. 586, and 1915, , 484, t ¢ The vibrations studied in that paper were those of the armature of a synchronous motor of the attracted-iron type when not in rotation, under the influence of the magnetic field due to an intermittent current. Phil. Mag. S. 6. Vol. 34. No. 200. Aug. 1917. -K 130 Prof. C. V. Raman and Mr. Ashutosh Dey on the resonance, are of the same form as those described by Klinkert in a paper on electrically-maintained vibrations *. It is noticed, however, that when the tension is such that the wire vibrates in two, three, or larger number of segments, and the electromagnet is not too tar away from the wire, the motion of the usual type first set up is unstable, and gradually changes form, the nodes ceasing to be points of rest, and the frequency of vibration changes to a value which is a submultiple of the frequency of the fork. For instance, if the wire initially divides up into two segments and vibrates with a frequency of 60, its centre, which at first is a node, gradually acquires a very considerable motion, and the frequency of the vibration alters to 30. Similarly, if the wire initially vibrates in 3 segments, the frequency changes to 20 when the instability sets in; when the initial vibration is in 4 segments, the frequency changes to either 30 or 15 according as the instability does or does not result in a movement of the centre of the wire, and so on. The rate at which the instability sets in and results ina change of type depends upon the position of the electro- magnet, its distance from the wire, and the strength of the intermittent current which excites it. Generally speaking, the rate of increase of the motion at the nodes is small, and it may take some minutes for the change to develop to the fullest extent. The gradual alteration of the form of the vibration may thus be closely studied, and this fact adds considerably to the interest of the experiment from the acoustical point of view. If the distance of the electro- magnet from the wire and the strength of the exciting current be suitably proportioned, the vibration with the altered frequency finally reaches a steady state, the ampli- tude of variation then attaining its maximum. If, however, the electromagnet be too near the wire, or if the exciting current be too strong in proportion to the distance, the motion continues to increase in amplitude till the wire finally comes up against the pole of the magnet. This occurs most frequently when the tension is small and the wire divides up initially into a considerable number of segments. To enable the frequency of the field to be compared with that of the motion set up by it, the vibration-curves of some selected point on the wire and of a small style attached to * G. Klinkert, Annalen der Physik, vol. lxv. (1898). For a practical application, in acoustics, of this class of vibration, see ‘Science A bstracts’ (1916), p. 483. Maintenance of Vibration by Periodic Field of Force. 131 the fork-interrupter are simultaneously recorded on photo- graphic paper *. This may be done, either at some stage in the progressive change of vibration-type, or when the motion reaches a final steady state. Six records obtained in the course of the work are reproduced in Plate I. They represent, respectively, cases in which the frequency of the vibration is equal to, 4, 4, +, 4, and 4 of the frequency of the field. Excluding the first, which is of the usual type, these records are typical of those secured at a fairly early stage of the progressive change of vibration-form, the electromagnet being placed at about } of the length of the wire from one end, and the point of observation being at a similar distance from the other end. The upper record in each case, which represents the vibration of the wire, shows a strong upper partial having the same frequency as the field. Other records (not reproduced) show that at a later stage the partial having the same frequency as the field becomes relatively less important than the others and is not then so obvious to inspection. It is clear, especially from the first two of the records reproduced, that the motion includes a considerable retinue of upper partials.- This is not sur- prising in view of the fact that the field due to the electro- magnet under the excitation of the intermittent current is practically of an impulsive character, as already shown in the paper quoted above (Phil. Mag. Jan. 1915). Theory (as will be shown below) indicates that the ordinary forced vibration which is excited when the tension of the wire is adjusted for resonance is not at all essential to enable a vibration having a frequency equal to a sub- multiple of the frequency of the field to be set up and maintained. This has been tested in the following way: two electromagnets are placed opposite different points on the wire, one or the other of which could be excited at pleasure. The first being placed opposite a point distant, say, 4 of the length from one end, and excited, the tension of the wire is carefully adjusted for resonance so that it vibrates in two, three, or larger number of segments as desired. The second electromagnet is placed exactly opposite a node of this forced oscillation, so that, in accordance with a well-known principle, it is incapable of maintaining a forced vibration of the ordinary kind when fed with inter- mittent current. It is observed that when the second * A method based on the optical composition of the vibrations of the fork and of a selected point on the wire could, no doubt, be used for the same purpose, as an alternative. K 2 132. Prof. C. V. Raman and Mr. Ashutosh Dey on the electromagnet alone is excited, the wire remains practically at rest. But this state of rest is unstable, and, gradually, a vibration develops and attains a large amplitude, its frequency being a submultiple of the frequency of the field. Investigation by the method of vibration-curves shows that, in the motion thus excited, the components having the same frequency as the field or any multiple thereof are practically or entirely absent. Records (not reproduced) have been obtained of the motion at various selected points of the wire for these and other cases. Theory of the Haperiments. The attractive force of the electromagnet in the experi- ments described is exercised over-a very small region of the wire which may practically be treated as a mathematical point. The essential feature of the case which enters into the explanation of the phenomena noticed above is that this attractive force is not a simple function of the time, but depends also on the position, at the particular epoch, of the point on the wire with reference to the pole of the electro- magnet. In other words, the expression for the maintaining force is not independent of the form of the maintained motion. For our present purpose, we may write it as the product of two functions, one of which involves only the time and the other is determined by the position of the wire in the field. ‘Thus, Force = F(y,)f@) ae 2ar nt = Ee an cos( ee én), n=0 1 where T/r is the periodic time of the field and y, is the displacement of the wire at the point a (opposite the pole of the electromagnet) from its position of equilibrium. y being positive when measured towards the pole, F'(yo) increases with yo, and may be taken to be unity when y=0. We may expand (yo), by Taylor’s theorem and write it in the form (1+ dy9+cyo’+ &e.). If the force varies inversely as some power of the distance between the pole and the wire * it may readily be shown that the constants ?, c, &., are all * From the measurements made by Klinkert over a limited range, it would appear that the attractive force on the wire varies inversely as the square root of the distance. Maintenance of Vibration by Periodic Field of Foree. 133 positive. The complete expression for the force, which may be assumed to act at the point 2» of the wire, is thus N= D ,9 1 ‘ (1+ bio + cy?+&e.) } a, cos ( oo ae en) n= We may now consider, first, the ordinary forced vibration. This may ne obtained by the method of successive approxi- mations. ‘l'o begin with, we may neglect the quantities by, eyo’, &e., in the expression for the maintaining force, which then assumes the simple form a, cos (27r nt/T—e,). Since the forced vibration is of negligible amplitude, except when the period of the field is more or less nearly equal to one of the free periods of the wire, the harmonic components in the motion may be determined, term by term, from the corresponding components of the impressed force. The forced vibration may therefore be written as Ba? SWOT imme 2nart ‘ > dnkn sin sin COS a en Cn | = a a a where « is the length of the string or of each vibrating segment, and kn, e,’ are quantities which, in respect of each harmonic, may be expressed in terms of the natural and impressed frequencies of vibration and of the decrement of the free vibrations. If x is equal to « or any multiple of it, the forced vibration becomes negligibly “small, the periodic force having an inappreciable effect when applied at a node. An aoe example in which the tormula given above may be applied is that of a single impulse acting at the point &y, once in each period of a iiasions / Elie cceficiont An is then the same for all the harmonics, and e,=0 for all values of n. It may readily be shown that if the period of forced vibration in this case is somewhat greater or somewhat less than the period of free vibration of the string, the form of the maintained vibration is practically the same as that of a string plucked at the point a. For the phase-con- stants e,’ are then practically all equal to zero and 7 re- spectively. Further, ky is then practically independent of the dissipation of energy (whatever this may be due to), and is inversely proportional to the difference between the square: of the natural and impressed frequencies. For ditferent harmonics, kn is proportional to 1/n?, and the 134 Prof. C. V. Raman and Mr. Ashutosh Dey on the expression for the forced vibration is then of the form Mier Tike | Zine sin cos—— > i and is thus similar to that of a string plucked at a) in the same direction as the periodic impulses or in the opposite direction, according as the natural frequency 1 is greater or less than the frequency of the impulses*. If the periodic force, instead of being impulsive in character, has a finite constant value during a part 28 of the period and zero at + sin other times, the maintained vibration in the two extreme cases assumes the form a Cionmeenme’) -.. nme 2nmrt in »S1n -, SIN —_— > cos : ee i a it i If 8 be small, this is practically of the same type as the expression for a plucked string in respect of the first few harmonics, but would differ appreciably from it in respect of the harmonics of higher order. The next step is to introduce a correction in the expression for the impressed force on account of the neglected terms byo, cyo’, &c. On substituting the value of y first found in these terms and simplifying the product F(y)) /(é), it is seen that the correction results only in alterations of the amplitudes and phases of the harmonic components of the impressed force, but no new terms are introduced of which the frequency is not the same as that of the field or a multiple thereof. This shows that the corrections cannot, by themselves, result in an alteration of the frequency of the forced vibration, so long as we assume, in the first instance, that yo has the same frequency as the field of force. They may, however, result in the impressed force (and therefore also the maintained motion) including such partial components as are absent in the field itself. A consequence of the preceding formule, which is of particular importance, is that, when the impressed force is of an impulsive character, the corrections by, cy,”, &c., when introduced cannot result in any alterations in the relative amplitudes and phases of the components of the maintained vibration. For the product F(yo)/(t) is zero at all times, except at the particular instant in each period at which the impulse acts, and, as these epochs are fixed, any change in * It assumed, of course, that the free periods of the wire form an harmonic series. This may be subject to modification if the wire is imperfectly flexible or yields at the cde Maintenance of Vibration by Periodic Field oj Force. 135 F (yp) can only result in the amplitudes of all the components of the impressed force being increased or decreased in the same proportion, their phases remaining unaltered. The non-uniformity of the field may thus affect the amplitude of the vibration, but cannot alter its form, it being assumed, of course, that the amplitudes are not so large as to alter the free periods. This peculiarity of the action of a non- uniform impulsive field is the explanation of certain interesting observations described, but not explained, by Klinkert in the paper referred to above. Klinkert experi- mented with two wires, both electromagnetically maintained, one of which was self-acting and the other was worked by a current on separate circuit rendered intermittent by the vibrations of the first wire. The vibration-curves of the two wires showed a marked dissimilarity, a special feature of interest being the fact that the vibration of the second wire when at its maximum was practically similar to that of a plucked string. In view of what has been said above, this result will be readily understood. The magnetic field is of ep on ble strength only during a small fraction of the period,'and may thus be regarded as of an impulsive character. When thereis an appreciable difference between the natural and impressed frequencies of vibration, the form of the motion approximates to that of a plucked string and this is what is actually observed when the exciting current is rendered intermittent by an independent interrupter. It is when the natural frequency i is somewhat greater than the impressed frequency that the vibrations of largest amplitude and those that show the closest similarity to the vibrations of a plucked string are obtained. For the vibrations are then nearly in the same phase as the impulses, and as an increase in tbe amplitude brings the position of the wire at which the impulses act closer to the electromagnet, and therefore still further increases the magnitude of the impulses, a vibration of large amplitude may be maintained in spite of the difference between the natural and impressed frequencies of vibration. The increase of natural frequency, due to a large amplitude, would also tend to encourage the assumption of this form of vibration and to make it stable. The conditions are, however, entirely different when the vibrating wire is a self-acting interrepter which determines the period and character of its own exeitation, and a detailed mathematical theory of the vibration-forms obtained with it must be reserved for separate consideration. We may now pass on to consider the cases in which the frequency of the vibration is not the same as the 136 Prof. C. V. Raman and Mr. Ashutosh Dey on the frequency of the field, but is a submultiple of it. To fix our ideas, we may assume the free vibration of the wire when it divides up into r segments to have nearly the same period as the field, that is T/r. The period of vibration of the wire, as a whole, is therefore T. EXxperiment shows that the forced vibration having the period T/r may be unstable, giving place to a vibration with period T. To explain this result, we may examine the effect, according to our equations, of superposing a small vibration of period T upon the ordinary forced vibration, if any, of period T/r. If byo, eyo’, &e., be neglected, there is no component in the impressed force having the period T, and the initial disturbance assumed would die away in the ordinary course. It is not possible, therefore, to obtain the resonance of submultiple frequency with uniform fields of force. With non-uniform fields, the additional terms by, cy”, &c., have to be taken into account, and it may readily be shown, on expanding the product F(y) f(t) in a series of sines, that there would be a term of period T in the expansion which would, under certain circumstances, be capable of magnifying the assumed dis- turbance continually till it assumes a large amplitude. For example, we may take r=2, and the initial disturbance to be, say, erage y sin yr The product Ant . byo Ay eos( “Tr —= a) 9 would contain a term b ie cost el 1) Gig SG COS: Saaqnel nen Pea aa Oi which, on being expanded, is seen to include a component Zari T ® This is proportional to the assumed disturbance, has the same period, and has a phase in advance of it by 90° It would therefore tend to magnify the assumed disturbance of period T till the latter reaches a considerable amplitude. An explanation of the phenomenon noticed is thus possible for the case r=2, in which no part is played by the com- ponent of yg having the same frequency as the field. For the cases in which r=3 or 4, &&., we have to proceed toa higher degree of approximation by taking into account, not 4 bayy sin e; cos se — | eee a Maintenance of Vibration by Periodic Field of Force. 137 only the assumed disturbance of frequency 1/T’, but also other subsidiary components whose frequencies are multiples of 1/T and play a part in the magnification or maintenance of the vibration of that frequency. If, in the distribution of the field F (yo), only the first correction term by is taken account of, the analysis proceeds practically on the same lines as that contained in the Phil. Mag. paper of Jan. 1915, except that, instead of the equations of motion for one degree of freedom, the general formule for the normal coordinates i in the forced vibrations of the wire would have to be used. The same general result would be obtained, that the components in the motion having the same be. quency as the field or any multiple of it, eae not play any part in the maintenance of the sietior of the kind now considered. We have already seen how this indication of theory may be verified experimentally. When, however, the correction terms of higher order, that is, cy,”, &c., are considered, some modification of this general statement might Uncen: necessary. Summary and Conclusion. The present paper considers experimentally, and theoreti- cally, a case of vibrations maintained by a non-uniform periodic field of force which is of some practical importance. It is shown that when a wire divides up into two or more segments and vibrates under the transverse attraction of an electromagnet, the motion which has the same frequency as the field may be rendered unstable by the non-uniformity of the field and then passes over into one, the frequency of which is a submultiple of the frequency of the field. Photographic records illustrating the first six cases of the kind are presented with the paper. It is also shown that a motion of this type may be set up and maintained even when the attracted point on the wire isa node and the ordinary forced vibration is therefore absent. The effects of the non-uniformity and of the periodic varia- tion of the field on the ordinary forced vibration are also considered in detail and the mathematical theory of certain effects noticed by Klinkert is set out. The experiments and observations described in the present paper were made in the Laboratory of the Indian Association for the Cultivation otf Science. Caleutta, February 2nd, 1917. Baas. | XII. On the Failure of Poisson’s Equation for certain Volume Distributions. By GanesH Prasap, M.A., D.Sc., Sir Rashbehary Ghose Professor of Applied Mathematics in the University of Calcutta * area? object of this paper is (1) to point out some typical volume distributions for which Poisson’s equation is invalid, and (2) to prove that Professor Petrini’s gene- ralization f of Poisson’s equation does not hold for every one of these distributions. It is believed that the limited scope of the validity of Professor Petrini’s generalization has not been pointed out by any previous writer. 1. Let po denote the density of the solid at any point P (a, y, 2) inside it. Then it should be noted that Poisson’s equation fails at P when V/V is either meaningless or has a value different from —4zrpo, V being the potential due to a small sphere of radius a and centre P. Some typical cases in which Poisson’ s equation fails. 2. Case I.t Let the density of the sphere at any point Q (&, 7, &) inside it be cos? @ log— where @ is the angle made by PQ with the axis of <, and » denotes the distance between P and Q. Then, remembering that the external and internal potentials due to a spherical shell of radius ¢ and surface density P,,(cos @) are Arter en drt P, (7 \’ — and ——— | 2n+1\r 2n+ VX E7 respectively, we have the potential at Q given by ri ee ee Ome Po (yy dt : (hake | cp Eee Ae i t lo ee Tt = s i . dt a \ 3 ee te mt he! < log * Communicated by the Author. + H. Petrini, “Les dérivées premiéres et secondes du _potentiel,” Acta Mathematica, b, Kx oe eS (11908), { «This case has been studied by Prof. Petrini (doc. cit. p. 186), but the treatment given by me is different from his. Poisson's Equation for certain Volume Distributions. 139 Hence Oe o) _ "(¢ dort? et dt ees Val 2 --\ ie an) |. b\ edo 1 og — Rilo or t t l He l6mr Ps) °8 r Anr Ge ae > hil 32h i log - log — a 7 S77 P, } ? br iL 1 f: log a where k, and ky are proper fractions dependent on 7. Therefore 1 10V(E7,6) _16rPs) Le __ Ak; _ 8rbeP2s (4, r fore aie eve 1 iy eles 1 los = low = 25 loos aa QF Op From the above equation it follows at once that, since the first differential coefticients of V are all zero at P, o7V VioOVGahey,.2), - Bom Yin a, ang IS OMG hss, cm Enh ee ing Eas Oy" amo ft Oy ; 27 SLA NGe eth) on == [pint 3 Omer +!) Of h=v h Oz Thus V?V has no meaning and, consequently, Poisson’s equation fails at P. percase LI.* . Let p=cos (log -). Then, by Newton’s theorem relating to the attraction of a spherical shell, at P =+2 OM y ar and at Q rr t2 cos (log L dt OV Be Ber. VRE ee bs 2 Uy Anerae ale) or e * For Cases II. and III. see my paper “ On the Second Derivates of the Newtonian Potential due to a volume distribution having a dis- continuity of the second kind” (Bulletin of the Calcutta Mathematical Society, vol. vi., 1916), in which are studied, for the first time, volume distributions for which lim p is non-existent. r=0 140 Prof. G. Prasad on the Failure of Now, putting t=e~*, we have r 1 90. ( t? cos (leg = dt ={<= cos vdu 1 log = a0 a ! cos (log 2 BL ai 5) i. Therefore, from (1), ~ ONE) ae: s(log = + tan ae (B) ov OY om Hence it follows at once that Sa one) oes are all non- existent. Thus V?V has no meaning and, consequently, Poisson’s equation fails at P. 4. Case. U1." Let pcos. Then, proceeding as in Case II., we find that anh ta cos dl Ib oVv( e Up i a’ 0 , eS) ia iL But, putting zee have { t? cos Gis — ae 0 t wi v NU ° ° ° ° d 1 ° which is numerically less than 27*, since as always positive and constantly diminishes as v increases. Therefore, from (2), Zl ou < 8r. ae Or.) 4 27 2 Hence it follows at once that Ss i eat c are all zero. Thus VV is zero and, consequently, Poisson’s equation fails at P unless we assign me value 0 to po. It should be noted that, in this case, pp may be assigned any value without affecting the value of VV. Poisson's Equation for certain Volume Distributions. 141 Petrini’s generalization of Poisson’s equation. 5. Professor Petrini has formulated the following gene- ralization of Poisson’s equation :— ‘‘La fonction AV existe toujours, méme si les dérivées Bee). OV p} 3 , , ° , e ae oy’ et ea n’existent pas séparément, si on définit le symbole /\ de la maniére suivante : ae OV(eth, y, 2) _ Ove, De. 2)) ey = him » ;(°* Tana AV h,=0 zg, YZ ho=0 hg=v Beh ie ot. lim “0 et déterminée.” hy Tt is easily seen from (A) of Art. 2 that the Bee cralaaeian holds for the Case I. For, let hy as hg ie hg CL Ve: (an where a, 8 are always different from zero as well as from infinity. Then Sar oe i Beelik ee = = mene) ous Na iE AV pan 5 log LG log a log — log — a ah o L " low ale | iS } at | 5G = 7 log 1 in the limit. Thus 142 Mr. H. A. Biedermann on the Energy Failure of Petrini’s generalization. It is easily seen from (B) that Professor Petrini’s generalization fails for the Case II. For : Acar il J oe mapa 41 = OM = Ae [ Vi0 : cos (log Sh, 55 lees oH tan 3) + c0s(log : + log eat tan~* ,) eos (1g! ie faim 3)t which exists only for special values of a, @. In fact, the necessary conditions for the existence of AY are the following :—— \ 1+ cos (log 1 + cos (log 2) = 0 in the limit, sin (Jog ~)+ sin (log = = Qin the dima When these conditions are satisfied, AV=0. Thus AV exists (and is zero) only when, in the limit, hs and u. are of the forms O he ¢ _ 29 ona e eeeier Tac A a 3) respectively, m and n being any integers. XIII. On the Energy in the Electromagnetic Field. To the Editors of the Philosophical Magazine. ES March 11th, 1917, Lt your March number Mr. G. H. Livens strongly criticises my reasons for putting forward the modified expression {H? + (div A)?}87 for the magnetic energy density in an electromagnetic field, aud says that my argument is convincing only when div A=0—that is to say, that it is entirely unconvincing. Mr. Livens obtains for the magnetic energy the ae a d= 32 er! mo “Ea wD) ia aye = = *(1,U, + Orv, + w,w,) — ae (div A)?dv, e 6 = wn the Electromagnetic Field. 143 where, for purposes of comparison, I have altered his notation to that of my paper. This expression, so far as it goes, is undoubtedly equivalent to my expression (6), though the last term in the form given by Mr. Livens is not ex- pressed explicitly in terms of the charges and their mutual distances, &c. Mr. Livens then states that in the special ease I considered—namely, that of two closed linear circuits — div A must be assumed to be zero at all points of the field, and therefore be s(t A)'dv=0. The whole point of my argument was that when this integral was evaluated it did not vanish—at least, that is what my aroument amounted to. To simplify matters for the moment, let us neglect terms i NO P. a\? containing (<) . Expression (6) of my paper may then be written l Ss ey" $ 9 2 Ss Cyl - 4 te (UO ee — (i, A 0 00,0.) P ; " ay Ep 2 12 Hel saa i ee » iss &re. ASE lutbe tee) HSE Slut ey} a‘ Comparing this with Mr. Livens’s expression above it is clear that 2 1 ‘ = > Cy 5 & jo CE ae (div A)?du=4>-—— (er eee tw?) + SSD 2 Clk 7 (UUs + UUs + W,Ws), as may easily be verified by direct integration. If Mr. Livens is correct in saying that div A must vanish at all points in the case of closed linear circuits, then the expression on the right should vanish either when applied to the two complete circuits or to each complete circuit separately. By subtraction it follows that 22 2 (uy ue+ v1 ¥,) should ey eas vanish, this representing the ‘‘mutual” part of = (div A)?dv for the two circuits. The above condition may be written v .. | (cose .» (((cos a; cos a, Fite 5 dsyds,— diya ds, ds.=0. a 7 If this were true, then expressions (7) and (8) of my paper 144 Mr. E. A. Biedermann on the finergy would be equivalent. It was precisely the apparent dis- crepancy between these two expressions which led me to put forward the modification of the usual expression for the magnetic energy density. If Mr. Livens can show that the above relation—a purely geometrical one—actually 2s true, so that (7) and (8) in reality are equivalent, there would be an end of the matter, and it would only remain for me to apologize for having raised the issue at all; but it certainly does not seem sufficient merely to say that we must assume it to be so,—for that, in fact, is what the assumption that div A=0 at all points really amounts to. I further suggested some reasons as to why expressions (7) and_(8) were apparently not equivalent—in other words, why div A does not vanish at all points,—and showed that this only occurred when the moving charge constituting the current was distributed continuously throughout the substance of the conductor instead of being, as it actually is, concentrated in discrete particles. Mr. Livens, 1 think, is under some misapprehension as to the generality I have attributed to the expression for T—a misunderstanding for which I am myself to blame. On p- 152 of my paperI said “It is suggested, therefore, that this (7. e. (H? +G*)/87) may be a correct representation in all cases.” I should have said ‘in all cases for which the original expressions for H and G hold good, namely all cases of slow uniform motion of the charges.” Mr. Livens’s further criticism still applies, however, and amounts to this —that because i 1 [ Cg Ve Creat 4 — | H’dv= — i > 4 223 a) Or 2c* da, Pa J for the special case of two closed linear circuits, I have assumed the expression on the right to be the correct general expression for the magnetic energy, which would, of course, be a most unwarrantable conclusion. Actually my argument il ; was, in effect, that because sa | Hae for the special case apparently does not equal the above expression, but is equal to oy) Pte ile sta ; 1 (som + 2332" (v'y,) } 7 | (div A)*du, 4 T 2G a, in the Electromagnetic Field. 145 where the integral apparently does not vanish, therefore the general expression for T should be ee 1 2 sal 8 dv+ s\¢ dv for the limited condition of slow uniform motion. For this condition div A=G, and if 2 1 Cape €,€5' x pals + @)de= | p Soe 22,2 —— (sve) | On p. 154 I expressly pointed out that expression (13) could not be regarded as of-complete generality, since it was derived from others for H and G, which took no account of the finite velocity of propagation, and, I might have added, expressions which were strictly true only for slow uniform motions. Neither did I intend to imply that SS was to be regarded as a complete expression of the potential energy. It is clear that the complete specification of the kinetic and potential energy explicitly in terms of the charges, their distances, velocities, and accelerations, would of necessity have to include the whole previous history of their motions. All that I showed in the remainder of the paper was that the general equations of the field were obtained from the limited special form of the kinetic potential of the system by substituting [p| for p, [eu] for pu, &e., and on p. 156 I pointed out that this could not be regarded as an independent derivation of these equations,—in fact that, as Mr. Livens says, it was not a method susceptible of strict mathematical specification. Finally, from a comparison of the general form of the expressions for H with the particular form assumed by these in the case of slow uniform motion, together with: the form assigned to the expression for @ under the same conditions, I concluded that the most general expression for @ was div A; so that the most general expression for the magnetic energy density was 1 2 ea \ is 2 1 2 3, (H +G )= g—i(curl A) + (div A?) }. This expression would, of course, only reduce to (13) under the limitations mentioned. The crux of the whole matter, then, lies in the equivalence Phil. Mag. 8. 6. Vol. 34. No. 200. Aug. 1917. L 146 Dr. 8. Chapman on the Partial Separation by or non-equivalence of expressions (7) and (8). If these are really not equivalent, then the disagreement constitutes an outstanding discrepancy, one solution of which would be the addition of the term = (div A)? to the magnetic energy density. am: Yours very truly, EK. A. BIEDERMANN. XIV. On the Partial Separation by Phermal Diffusion of Gases of Equal Molecular Weight. By 8. CHAPMAN, iM Acs DD So% N a recent memoirt I have shown mathematically that a temperature gradient in a mixture of two gases is in general sufficient to produce diffusion, independently of any non-uniformity of composition or of the action of external forces. ‘T'his phenomenon had apparently escaped the notice of experimental workers on gases, as well as of previous theoretical writers, but in a recent joint note{ with Dr. F. W. Dootson, an account of some experiments has been given which affords satisfactory qualitative confir- mation of the theory§. Further experiments are now in pro- gress with the aim of obtaining a close numerical comparison of observational data with the chief features of the somewhat complex theory of the phenomenon. According to the theory, the amount of the effect is greatest when the gases are mixed in nearly equal proportions by volume, and also is greater the more unequal are the masses and diameters of the gas molecules. It depends, moreover, on the nature of the molecules. It seems to be greatest for rigid elastic spherical molecules, while it vanishes altogether for Maxwellian mole- cules, 2. ¢. for point centres of forces varying inversely as the fifth power of the distance. Italso vanishes when the masses and diameters (or laws of inter-action) of the two sets of molecules are alike. * Communicated by the Author. + Phil. Trans. A. 1916 (unpublished); an abstract is given in Proc. Roy. Soe. A. xci. p. 1 (1916). { Phil. Mag. xxxiii. p. 249 (1917). § Since writing this Note I have received an Inaugural Dissertation (Upsala, 1917) in which Dr. D. Enskog, by a different method of analysis, has arrived at the same theoretical results as are contained in my recent memoirs (Phil. Trans. A. cexvi. p. 279, 1915, and A. cexvii. p. 1, 1916), including those relating to thermal diffusion. Thermal Diffusion of Gases of Equal Molecular Weight. 147 The object of the present note is to point out that “ thermal diffusion ” offers a physical means of partially separating two gases of equal molecular weight, provided that their diameters or laws of inter-action are not identical. This property may be of much or little practical importance, but it seems worth while to call attention to it on account of its theoretical interest. If we have two vessels containing numbers of identical molecules, and open a channel of communication between them, the random molecular motions will soon bring about a mixture of the two sets of molecules such as is beyond our powers to disentangle. If, however, the two sets of molecules are unlike, in such wise that we can apply unequal external forces to the members of the two sets, a partial separation by means of “forced diffusion ’’ can be effected. This case is realizedin practice when a vertical jar containing a mixture of two gases of unequal molecular weight is left to itself under the action of gravity, which acts unequally on the molecules of the two kinds. If one set of particles in a gas-mixture be charged, an electrical force will similarly effect a partial separation. In my memoir already cited I have shown that a pressure gradient, in whatever way pro- duced, also results in diffusion of the two gases, so that if a force similar to that of gravitation (but acting equally on all the molecules) could be used to establish such a gradient, the molecules might be partially sorted out by “pressure diffusion.” This effect, however, is directly pro- portional to the difference between the molecular masses, so that actually gravitational separation hardly affords an in- stance of this phenomenon as distinct from that of forced diffusion. A better example is to be found in a horizontal layer of gas contained in a vessel which is rapidly revolving about a vertical axis: the heavier molecules will then, of course, be proportionately more numerous at greater than at smaller distances from the axis. When the two sets of molecules are of equal molecular weight, however, and when they are electrically neutral, there has hitherto seemed to be no means of gaining any control over their relative proportions in a mixture, after this has once become uniform. The equality of mass requires an equality of the mean random motion of the molecules, and pressure diffusion, which depends upon an inequality of this random motion, is unable to act. The discovery of thermal diffusion, however, opens a mode of possible action which is more subtle than any of those yet mentioned, in that it depends essentially (when the molecular masses are equal) on the nature of the inter-actions between molecules during 148 Dr. 8. Chapman on the Partial Separation by their mutual encounters. If both sets of molecules obey the Maxwellian law, whether or not their force constants are the same, or if they obey any other law during encounter, provided their force constants or diameters are identical, thermal diffusion is not produced. But if their laws of inter-action (not being Maxwellian), or diameters, are dif- ferent, a temperature gradient will cause interdiffusion. When the molecular masses are unequal, and the diameters equal, the heavier gas diffuses towards the cooler regions. When the masses are equal, but the diameters unequal, the larger molecules diffuse in that direction. In order to gain some idea of the numerical magnitude of the separating action of thermal diffusion, we shall consider a mixture of vy; molecules of radius a, with v, molecules of radius o, per unit volume, the masses of each being m. Both will be supposed monatomic, in order that the mathematical theory may apply strictly, though the non-fulfilment of the condition is not likely to affect our estimate seriously. We shall consider only the first approximation to the coefficient of thermal diffusion D,, as given in the memoir and abstract already mentioned. Second and third approximations to D,; increase its value only very slightly. If Dj, is the ordinary coeflicient of diffusion, the relation between the approxi- mations to D; and Dj, is as follows :— Ds=k:Dy, where, in the notation there explained, L= €,(1—é) MA Ry oe Az) (01 — oy)” mato? VY \ 167 Ayo, Lt BY 15 { a a 7 the 7. oe = ties) 4 “5 (01+ 22) in the present case of ss molecular masses. In the last equation A, and A, are defined as follows :— M=Nn/(M+M), Ae=V2/(%+ V2); so that Ay tA,=1, | Ar Az | des bl Clearly &; is always positive if o;>o,, whatever be the values of A, and Ay. Since the equation of thermal diffusion is (J. ¢.) 0 log T ug = — D: - (where uy is the velocity of diffusion of the molecules (1), and T is the absolute temperature), and since D; is positive Thermal Diffusion of Gases of Equal Molecular Weight. 149 if a; >o2, it follows that, as above stated, the larger molecules diffuse towards the regions of lower temperature. As a numerical example, let us consider a mixture of ethylene (C,H,) and nitrogen (N,): the atomic weights of hydrogen, carbon, and nitrogen being approximately 1, 12, and 14, the molecular weights of these two gases are approximately equal, viz. 28. The molecular radii, deduced from the coefficients of viscosity on the hypothesis that the mojecules are rigid elastic spheres, are* 2°39 . 10-8 em. (C2H,) and 1°84.10-§ cm. (Ne). On substituting these values into the above expression for k;, we find that in this case (the gas 1 being taken as ethylene) __ 4-64 4 0°302(Ay — Av) ky eymeeet30 2 93-2 V9 Vy} This has its maximum value when vy, and v2 are nearly equal. The following are three typical values of k; for various relative proportions of the two gases :— V1 2V5 anes beef Dik i a 0:0094 0°0122 00078 In order to understand the physical significance of these numbers, we may consider the steady state of the mixture when placed, in originally equal proportions (say) by volume, in a tube in which a permanent temperature gradient is maintained from Ty at one end to Ty’ at the other. The equation of steady state (J. ¢.) is (Om OAe 0 log-T 62) rea oa We may assume &; to be nearly constant, so that the change in A, or A, between the two ends (due to the balance of influence between thermal diffusion, which tends to separate the two sets of molecules, and ordinary diffusion, which tends to restore uniformity of composition) is equal to kylog. T,/To. If, for instance, T,’= eT, (say Tp= 200° abs., and Ty’=544° abs.), the percentage difference of compo- sition between the two ends would be 100;, or 1°22 per cent. in the case suggested. The separating action is clearly not at all powerful, although by successive operations of thermal diffusion a very appreciable difference of relative proportions could be produced in the mixture. It should be added, however, that the effect would in fact probably be only one half or one third of this caleulated amount, on account * Cf. Phil. Trans, A. cexi. p. 476 (1912). 150 Partial Separation by Thermal Diffusion of Gases. of the difference between the behaviour of real molecules at collision, and of the elastic spherical molecules which have formed the basis of our calculation. For corresponding gradients of log p and log T, pressure diffusion is usually more powerful than thermal diffusion, but the former vanishes altogether when the molecular masses are equal. The equation of state for pressure diffusion is OAD ENON, Oe ellos Ow On. where pe AA2(m, — My) A1M1 + AgMs It is interesting to compare k, and k; in the above case, where we have supposed m, = mg, a relation which is, however, not quite exactly fulfilled. From the Smithsonian Physical Tables (1910 ed.) the following exact values of the relative ia weights of hydrogen (1), carbon, and nitrogen are taken :— Carbon 11:99, Nitrogen 13°90. Hence the molecular weight of C,H, is 27:98, and of N,O is 27°80, so that for the proportions ), and A», already con- sidered, the following are the calculated values of ky:— V12V» IP) eral Oo: ie em Or OOM CAE 0:00161 0:00121 p In this case, therefore, £, is only about one eighth as great as k; If our atmosphere were composed of roughly equal proportions of C,H4 and No, the settling out under the in- fluence of gravity would only amount to 1 per cent. in about 100 kilometres, supposing there were no convection. Thermal diffusion clearly gives much more control over the mixture, in a case like this, than does pressure or forced diffusion (under gravity). There are many pairs of gases of very nearly equal mole- cular weight, but their diameters are usually less different than in the above case, when the numbers of atoms in the molecule are nearly alike. For instance, the diatomic molecules CO and No, of equal molecular weight, have diameters 1°89 .107® and 1°88 .10~* cm. respectively. Simi- larly, the triatomic molecules CO3 and N2O, of equal mole- cular weights 44, have diameters 2:27.10 and 2°30.107% em. respectively. The large difference of diameter in the case of C,H, and N$'seems to be due to the large difference of atomicity between them. Except in such cases the general similarity between molecular diameters, Just noted, seriously Geological Society. 151 limits the application of thermal diffusion as a separating agent. I venture to publish this note, however, in the hope that, apart from its theoretical interest, it may, perhaps, at some time enable a partial separation of two gases to be produced in difficult cases where this means, though a weak one, may be the best available. Whether it has any application to the radium products of nearly equal molecular weight, which Prof. Soddy has stated to be difficult to separate, I am personally unable to say. ) Greenwich, March 1917. XV. Notices respecting New Books. X-Rays and Crystal Structure. By W. H. Braee, M.A., D.Se., F.R.S., and L. W. Braee, B.A. Pp. viii+229. London: G. Bell & Sons, Ltd. 1915. ASX admirable presentation of the theory and of the experimental side of this recently discovered domain of diffraction phe- nomena is given in an easily readable and very attractive manner. The greater part of the volume is dedicated to the investigations of the authors, whose merit in this rich field of research cannot be overestimated. The volume can be warmly recommended to the physicist as well as to the crystallographer and the chemist. XVI. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. (Continued from yol. xxxiil. p. 535. ] February 16th, 1917.—Dr. Alfred Harker, F.R.S., President, in the Chair. TEXHE PrestpEnv proceeded to read his Anniversary Address, in- cluding first obituary notices of Jules Gosselet (elected Foreign Member in 1885), J. W. Judd (el. Fellow 1865), J. H. Collins (el. 1869), C. T. Clough (el. 1875), Clement Reid (el. 1875), Bedford McNeill (el. 1888), H. Rosales (el. 1877), W. E. Koch (el. 1869), C. Dawson (el. 1885), T. de Courcy Meade (el. 1891), and others. The remainder of the Address dealt with some aspects of igneous action in Britain, and especially its relation to crustal stress and displacement. ‘This relation appears not only in the distribution of igneous activity in time and space, in the succession of episodes, the habits of intrusions, etc., but also in the petro- graphical facies of the igneous rocks themselves. The cause of such relation was sought in the existence of extensive inter-crustal regions in a partially molten state: that is, with some interstitial fluid magma, which must normally be rich in alkaline silicates. There will be a continual displacement of the interstitial magma from places of greater stress to places of less stress, and certain 132 Geological Society. broad differences in chemical composition are therefore to be expected between the igneous rocks of orogenic belts and those erupted in connexion with gentle subsidence. The Archzan plutonic rocks were intruded in close relation with powerful lateral thrust, and they accordingly include no alkaline types; but the Dalradian sediments were deposited in an area of tranquil subsidence, and the lavas intercalated in them are of the spilitic kind, rich in sodic felspars. The Lower Paleozoic formations were laid down in a geo- syncline, which for a long time experienced merely a slow depression, and the late Cambrian and early Ordovician eruptions, situated chiefly along the borders of the area, had a pronounced sodic facies. In Mid-Ordovician times there entered a certain element of lateral thrust, and accordingly in the Llandeilian vuleanicity the spilitic type gave place to the andesitic ; but the scattered outbreaks of Bala and Silurian age often afford evidence of a reversion to the earlier facies. Following upon the great Caledonian crust-movements there was, in the Scottish Highlands and elsewhere, a copious intrusion of plutonic magmas, all of ‘calcic’ as contrasted with alkaline types. The same characteristic belongs to the igneous rocks of the Lower Old Red Sandstone, which were extruded and intruded in connexion with the later Caledonian folding, while the country was still in a condition of stress. With the dying-out of this stress a more alkaline facies supervened, and the Lower Carboniferous igneous rocks of Scotland, though developed largely in the same synclinal folds as the preceding series, present a strong contrast in petrographical characters. They indicate a certain richness in soda, and this feature becomes more pronounced, until it culminates in the Permian of Ayrshire and Hast Fife in highly- allkaline rock-types. In Southern England, remote from the main Caledonian dis- turbance, the Devonian and Carboniferous lavas are of the same spilitic type as those of the early Ordovician. Later, this part of the British area was involved in the Hercynian crust-movements, which were accompanied by the intrusion of the Cornish granites and their satellites. In Mesozoic times our country experienced no orogenic dis- turbance of a pronounced type, and there was a prolonged cessation of igneous activity. The Tertiary Era introduced a new factor in the form of very extensive plateau-faulting, bearing no relation to the structure of the country. This movement, generally of the nature of subsidence, affected a vast area, of which Northern Britain is only a small fraction, and was attended by igneous action on the same extensive scale. The mechanism of extrusion and intrusion differed in important features from that illustrated by the Paleozoic eruptions. The Tertiary igneous rocks, as a whole, are decidedly, though not strikingly, rich in soda; but this alkaline character is lost in the neighbourhood of isolated centres, where there is evidence of locally-developed stresses of an acute type. ths 25 po ee Raman & Dev. Phil. Mag. Ser. 6. Vol. 34, PI. I. ner E, ; 4 e ~ = ws w - - a ¥ ad sd w . : es iw we J oa sd ial z ; = Ag a ee 3 a ~ ~ ~ we a cd dl Ilustrating vibratory motion in a periodic field of force; the upper curve in each case representing the vibration set up by the field, and the lower showing the frequency of the field for comparison. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. Fernie a | & - | | oe. [SIXTH SERIES.] ( \ ZoEP 9 x NX >, ©7075 SHEP THMB ER’ 1917. S4re, ji STENT OFEyCe Tube. By Sir E. Ruruerrorp, F.R.S., Professor of Physics, University of Manchester*. HE present paper contains an account of some expe- riments made to determine the maximum penetrating power of the X rays excited by high voltages in a Coolidge tube, using lead as the absorbing material. Owing to the lack of time at my disposal, the experiments, made a year | ago, are incomplete; but they may prove of interest in indi- | cating the penetrating power of the X radiation that can be | Ya } | | XVII. Penetrating Power of the X Radiation from a Coolidge { } | obtained from this source under practicable conditions, and in throwing light indirectly on the probable frequency of the very penetrating gamma radiation from radioactive bodies. In these experiments, the absorption of the X radiation by lead has been examined over a very much wider range of intensity and of thickness of absorber than in the original experiments of Rutherford, Barnes, and Richardson f. To excite the radiation, a large induction-coil of 20-inch spark was used, actuated by a mercury motor-break in an atmo- sphere of coal-gas. The heating current through the tungsten spiral was adjusted to give a radiation of maximum intensity at the voltage required, which was fixed by an alternative spark-gap between points. The radiation was found to be most constant when a fairly rapid stream of sparks passed between the points during the measurements. The well- insulated Coolidge tube was placed inside a large lead box, and the X rays, issuing through a rectangular opening in * Communicated by the Author. + R.. B., and R., Phil. Mag. xxx. p. 339 (1915). Phil. Mag. 8. 6. Vol. 34. No. 201. Sept. 1917. M a)” i adi eat 154 the box, passed into the measuring vessel which was placed close to the opening. The ionization current was measured by means of lead electroseopes of the self-contained type used for gamma rays. Three of these electroscopes, of cubical form, respectively 11 cm., 10 cm., and 12 em. side, were employed in the course of the experiments. For determining the initial absorption, the lead face of the electro- Sir EH. Rutherford on the Penetrating Power of scope was cut away, and replaced by thin aluminium-foil. For greater thicknesses of absorber, a lead electroscope with sides 3 mm. thick was used; while for still greater thick- nesses, a lead electroscope 8 mm. thick was used in some experiments. In order to avoid disturbances due to stray radiations, the windows of the electroscopes were of thick plate-glass, and still further protected by lead extensions. Such precautions are essential when, as in the present expe- riments, the intensity of the end radiation under measurement was in some cases less than one millionth of its initial value. In order to make experiments over such a wide range, the heating current through the tungsten spiral was adjusted to give a convenient rate of leak in the electroscope in each experiment. | The voltage corresponding to the alternative spark-gap was determined by comparison with the sparking potential between two large brass spheres 20 cm. in diameter.. The absorbing lead plates, which were of much greater area than the face of the electroscope, were placed close to the electroscope. In such a case, the greater part of the radiation scattered in the absorber in a forward direction enters the electroscope. The average absorption coefficients w for different thicknesses of the absorber were determined inthe usual way. The results for different voltages are given in the following table:— Range of ae ‘ M Range of AbseROE Mass Max. | thickness onpaon ee Max. | thickness pea eas ey Voltage.| in lead, Coy Abs. Coef. Voltage.| in lead, Coscia ae Coef. Ra TEN 00 /0. Ta pe. Cm. p/p. 79,000 | 1:°8—2°5 27 2°37 | 183,000 | O'7— 1:3 26 2°28 2531 26 2:28 1:3— 2:0 24 P11 92,000 | 1:-8—2°’5 25 2°19 2°4— 4:0 20°5 1:80 2°5—3°4 24 2a ial 4-0— 4:6 18 1:58 3°1—3°7 24 att 46— 53 15 1°32 105,000 | 2‘7—3:3 23 2-02 53— 6:4 13 1:14 | oo 39 22 1:93 6:4— 7:0 12 1:05 3:9—4:6 22 1:93 196,000 | 4:3— 5:5 138 1:14 | 118,000 | 2°7—3°3 22, 1:93 55 — 64 12 1:05 Sioa 22 1:93 64— 78 lis! ‘96 144,000 | 2:°7—3°3 22 1:93 7:8— 9-2 10 88 3°4—4'6 22 1:93 | 8:8—10:0 8:5 yo 170,000 | 3:1—3°7 18 1:58 | | 37-45 Li 1:49 | 4:3—5'5 | 18) 1°32 the X Radiation from a Coolidge Tube. 155 It will be seen from the above table that the thickness of lead through which the radiation was measurable increased with the voltage applied. This is a result not only of the increase of the penetrating power of the radiation, but also of the large Increase with voltage of the intensity of the radiation. With a voltage of 196,000, the radiation was detected and. measured through 10 mm. of lead. In this case, the intensity of the radiation after passing through this thickness of lead was considerably less than one millionth of its initial value. No doubt. by the use of still more powerful rays and more sen- sitive methods of measurement, the radiation could be detected through a still greater thickness. The maximum voltage applied (196,000 volts) was about the limit of capacity of the induction-coil under the working conditions. In addition, I should adjudge this voltage to be about the limit of safety for the bulb itself, so that no attempt was made to examine the penetrating power of the radiation for still higher voltages. Certain interesting points arise in considering the results given in the table:— (1) There is not much change in the value of w for the end radiations between 79,000 and 144,000 volts, and no observable change in uw between 105,000 and 144,000 volts. (2) Between 105,000 and 144,000 volts the radiation is absorbed nearly exponentially with a value of »=22. Above 144,000 volts the absorption is no longer exponential, but the value of w decreases progressively with increase of thickness of absorber. This is best shown by the results for 183,000 volts, in which the value of mw decreases from 26 to 12 as the thickness of absorber is increased from °7 to 70 mm. These results, which are at first sight peculiar and un- expected, can be very readily explained by taking into account the absorption of rays of different frequency by lead. In a recent paper™, Hull and Miss Rice have carefully examined the absorption coefficient of lead for X rays of different wave-lengths, obtained by reflexion from,a rock- salt crystal. Tor wave-lengths greater than 0-149 ALU., the absorption in lead obeys the law u/p=430A3 + 0°12, w HEHE is the wave-length in Angstrom units and 0:12 is the assumed mass- scattering coefficient, o/p. The value of p/p suddenly increases for “values of X below 0-149 A.U. owing to the presence of a characteristic absorption-band in lead. The presence of this sharp absorption-band has been shown also photographically by Hull and Miss Rice and by * Hull and Miss Rice, Phys. Rev, viii. p. 326 (1916). M 2 156 Sir E. Rutherford on the Penetrating Power of De Broglie. By plotting the logarithm of X (fig. 1) against the logarithm p/p for lead, Hull has shown that the curve is nearly astraight line AB. At B, where o=0°149 A.U., the absorption suddenly increases, shown by the nearly horizontal line BC. Assuming that the law of absorption after passing Bisel, 12 14 8 T6 £8 0:0..+=—-2 4 6 ‘8 Log Ai for lead through the absorption-band is similar to that observed before, the line CDE should represent the new portion of the curve. The circles represent values actually found by Hull and Miss Rice. Taking the quantum relation, \=0:149 A.U. corresponds to 83,000 volts, and the minimum corresponding value of u/p for lead found by Hull was 1°30, 2. e. w=17'5. From the dotted portion of the curve the radiation emitted between A=0°149 A.U. and 7~=0:098 ASU 1. e. between 53,000 and 125,000 volts, should be more absorbed than that emitted for voltages slightly less than 83,000. We should thus expect the value of mw for the end radiation to be sensibly constant for the above range of voltages. Actually we find p nearly constant between 92,000 and 144,000 volts. This difference is not important, and is to be anticipated from the nature of the measurements. A radiation more penetrating the X Radiation from a Coolidge Tube. Loy than w= 22 must be present in some quantity before its pre- sence can be detected by absorption methods. The minimum value found by Hull, w=17°5, is somewhat less than the value, .= 22, found in these experiments, but the difference is no doubt to be ascribed to the difficulties of accurate measurement of w in both cases. From the dotted portion of the curve, the minimum value of pw for lead at 196,000 volts (W=:063 A.U.) should be about 5. The observed value is 8°5. Taking into account that the minimum value of w for 196,000 volts must be somewhat less in any case than 8°5, and that the actual curve of absorption is probably somewhat steeper than the dotted portion of the curve, there is not a marked divergence between the observed and the calculated results. Taking these factors into consideration, the absorption measurements are not in themselves inconsistent with the view that the maximum frequency of the radiation from a Qoolidge tube is given by the quantum relation, H=hv, over the range of voltage examined. Hull and others have already shown by crystal methods that this relation certainly holds up to 100,000 volts and probably up to 150,000 volts. The peculiarities of the absorption by lead of X rays of different frequencies affords a simple explanation of the results obtained by Rutherford, Barnes, and Richardson *. In their experiments the absorption of the end rays by aluminium was found unchanged between 142,000 and 175,000 volts after the rays had passed through 2°49 mm. of lead as absorber. A reference to the table shows that under these conditions the issuing radiation consisted mainly of the characteristic radiation of lead with a value of w= 22, and no observable change in the absorption by aluminium is to be expected under the experimental conditions. Absorption by Aluminium. A few isolated and approximate measurements were made of the absorption of the rays by aluminium under different conditions. In order to avoid complications due to the characteristic radiations of heavy elements like lead, the greater part of the radiation was first absorbed by its passage through an element of low atomic weight like iron. Under such conditions, the absorption results should not be seriously influenced for frequencies much higher than that of the K radiation of iron. The following results were * Loe. cit. 158 Sir E. Rutherford on the Penetrating Power of obtained for the absorption by aluminium of the end radiation after passing aa iron :— Volts. p/p. 92,000 38 14 144,000 *30 “11 183,000 23, \ Ades The corresponding values of w were found to be higher if lead were used as initial absorber instead of iron. The absorption was measured by placing the aluminium plates close to the electroscope between the latter and the iron plate. Under such conditions the greater part of the forward scattered radiation enters the electroscope, and consequently the absorption coefficient as measured is inter- mediate between » and w+o (where yp is the true absorption coefficient and o the scattering coefficient), and probably closer to the former. The value of » as given by Hull and Miss Rice corresponds to ~ +o in the above notation. In a recent paper, 8. J. Allen and Alexander” have examined the absorption of X rays from a Coolidge tube when different metals are used as filters for the rays. With a tin filter, they found that the absorption coefficient in aluminium for the issuing rays was lower than for any other metal. The value, «/p=0:12, for aluminium was observed with a steady voltage of about 120,000 volts; with an iron filter w/p=0'134 under the same conditions. These numbers are in good agreement with those found by the writer. Application to the wave-lengths of gamma rays. The observations on the absorption of X rays in aluminium and lead throw important light on the difficult question of the probable wave-lengths of the penetrating gamma rays from radioactive substances. or convenience, the approxi- mate results so far obtained are collected in the following table. The minimum wave-length is deduced from the voltage or vice versaon the assumption that the quantum relation, H=hy, holds. The rows with an asterisk give values of u/p obtained by Hull and Miss Rice (loc. cit.). In their case, the values of lp include the effect of scattering as well as absorption, and are consequently not strictly comparable with the values found by the author for aluminium, in which the correction for scattering is less important. ‘The values of w/p for the * Allen and Alexander, Phys. Rev. ix. p, 198 (1917). the X Radiation from a Coolidge Tube. 159 penetrating gamma rays from radium C are those given in a recent paper by Ishino *, where the coefficients of absorption and scattering were separately determined. The values of the mass-scattering coefficients, o/p, tor the gamma rays were found by him to be ‘045 for aluminium and ‘034 for lead— values much smaller than those previously found for ordinary X rays. | Woliaces Wave-length in | Mass Absorption Coefficient p/p (volts). Aw, | rN. in Aluminium. in Lead. *84,000 0-147 07154 1°50 92,000 0-135 Ue: aaa Th I | *102,000 0) LOD) hee sec 3°00 144,000 O86 | O11 ae | 183,000 ‘068 | ‘085 1:05 196,000 | Beak)! eee Cre | 0-75 Gamma rays | } 9 ; from radium C j e jee | oe | Tt will be observed from the table that the value of p/p in aluminium decreases very slowly between 84,000 and 196,000 volts, even at a slower rate than the first power of the wave-length; while for longer waves it is well known that the value of w varies approximately as the cube of the wave-length. As we should expect, the variation in p/p with wave-length is much more rapid for lead than for aluminium over the samerange. It will be noted that, while the value of u/p for aluminium for X rays generated at 183,000 volts is only 3 times the value for the gamma rays, the corresponding ratio in the case of lead is more than 20. The general results suggest that when the value of p/p becomes of the same order of magnitude as that of o/p, the former coefficient varies slowly with the wave-length, the latter probably remaining constant. In addition, it appears not unlikely that there is a definite connexion between absorption and scattering, and that, for very short waves, the absorption like the scattering may ultimately reach a minimum value independent of wave-length. From some points of view such a connexion between these two quantities is not improbable, but unfortunately no waves of sufficiently short wave-length are available to test the relation expe- rimentally. The two shortest wave-lengths of the gamma rays observed * [shino, Phil, Mag. xxxiii. p. 129, January 1917. 160 Sir H. Rutherford on the Penetrating Power of in the experiments of Rutherford and Andrade* were ‘072 and ‘099 A.U., corresponding on the quantum relation to waves excited by 174,000 and 125,000 volts. The values of w/p for aluminium corresponding to X rays excited at these voltages are about ‘09 and ‘12 respectively, while the observed value of u/p for the penetrating gamma rays from radium ( is much less, viz. ‘026. Since undoubtedly for such high frequencies, «/p varies very slowly with frequency, it is clear that the wave-length of the more penetrating radiation is considerably smaller than that of the shortest waves observed by Rutherford and Andrade. In other words, the wave-length of the main gamma rays is much shorter than was previously supposed. This conclusion is still more strongly confirmed by the observations on the absorption of the radiation by lead. For a voltage of 196,000 volts, corresponding to a still shorter wave-length than the shortest observed by Rutherford and Andrade, the observed value of w/p in lead was 0°75, while the value of p/p found by Ishino for the penetrating gamma rays was ‘(042— a ratio of nearly 20 times. Even allowing that the true value of y/p for waves generated at 196,000 volts is some- what smaller than the value observed, the largeness of the ratio shows that the gamma rays must be much shorter than those generated at 200,000 volts, 7. e. much shorter than N— 06205): In our present ignorance of the law of variation of p/p with frequency in this region of the spectrum, it is only possible to estimate the actual wave-length of the most penetrating gamma rays. It is clear, however, that the waves are at least three times and may be ten times shorter than those which correspond to 200,000 volts, 2. e. they correspond to waves generated by voltages between 600,000 and 2,000,000 volts, and thus lie between :02 and -007 A.U. It is thus clear that the gamma rays from radium © consist mainly of waves of about ;4, the wave-length of the soft gamma rays from radium B, and are of considerably shorter wave-length than any so far observed in an X-ray tube, with the highest voltages at our disposal. Another very interesting and important point arises from this discussion. It is well known that the @ rays from radium B and radium C when examined in a magnetic field give a veritable spectrum of bright lines corresponding to definite groups of @ rays, each group consisting of electrons expelled with a characteristic and definite velocity. The * Rutherford and Andrade, Phil. Mag. xxviii. p. 263 (1914). the X Radiation from a Coolidge Tube. 161 energy of motion of each of these groups of electrons have been measured by Rutherford and Robinson *, and the more intense groups (labelled with letters in the original paper) are given in the following table:— 8 rays from Radium B. | Energy Group. | Intensity. = 1015, \Voltage(volts).|| Group. | Intensity. PsaQise. Voltage(volts). pN Ss. 3°332 333,200 A m.f, | 21:02 2,102,000 B v.58. 2-610 261,000 B mite, \o.dgok 1,751,000 C V.S. 2°039 203,900 C m. 06 71 1,671,000 D v.S. 17519 151,900 D m. | 14:09 1,409,000 E s. "503 50,300 EH m.s. | 13°28 1,328,000 is v.s. 376 37,600 F m. | 11°49 1,149,000 & m.s. | 103i | 1,031,000 H m.s. | 5:94. 594,000 K s. | 516 516,000 1, m. 2°96 296,000 | M m. 2:59 | 259,000 | N m. 181 181,000 | I | | 8 rays from Radium C. | Energy | / The column headed “ voltage ” gives the potential difference in volts between which the electron must move to acquire the observed energy. Apart from the low-velocity groups L, M, N, the 6 rays from radium C consist mainly of groups lying between 500,000 and 2,000,000 volts. This is about the same range of voltage as we estimated to excite the penetrating gamma rays from consideration of the absorption of X rays and gamma rays by aluminium and lead. It would thus appear probable that the ebserved groups of 8 rays are due to the conversion of the energy, H=Ay, of a wave of frequency v into electronic form, and that consequently the energy of the 8-ray groups may be utilized by the quantum relation to determine the wave-lengths of the penetrating gamma rays. Such a conclusion is borne out by consideration of the groups of rays from radium B. H. Richardsont has determined the absorption of these rays by lead, and concluded that they could be analysed approximately into three component groups for which the absorption coefficients, w, in lead were 45, 6, and 1:5 em.~" respectively. From the observations with a Coolidge tube, the value, ~=6, should correspond to waves excited at about * Rutherford and Robinson, Phil. Mag. xxvi. p. 717 (1918). + Richardson, Proc. Roy. Soe. xci. p. 396. 162 Penetrating Power of X Radiation from Coolidge Tube. 200,000 volts, and it is to be noted in the table that three strong groups, B, C, and D, of @ rays from radium B corre- spond to voltages between 261,000 and 152,000 volts, an average of about 200,000 volts. The value of w=1°5 may correspond to group A ora still swifter group, of voltage about 500,000 volts, observed in the spectrum of 8 rays excited in lead by the gamma rays from radium B and radium C together *- ay The results as a whole snggest that the groups of @ rays are due to the transformation of the gamma rays in single and not multiple quanta, according to the relation H=hv The multiple relations observed between the energy of some of the groups of 6 rayst must on this view indicate approximate multiple relations between the frequencies of the gamma rays. With the assistance of Mr. J. West, B.Sc., I have made some experiments to see whether it is possible to detect by the crystal method the presence of waves shorter than those observed in the experiments of Rutherford and Andrade (loc. cit.). A narrow pencil of gamma rays and strong sources were employed, but no certain evidence of the existence of such waves was obtained. This may be due either to the overlapping of the numerous lines that should be present, or to the failure of the crystal to resolve waves whose length is very small compared with the grating space. If the single quantum relation should prove to hold generally for the conversion of y rays into 6 rays, the mag-. netic spectrum of ® rays shovld afford a reliable method of extending the investigation of X-ray spectra into the region of very short waves where the crystal method either breaks down or is practically ineffective, and thus places in our hands a new and powerful method of analysing waves of the highest obtainable frequency. The complexity of the @-ray spectrum for radium B and radium C indicates that the spectrum of the gamma rays, and presumably the very high-frequency spectra of heavy elements in general, are as complicated as the ordinary light spectra of such elements. | University of Manchester. May 12, 1917. * Rutherford, Robinson, and Rawlinson, Phil. Mag. xxviii. p. 285 (1914). + Rutherford, Phil. Mag. xxviii. p. 305 (1914) des), | XVIII. Astronomical Consequences of the Electrical Theory of Matter. Note on Sir Oliver Lodge’s Suggestions. By Prof. A. 8. Eppineton, .A., F.R.S., Plumian Pro- fessor of Astronomy in the University of Cambridge ie 1 the Philosophical Magazine for August, p. 81, Sir Oliver Lodge offers an explanation of the celebrated discordance of the motion of the perihelion of Mercury. His explanation is comparatively simple, and on that account will be widely preferred to the recent theory of Hinstein, which introduces very revolutionary conceptions, provided that it meets certain other astronomical requirements which seem necessary. In removing the discordance for Mercury, it must not introduce discordances for Venus and the Harth, which at present satisfy gravitational theory. Ifthe explanation breaks down under this further test, the discussion will make prominent a feature of the success of Hinstein’s s theory which has perhaps not been sufficiently emphasized. It will be recalled that Lodge makes the hypothesis that the extra electrical inertia due to the motion of matter is not subject to gravitation. Hence, for a planet moving in a circular orbit, gravitation remains constant, but the inertia alternately increases and decreases according as the orbital velocity compounds positively or negatively with the uniform motion of the solar system through space. The latter motion is entirely unknown, and any value within reasonable limits may be considered plausible. This theory is shown by Lodge to lead to changes in the perihelia and eccentricities of the orbits. He suggests that “through a comparison of the outstanding discrepancies between theory and observation for different planets, if they were definite enough, it might be possible to get some indi- cation of the direction as well as the magnitude of the Sun’s true motion through the ether of space.” It will be shown that the astronomical data claim to be quite definite enough to follow up the theory in the way he proposes. I give below the present discordances between gravitational theory and obser- vation for the four inner planets+. Here a@ is the longitude * Communicated by Sir Oliver Lodge. + W. de Sitter, ‘ The Observatory,’ vol. xxxvi. p. 297 (1913), with some small corrections taken from ‘ Monthly Notices,’ vol. Ixxvi. p. 728 (1916). I have transformed the mean errors there given into probable errors. It may seem strange to those unfamiliar with astronomical practice to find _& pure number de expressed in seconds of arc; the angle is to be identified with its circular measure. 164 Prof. A. &. Eddington on Astronomical of perihelion, e the eccentricity, and d represents change in a century. eda. de. Mercury teen + 8:24 + 0-29 — 0°88 SE 0:33 Venus......... —0°06+0°17 +0:21+40°21 arth yee +0:07+0-°09 +0:02+0°07 Mars’: eee + 0°64+0°23 +0°294 0°18 It may be desirable to explain why we use ed@ instead of _ simply the centennial motion of perihelion dw. Ina circular orbit w is indeterminate, and when the eccentricity is small the direction of the apse line is difficult to determine with accuracy. Multiplying da by e, we obtain a quantity which can be observed with the same degree of accuracy whatever may be the eccentricity. In fact, eda measures a distortion of the orbit of the same nature but at right angles to that indicated by de; and it will be seen from the table that for each planet de and eda have been found with nearly the same probable error. They are actually rectangular com- ponents of a vector (like dr and rd@ in polar coordinates). After the perihelion of Mercury, the next largest dis- cordance (in comparison with the probable error) is in the perihelion of Mars; but this can scarcely be considered a genuine discordance, since the theory of errors predicts a residual of about this size among eight residuals. However, Lodge’s theory gives complete agreement for the perihelia of Mercury and Mars. We have to examine whether it can accomplish this without spoiling the agreement of theory and observation for the other six elements. It is clear that the initial chances of success are less favourable to Lodge’s than to Einstein’s theory. The latter makes no theoretical change in the eccentricities, so that the agreement of these with observation is automatically pre- served ; but Lodge’s theory predicts changes usually of the game order as in eda, and it could only be by a particular arrangement of the orbits that a discrepancy could be avoided. Secondly, his theory (besides less important differences *) has eda where Einstein has dw. Now, for Venus and the Earth, whose orbits are nearly circular ( unlike Mercury and Mars), Hinstein is saved, because his rather large corrections de have to be multiplied by the small factor e; on Lodge’s theory we have no such opportunity of escape. In fact, * A further difference is that Einstein has v’ instead of » in his co- efficient, so that the effect for the outer planets diminishes more rapidly, Consequences of the Electrical Theory of Matter. 165 Hinstein’s formule point at once to the perihelia of Mercury and Mars as the only elements likely to be observably affected, whereas on the present theory a more minute discussion is required. Let V be the velocity of translation of the solar system resolved in the plane of the planet’s orbit. Resolve V into components V cos w along the major axis and V sin @ along the minor axis, the longitude of perihelion @ being measured from the direction of V as zero- point. The component V sin a is the one which produces a correction to eda; Vcos a produces a precisely similar correction to de. This can be seen from Lodge’s formula (3), p. 86, viz.: ( Ww wv COs r u= 7 {1— ae +ecos (@—a) — Oa ésin 8}. The small constant term (w?+ v”)/2c” does not here concern us. Omitting this, and remembering that corresponds to 90° + a, Sealeecs). to V in our notation, this gives — f{ i tesin (@—=)— 34 6sino |. Write sin @=sin (9Q—w) cosa +cos (9—z) sin a, Then w= 5 {14 (¢ -— EV cos) sin (0— wa) — The equation of the undisturbed ellipse is* U= 7; s1+esin (@—a)}, and if small variations de and dw are given to e and a, F {1 +(e+de) sin (@—a) —eda cos (—a)}. (2) = p3 Coniparing (1) and (2), we evidently must have 1 do ia V cos a, oe a cee eS BY; (3) eda = +53 V sino. The second of these is, of course, equivalent to Lodge’s * The sine appears instead of the more usual cosine, because Lodge makes @ refer to the tangent instead of the radius. 166 Prof. A. 8. Eddington on Astronomical formula; the first was not given by him explicitly, but is almost implied by his remarks on p. 88. Since we are taking de and edw to be centennial changes, @ must be taken as the angular motion of the planet ina century. Hence the coefficient v6/2c? is proportional to the linear velocity x angular velocity of the planet—z. e. to the acceleration, or to the inverse square of the radius of the orbit. If ris the radius of the orbit, K a constant for all planets, we have de= — mV cos =, oa eda= + wa sin or. Let (ds)? =(de)’+ (eda); then ds measures the total dis- tortion of the orbit, which may be due partly to de and partly to eda. We see that (he | The angle between the perihelion and the direction of V does not affect ds, but determines in what proportions ds is to be resolved between de and eda". : For the moment we shall suppose the orbits of the four inner planets to be coplanar, so that V is the same for all, : 1] and hence ds varies as —.. The values of r? for the four r planets are in the ratios 1:3°5:6°7:15°5. In order to account for the motion of the perihelion of Mercury, we must assume a value of V such that ds for Mercury is about 8'’. (According to the table at the beginning of the paper the required value of ds is 8'"3; Lodge’s value of the solar motion on p. 91 would give ds somewhat larger.) The corresponding values for the other planets are then: ds= »/ }(de)?+(eda)? $s Mirernenneayps o-.5 See eee 3) AV ervey uk es ee Diet Berita eae i ba ol eee IIL) BV IGS Ole s, cislc sc cc) nrc ees 0°51 * The following geometrical construction is useful. Represent the eccentricity by a vector drawn in the direction of aphelion. Compound with it the vector ds=KV/r? drawn in the direction of V. The resultant represents the new eccentricity and direction of aphelion. Consequences of the Electrical Theory of Matter. 167 Thus any value of the solar motion which will account for the perihelion of Mercury must introduce corrections either to de or edw (or compounded from both) amounting to 23 in the case of Venus and 1'':2 for the Earth. A reference to the table on p. 164 shows that such corrections are quite inadmissible, being far outside the probable errors of observation. The introduction by the theory of these new large discordances leaves the position almost worse than it was originally. To make the proof complete, we must consider the incli- nations of the planes of the orbits. That of Mercury is inclined 7° to the ecliptic, of Venus 33°. Consequently, a very large component of solar motion perpendicular to the ecliptic would have an appreciable effect on Mercury, a much reduced effect on Venus, and none atall on the Earth. But the perihelion of Mercury lies 29° from its node on the ecliptic, and hence a component motion normal to the ecliptic would affect eda and de in the proportion cos 29° to sin 29°, or 1:0°55. Thus if we wish to obtain a correction of 8'' to eda in this way, we cannot help getting a correction of 44 to de—which is out of the question. A small cor- rection to de (0’’°88) is suggested by observation, but it is actually in the opposite direction. We have thus shown that a solar motion of the amount necessary to produce the motion of the perihelion of Mercury would make the elements of the Earth discordant if it were in the ecliptic plane, and would make the eccentricity of Mercury discordant if it were normal to the ecliptic. And, since the effects of component velocities are additive, it is easily seen that an intermediate direction has no better success. It is disappointing to find that this interesting suggestion, which gives a simple explanation of the most celebrated discordance of gravitational theory, is apparently unable to satisfy the more stringent test proposed. In the course of correspondence with Sir Oliver Lodge, some unexpected features have arisen with regard to the dynamics of the problem, and a further note on the subject will appear in October. [16844 XIX. Thermodynamic Cycles with Variable Specific Heat of Working Substance. By Wu. J. Watxer, 5.Sc., School of Lechnology, Manchester™. SumMMaARyY.—The paper deals with those thermodynamic cycles which are of particular interest in internal combustion engineering science. Approximate expressions for the efficiencies of the Con- stant Volume and Constant Pressure Cycles, assuming variable specific heat, have been obtained by other writers. In the present paper the expressions for these efficiencies have been cast in another and simpler form by first obtaining the general expression for the efficiency of the Dual Combustion Cycle and then deducing the efficiency expressions from that, for the Constant Volume and Constant Pressure Cycles as particular cases. In conclusion, an interesting relationship is pointed out between the efficiency ex- pressions for these cycles as given, and that for the Carnot or Constant Temperature Cycle. TEXHE following notes represent an attempt to obtain some simple expression for thermal efficiency, involving variable specific heat, which might be used as a standard of comparison in estimating the relative performances of internal combustion engines operating on different thermo- dynamic cycles. Although thus primarily intended for practical application, it is hoped that the results obtained are of sufficient interest from the physical and theoretical stand- points to warrant their being given here. Expressions for these efficiencies have already been ob- tained by Wimperist and Leest for the Constant Volume Combustion and Constant- Pressure Combustion Cycles respectively. The method of analysis adopted here is an extension of that applied by Lees. The general expression for the efficiency of the thermo- dynamic cycle illustrated in fig. 1 will first be derived. This type ot cycle is typified by that on which the Blackstone oil-engine is operated, aud is here called the Dual Com- bustion Cycle, some of the heat being imparted to the working fluid at constant volume and the remainder at constant * Communicated by the Author. + ‘The Internal Combustion Engine,’ p. 81, by H. E. Wimperis. } ‘Engineering,’ Jan. 1st, 1915, Thermodynamic Cycles and Variable Specific Heat. 169 pressure. The various operations in the cycle are as follow :— 1. Beginning at T the working fluid is compressed adia- batically from volume v to volume v, and temperature Tp. 2. Heat is imparted to the working fluid at constant volume from T, to T,, the pressure changing from p, to 7}. 3. Heat is imparted to the working fluid at constant pres- sure from T, to T,, the volume changing from 2 to U. 4, Adiabatic expansion of the working fluid takes place trour |, to Ts. 5. Heat is rejected at constant volume from T; to T, the pressure changing from p; top. This brings the working fluid back to its initial state. Fig. 1. + oweaece ane! a - Ve % VOLUME aU aa Assuming now that K,,=specific heat at constant vllume=B+ST, where B and S are constants and T=absolute temperature, and that K,= specific heat at constant pressure=A+ST, where A is a constant: Phil. Mag. 8. 6. Vol. 34. No. 201. Sept. 1917. N e 2 @ 170 =Mr. W. J. Walker on Thermodynamic Cycles The heat imparted to the working fluid in the Dual Com- bustion Cycle is given by Ty, T, He { K,dT + { K,aT Ty Ty aae (T:—1,)(B + eT, oP T; 58 (T,—T,) (A + =e +T, ) The heat rejected = H, Ts S =|Kar= (T;—T) (B To T; +T). The thermal efficiency 7,,=1— fo) the suffix ‘“‘m”’ denoting the Dual Combustion Cycle. : (.—1)(B+ $1 +7) Nn=1- “¢ Expressing all the temperatures in the cycle as functions of T, the suction temperature, the following relationships are obtained, denoting the ratio - by A, T= Try—-1 A(T —T,) where v T= fme9 Vo i A US 8 and e is the base of the hyperbolic logarithms. This becomes, to a first approximation, Ty=Try-'(1+4aT1—r7—), since is very small; therefore Tomley-1(14+AD),). 2. ey where D=T(1L—77~') ; Ter T+ AD), where ea Po T=part TL+AD)) 2 a ee (T, —T,) (B+ 5 T, +7.) + (T2—-T)(A+ 5+) (1) with Variable Specific Heat of Working Substance. 171 where he. Uy and T,=apYT(14+T1—77—) + apr?! — apy)... Sopu (impede) vee) se! (D) where E=T(1—ry—1 + apr¥—1—apy). Substituting (2), (3), (4), and (5) in (1), and neglecting powers of X higher than the first, the efficiency bcomes Vn = Bek (epY—1)AT ee) Im 2r?~fa—1+ay(p—1)} ap’ —1 ee et 6) — (ap" —1) + 2(r7-11)— Bea ett) where 7m'=efficiency of the Dual Combustion Cycle assuming Constant Specific Heat ae 1 f apy —1 mg i a—1-+rya(p—1) If p=1, as in the Constant Volume Combustion Cycle, the expression becomes AT see Ny=Ny — > (4+) (1- aie PON ahah (7) where foe 1 Ny =1 yt = Constant Volume Combustion Cycle effi- ciency with Constant Specific Heat. Jf «=1 as in the Constant Pressure Combustion Cycle, then ppt LAE alt) ae an ayr?—(p—1) (Ee ea —(@r—H +2071 FOF og here 7,/=efficiency of Constant Pressure Combustion Cycle a with Constant Specific Heat 1 pY—1 In (7) and (8) the coefficients of the term AT in each case N 2 VALUES w “CORRECTION FACTOR 172.) Mr. W. J. Walker on Thermodynamic Cycles are known as the variable specific heat “Correction Factors.” It is not possible to transform (8) to a simple form such as is obtained in (7); buton plotting the graphs shown in fig. 2 6 Fig. Constant. Vola me Gyele Correction i ) 2 3 VALUES of © or OA to represent the variation of the so-called “ correction factor ” with @ in (7) and with p in (8), it is seen that the two graphs practically coincide up to the value of « or p=4. The actual figures, assuming r=8, are:— aor. 10 le, iene 2. 3. 4. Correction factor in (7)...; °565 | 593 "850 Lis 1:41 i$ so.) | tS) eee eoeOOO 590 S80 ja AbD 1:39 From the nature of this practical coincidence between the two graphs, itis evident that the thermal efficiency of the two cycles may be written in the forms: namin «+p 2 ie m= {1-5 (e+) }, . MP from which the efficiency,in the general case, i."e. the Dual Combustion Cycle, may for all practical purposes be written m= {1—" (ate) J) 6)) gay aa with Variable Specific Heat of Working Substance. 173 It is to be observed that the Constant Volume, Constant Pressure. and Dual Combustion Cycles are capable of some modification with regard tothe manner in which rejection of heat is carried out, since this may take place at constant volume, constant pressure, or a combination of both. If, however, the factors 7,', 7,',and yn’ represent the efficiencies of the modified cycles of each class, under the constant specific heat assumption, then 7, np, and nm, as given by (9), (10), and (11), will be the corrected efficiencies complying with variable specific heat conditions. =athe simplicity of the expressions (9), (10), and (11), together with their close approximation to truth, would appear to make them of some value as standard expressions. Further interest attaches to the form of the expressions given, by comparing them with that for the well-known Carnot or Constant Temperature Cycle. This cycle is shown Fig. 3. Isethermal yo ; a lines ey a ee ee bdS5@ 0D 0° | MED RRS —— PRESSURE Ce oe 4 Us 7 VOLUME in fig. 3 from which the expression for the efficiency is readily obtained as follows :— — 1 pats rY—-1 AT — To), (d) . . . . where r= — =adiabatic compression ratio ; 2) ( tL e ° meee he = (1— 7} (1—XT) toa first approximation. (12) 174 Prof, A, Anderson on some Properties of the It is interesting to note that expressions (9), (10), and (11) reduce to (12) when « and p are each taken equal to unity. In the limiting case of the Dual Combustion Cycle, when a and p each equal unity, the repeated cycle becomes one of alternate compression and expansion along the same adiabatic line. Clearly this case may be taken also as equivalent to the Carnot or Constant Temperature Cycle between two infi- nitesimally close adiabatic lines. This gives further evidence of the approximate truth of the simple efficiency expressions obtained in the paper. XX. Some “Properties of the Nul Point of Thin Axial Pencils of Light directly refracted through a Symmetrical Optical System. By Prof. A. ANDERSON™. | a short paper in the Philosophical Magazine of January last I showed that a point, which I now venture to call the nul point, and which is easily found by experiment, could be used with advantage in determining the constants of a lens-combination. It may be defined as the point of inter- section of the axis of a symmetrical optical system with those axes perpendicular to it about which a small rotation of the system has no effect on the position of theimage. Rotations in the same direction about perpendicular axes on opposite sides of it produce displacements of the image in opposite directions. It was proved in the paper just referred to that, if O be the nul point and P,, P. the axial positions of the object and image, OP,/OP; is the magnification. In other words, all straight lines joining corresponding points of the object and image pass through O. It is, in fact, the centre of perspective of the object and image, and changes in position with a change in the position of the object. In fig. 1 the letters are those usually employed to denote * Communicated by the Author. Nul Point of Thin Axial Pencils of Light. 175 the various points of an optical system. Let F,Pi;=%, F,P,= Xo, H,F,=A, HF, =f: 2, H,P)= v1, H,P2=%, H,Hi=a, and let m be the magnification. We have BOR 2 @y es mr OP, F. a Hence m= OP, _ PoP, OF = OH, +/2—4 OP, HF, OPi+H.F, OHi+atAth Therefore (1—m)OH,=a—fo+m(fit+fot #1) =a—fo+m(fitfr)—f =a—(1—m)(fitfa), OH= p=, — its), and ma ga —(At/)- OH,=OH,-a= If m=0, O is at a distance f,+/, to the right of H,; and, if m=o, at a distance f,+/, to the right of H,. These positions of Oare the nodal points of the system. In the case where the index of refraction of the first medium is the same as that of the last, f,+/.=0; and, consequently, O coincides with H, when m=0, and with H, when m=oo. In the former case the incident light is a pencil of parallel rays, in the latter the emergent pencil is parallel. The above formule for OH, and OH, may be used in determining the value of a, or H,H;, by experiment. For suppose we measure the magnifications m and m' for two positions O and O' of the nul point, we have 1 1 pe 1—m’ 00'=a( m—m' Thus, by measuring OO’, we determine a, and, in the case where f,; +/,=0, the actual positions of H, and H, are also found. If f,+f2 is not zero, we may proceed to find the positions of H, and H, as follows. 176 Nul Point of Thin Axial Pencils of Light. Turn the optical system round O (fig. 2) through two right angles so that H, comes to H,’ and H,to H,’. O ceases to be the nul point; but if the system be displaced to the oa a distance a+20H,, H,',and H,’ will oceupy the Fig. 2. ; Hi H5 fe) Hoes H, former positions of H, and H,. O will be again the nul point with the same magnification as before. Thus, since a+20OH, can be measured, and since a has been already determined, OH, can be found, and consequently the positions of Hi, and Tele. We can now find easily the positions of F, and F,, and thus the focal lengths f; and f;. Through O (fig. 1) draw any line intersecting the object and image planes in Q, and Q,, and through Q, draw a line parallel to the axis meeting the first principal plane in K,. Take H,K, equal and parallel to H,K,. Then K,Q, intersects the axis in Fy. Similarly, by beginning with Q,and ending with Q,, F, may be found. Also, since F,N,=H.F,, and F,N,=H,F,, the nodal points N, art N, are determined. Or, f; and fp may be found from the formulze Die eco A= 1 —m f= l—m The nodal points N,; and N, may, however, be determined directly. We have ON,=OH,+ H,N,=O4, + HOF, + HF, =OH,+ ft/s _ ma ee and ON, =OMi+it+/2= I Thus ON,/ON,;=m. When the last medium has the same refractive index as the first, N; and N, coincide with Hy, and H,, and we have OH,/OH,=m. The method described above for determining H, and H, becomes very simple in this case. In fact, the distance d through which the system is moved is —m Lem H,H,, and, therefore, l1—m l—m ma, : eas are OH.= 7» = Tem On the Ionizing Potential of Sodium Vapour. 177 A simple relation exists in the general case between the distances OH, and OH,. Let = —— a ae ia A 2 Me My p and ps being the refractive indices of the first and last media. U Then OH;= — —(H2— pa); yi hes U or OH, Ska f ae (2 — #4). Al OH Be: a. = - — (Me — f1)3 therefore OH, OH, _ 1_1\_ spe of uf ie (Ha Ha) (- % ee ie ; OF, Olam or i oe, eg 2 If =e, OH,= af and OH,= a, v U XXII. On the Ionizing Potential of Sodium Vapour. By R. W. Woop and 8. OKano*. | experiments of Frank and Hertz, M*Lennan and others have shown that, when the vapours of mercury, cadmium, zinc, and magnesium in vacuo are bombarded by electrons from a hot cathode, a single line spectrum is emitted, provided the kinetic energy of the electrons does not exceed a certain critical value. In the case of mercury, Frank and Hertz showed that the single line X=2536°7 appeared when a potential difference of 4:9 volts was applied between the hot cathode and the anode wire, this being the voltage required by the quantum relation for the frequency of the line 2536-7. It is of course very important to ascertain whether the quantum relation holds in the case of all metallic vapours, as it has been found to do in the case of the four above enumerated, and we have accordingly carried out an extensive series of experiments with the vapour of sodium, which does * Communicated by the Authors. 178 Prof. R. W. Wood and Mr. S. Okano on not appear to have been investigated up to the present time. A resumé of our results was presented at the annual meeting of the American Philosophical Society in May. In the majority of our experiments we have used bulbs made of the new pyrex glass manufactured by the Corning Oo. This glass shows much less discoloration from the action of sodium vapour at high temperatures than any of the glasses which have been on the market heretofore, and on account of its very low expansion coefficient requires little or no annealing, being almost as satisfactory as fused silica in this respect. It can be obtained in the form of tubing of all Fig. 1. sizes, flasks of various forms, beakers, &c. ; consequently it is possible to make very elaborate apparatus with little difficulty. The apparatus with which the minimum ionizing potential of sodium was measured is shown in fig. 1. the Ionizing Potential of Sodium Vapour. 179 The cathode was a spiral of fine tungsten wire (5 mils. diameter) attached to two stout copper wires } and c, the ends of which were split and then squeezed together. The sodium was placed ind, and after exhaustion ‘of the bulb, distilled into A, after which it was sealed off. The anode a was of platinum. The electrodes were sealed in with sealing-wax, the lateral tubes through which the cathode wires pass having a bore only slightly larger than the diameter of the wire to ‘prevent the diffusion and condensation of sodium vapour on their walls. There is bound to be a loss of the vapour through the tube leading to the pump, but this cannot be helped; for if we seal off the tube from the pump the vacuum is rapidly impaired by the liberation of hydrogen from the sodium. Experiments on the resonance radiation of sodium vapour have shown that it is practically impossible to remove all of the hydrogen from the metal by repeated _ distillation in vacuo, for the metallic vapour carries down hydrogen with it, when it condenses on the wall. The tungsten wire was heated by the current from a small storage-battery, and the potential applied as shown in fig. 1, by a potentiometer, consisting of a wire of 10 ohms resistance stretched on a metre stick, and from one to three or more dry cells. In our first experiment we started out with an applied potential of 6 volts between the cathode spiral and the anode wire. The tungsten was raised to normal incan- descence, and the bulb heated by brushing its surface as uniformly and rapidly as possible with the flame of a Meker burner. A bright yellowish glow appeared around the anode, and the spectroscope showed, in addition to the D lines, the red ‘and green lines of the subordinate series. On diminishing the applied potential we found that the subordinate series faded gradually and disappeared entirely at 2°34.V. The Dlines, however, remained bright. On still further reducing the potential we found that the yellow glow at the anode wire disappeared at a potential of 0°5 V., though we could still see the D lines in the spectroscope at still lower potentials, or even with the connexion at A broken. This we subsequently found was due: to the fact that the potential drop along the tungsten filament was sufficient to cause a glow around the positive leading-in wire, and some of this light was reflected into the spectroscope from the wall of the tube. If the potential difference between the terminals of the filament exceeds about three volts, arcing takes place when the bulb is heated, without the application of any potential between a and ¢, the yellow glow filling the 180 Prof. R. W. Wood and Mr. 8. Okano on greater part of the bulb, and showing the subordinate series as well as the D lines. In the case of vapours for which the single line emitted lies in the ultra-violet, and evidence of its presence is obtained by the speetroscope, it is obviously necessary to make sure that the potential difference between the terminals of the hot cathode is not responsible for the appearance of the line. When the single line lies in the visible region there is less trouble, since, with diminishing voltage, the glow contracts to a thin skin covering the positive electrode, which may be either the auxiliary anode ‘‘a” or the positive terminal “ }” of the hot cathode, according to whether the applied potential, or the potential due to the drop along the cathode filament, is responsible for the emission. By using a very short filament we succeeded in reducing the potential difference between its terminals to about one volt; but even in this case we detected the D lines when the spectroscope was directed towards the terminal “0.” They were so faint, however, that there was not much chance of their being seen by reflexion from the walls of the bulb. The anode wire in this case was perfectly straight, and by viewing it ‘end on” the visibility of the faint luminous glow surrounding it was enormously enhanced. To still further increase the sensibility of the method, we formed an image of the end of the wire on the slit of the spectroscope by means of a lens (see fig. 1). This was accomplished without difficulty by throwing an image of the sun on the wire. On darkening the room we found that we could observe the D lines at the wire “a” until the applied potential was reduced to 0°5 volt, or perhapsa little less than this. The exact point at which the line disappears depends of course upon the condition of the eye. To remove entirely the possibility that the potential drop along the tungsten wire was contributory, we employed two methods. In the first or stroboscopic method, we employed a brass disk with wide teeth and small holes as indicated in fig. 2. This wheel interrupted the heating current, and by viewing the tube through the apertures we observed the condition at the auxiliary anode only at the moments when there was no potential difference between the ends of the tungsten filament. (The circuits were of course simpler than indicated in fig. 2.) This method gave good results when used with the steel tube (which will be described presently), the Ionizing Potential of Sodium Vapour. 181 as in this case the sodium glow could be viewed against an absolutely black background and the light of the filament eut off by means of a suitably placed screen. Fig. 2. The second method was very similar except that we arranged the circuits as shown in fig. 2, the rotating wheel applying the potential between the hot cathode and the anode wire only for the time intervals during which the heating current was shut off. The condenser C was inserted to lessen the spark at the break. In both cases the wheel was turned at a speed sufficient to maintain the filament at a constant intensity without any visible flicker. Our earlier observations were confirmed by both of these experiments. The D lines appeared when the potential between the hot cathode and the wire was 0°5V. or greater. As this value is so much below that required by the quantum relation (2°1 V.) it was necessary to make sure that the emission of the D lines was not due to some secondary action, the light from the incandescent filament for example. To test this point we mounted the tungsten filament at the centre of a steel tube, as shown in fig. 3. Immediately 182 Prof. R. W. Wood and Mr. 8S. Okano on below the filament and close to it was a very thin film of mica, which stopped the electrons but transmitted the light. The sodium was placed below the filament and vaporized by a smal]l bunsen flame. The cathode end of the tube was Fig. 3. GLASS PLATE wrapped with black cloth, and as the sloping walls of the glass tube reflected no light in the direction from which observations were made, the background was practically black. In this experiment the steel tube was made the anode, and with a potential difference of 0°5V. between it and the cathode, the yellow glow of the sodium vapour appeared above the mica plate but not below it. This indicated that the light from the cathode played no part in the production of the phenomenon. We also tried illuminating the tube shown in fig. 1 with a concentrated beam from the arc, but no difference in the brillianey of the D line or the potential at which it appeared could be detected. There remained apparently only the possibility that the effects might be due in part to a contact difference of potential. We made a number of experiments to test this point, using various materials (copper, platinum, tungsten, &c.) as anodes, and obtaining always the value 0'5V. The sodium vapour, however, usually condenses on the anode, making the experiment inconclusive. In one case, to prevent the condensation, we mounted a tungsten anode wire along the axis of the incandescent spiral cathode wire. This gave a value of 1°4V. for the minimum potential; but we feel certain that the higher value was due to the deflexion of the electrons away from the anode by the magnetic field of the spiral. | The most conclusive experiment was made with a cathode of the form indicated by fig. 4. Ihe portions C are of copper wire, while the spiral and loop (W and W’) are of 5 mil. . tungsten wire. | In this case we are not using an auxiliary electrode, and | the Ionizing Potential of Sodtum Vapour. 183 the potential difference results from the drop across the spiral. The glow appears around the tungsten loop W! and its copper supporting wires, which are at a temperature suffi- - ciently high to prevent any condensation of sodium. W!' remains below a red heat, while W is at incandescence. Fig. 4. 7 The potential difference can be varied somewhat by changing the heating current, and consequently the temperature and resistance of the spiral. We found in this case that the D-line glow appeared and disappeared at the same instant on the tungsten loop W!' and its two supporting wires of copper: this appears to indicate that contact difference of potential plays no part in the production of the D-line glow. The conclusion that we have reached, as the result of all of our experiments, is that the D-line emission results from the application of a potential of 0-5 volt or more, and that the subordinate series appears at 2:3 V. The two points are determined, however, by the visibility of the lines in a Schmidt & Haensch pocket spectroscope, which is a most efficient instrument, but the lines fade gradually in each case, and there seems to be no point at which there is a discontinuity. ‘This is perhaps to be expected, as a result of the wide variation in velocity of the electrons expelled from the hot cathode. The average velocity of these, in the case of an incan- descent tungsten wire, is the equivalent of a potential drop of about 0°4 volt. Assuming the Maxwell distribution, there must he a con- siderable number moving with a sufficient velocity to excite the D-line emission without the application of any electro- motive force. It-seems desirable to arrange an experiment with sodium vapour in which we can deal with a stream of electrons in which all are moving at very nearly the same velocity. This could be accomplished perhaps by magnetic separation. Johns Hopkins University. - May 1917. Le XXII. A Note on the Relation of the Audibility Factor of a Shunted Telephone to the Antenna Current as used in the Reception of Wireless Signals. By BATH. VAN DER Pot, ' Jun., Doct. Sc. (Utrecht) *. [* the measurements of the strength of wireless telegraph signals, it is usual on board ship to employ a shunted telephone receiver and to measure the signal strength by the value of the shunt required to reduce the signal stren eth to a point at which dots and dashes may be just differentiated. Such measurements have been made by Dr. L. W. Austin + and by Mr. J. L. Hogan, Jun.{, to find experimentally the law according to which the received antenna currents vary with the wave-length and distance from the sendin g station. The above results have been criticized by Prof. A. E. H. Love, F.R.S.§, who has raised doubts how far Hogan’s audibility factor (R+58)/S, where R is the resistance of the telephone and § that of the shunt, is proportional to the square or to the simple value of the antenna current I. Love has suggested that certain results of Austin and Hogan indicate that (R+8)/S is proportional to I? for large values and to I for small values, in which case the theoretical results of Love! and Macdonald {] would be in close agree- ment with the corrected experimental data of Hogan. | As long as the phenomena occurring at the contact of the two substances of a crystal detector are neither qualitatively nor quantitatively known, we cannot calculate the relation that exists between the antenna current and the value of the shunt S required to reduce the signal strength to just audibility. Direct measurements of the antenna current at great distance from the sending station cannot be made, for no ammeter exists with which it is possible to measure high frequency alternating currents of the order of 5 micro-amp. The only way, up to the present, to find the value of the receiving antenna current at great distances is to determine the shunt 8. Of course an experimental way must then be found to compare the values of the required shunts S with the actual antenna currents. * Communicated by Prof. J. A. Fleming, F.R.S. + ‘Bulletin of the Bureau of Standards’ (Washington), vol. vii. No. 3 (1911), p. 315, and ‘Journal of the Washington Academy of Sciences,’ Dec. 4, 1914. { ‘ Electrician,’ vol. lxxi. p. 720 (1918). } § Phil. Trans. Roy. Soc. Lond. vol. cexv. A, p. 105 (1915). ) Love, see paper cited. 4] H. M. Macdonald, Proc. Roy. Soe. (ser. A), vol. xc. (1914) p. 50. Audibility Factor of a Shunted Telephone. 185 From the two papers of Austin * it is not quite clear how this calibration was made, and Hogan f assumes that his “‘audibility factor ” _R+S A,=—a-, where R is the resistance of the telephone, is proportional to the square of the antenna current f. It was suggested to me by Dr. W. H. Eccles that this point could be experimentally examined in the laboratory. Dr. J. A. Fleming, F.R.S., thereupon kindly placed at my disposal the means for doing this in the research department of the Pender Electrical Laboratory of University College, London. The following is a short account of the experiments so far conducted. To generate high frequency oscillations of a steady character a double Fleming and Clinton commutator f, Fig. 1, mounted on the same shaft, was used. This commutator, though originally designed for measuring small capacities, but now connected in the circuits as shown in fig. 1, was * See papers cited. + See papers cited, and also Love, Phil. Trans. Roy. Soc. Lond. vol. ecxv. A, p. 128 (1915). t J. A. Fleming, ‘ Principles of Electric Wave Telegraphy and Telephony,’ 3rd ed., p. 205. Phil. Mag. 8. 6. Vol. 34. No. 201. Sept.1917. 186 Dr. B. van der Pohl on the Relation of Audibility found to produce very steady oscillations, of which the ampli- tude, wave-length, damping, and number of trains per second could be varied at will within considerable ranges. In fig. 1 H isa high tension battery of secondary cells, C, and C,' are two variable air-condensers of approximately equal capacity. The working of the two commutators is such that A is connected to D at the same time that B is connected to F. At this moment the condenser C,’ becomes charged to the voltage of H and any charge on Q, is discharged through the self-inductance L,, so that in the circuit C,L, high fre- quency oscillations are set up. At the moment when the commutator has revolved through an angle of 90°, C, becomes charged and C,' discharges through the same inductance Ly, for at this moment A is connected to H and BtoG. The capacity of C,' was varied, so that with its leads and L, it was in tune with C,, its leads, and L, The leads C,A, C,/B, HE, and L,G were made up of 20 strands, each 30 metres long. The commutator, which was driven by an electromotor at a speed of approximately 3500 R.P.M. (467 discharges per second), was not in the same room where IL, with the receiver circuit No. 2 was placed, to avoid the noise of the commutator interfering with the small sounds to be heard in the receiving telephone. The inductance L, con- sists of a flat vertical coil of one layer of 15 turns with an inside diameter of 8°5 cm. and an outside diameter of 21 cm. Coil L,, whose distance from L, can be varied by known amounts up to 100 cm., has the same dimensions as Ly, but consists of two layers. lL, forms, with the variable air-con- denser C., a high frequency circuit from which the signals, produced by tapping the key Ik, can be heard in the tele- phones T which, in series with a Perikon detector [ Chal- copyrite Cu FeS,—Zincite ZO], are connected across the terminals of GC), A headgear of two Sullivan telephones [resistance R=1240 © in series] was used. These tele- phones were shunted by a variable resistance S as shown hoy anions, IL The H.M.F. induced in Lz, is proportional to MI, where M is the mutual inductance between the parallel coils L, and L, and I the current amplitude in circuit No. 1, when only the couplings coefficient k= M/ / L,L, is small enough not to allow for any appreciable reaction from circuit No. 2 on circuit No. 1. Varying therefore the mutual inductance at a constant primary current has the same effect on circuit No. 2 as varying the primary current I at a constant coupling. By moving L, to and fro the distance between L, and L, Factor of a Shunted Telephone to the Antenna Current. 187 can be determined for which, with a certain shunt S, the sound of the signals, made by tapping the key K, can just be differentiated. The value of the mutual inductance M was then found by the use of a curve calculated from the formula of Maxwell-Rosa *. M = 277?nyngV aA 13?{1 + 3/82 +15/64k4.. . ., where 7, 7%. are the number of turns of L, and I,, a, A are the mean radii of the coils L, and La, and ps (a+A)? — {A (apApP+a+d}’ where d is the distance between the flat coils. In the experiments coupling coefficients were used only between the limits k ='003 and ‘0002. The wave-length used was about 1125 metres. The results of these measurements are recorded in the following table :— | | R+S R+S | | log 3° log M. log ae log M. 0816 1-419 Ot 1s. OBI "0962 1-446 @oi3 || | 902 | 1173 1-47] eogl =|} 1-980 poe el a02 | 1°520 Bios | 2060 "2095 1°583 ALT |, 2102 ‘2617 1-628 13359 | 2-164 3502 1-680 RAG | | 2197. 4065 | 1-704 Ma0ol | 2-25] 4427 175 16266 | 2313 | +4869 1-762 179938 | 2°394 fp wotl6 1-804 1-9227 2°473 6128 1°813 20969 2515 A curve (see fig. 2) was then plotted for which the ordinates indicate the logarithms of the mutual inductance for a constant current [which is proportional to the antenna current for a constant coupling], and where the abscissae give the logarithm of the audibility factor (R+8)/S. For values of log (R+8)/S from °6 to 2-2 (audibility * See Bulletin of the Bureau of Standards, vol. viii." 188 Audibility Factor of a Shunted Telephone. factors from 160 to 4) our experiments indicate the relation R+8 op S for between these limits the curve is a straight line with an inclination angle = tan7!°5. Fie. 2. 20 For values of log (R+8)/S from :08 to °6, i. ¢. for values of (R+8)/S from 4 to 1:2, the audibility factor is propor- tional to a power of I ranging in this interval from 2 to °7. Summary. The assumption that the audibility factor of a shunted tele- phone is in general proportional to the square of the antenna current, as stated by Hogan, seems to be invalid. For large values of the current we found a proportionality between the audibility factor and the square of the current, and for weaker signals the audibility factor was found to be pro- portional to a power of the current varying between 2 and °7, in agreement with Love’s expectations. . The author hopes to pursue these investigations further and to give an account of the results arrived at. In conclusion, I should like to express my thanks to Prof. J. A. Fleming for the facilities afforded to me for conducting the experiments here described in his laboratory. Pender Electrical Laboratory, University College, London. BRE 1 XXIII. The Numerical Calculation of the Roots of the Bessel Function J,(a) and its first derivate J,'(x). By JOHN R. Arrey. W7.A., D.Sc.* JORMULA for the higher roots of J,(x) and J,'(2) have been derived by McMahon from the asymptotic expansions of these functions. From Debye’s results, it has been shown t!-at p,, the pth root of J,,(z)=0 can be found by the method of successive approximations from PEC. ., ihe AG ea a CARN Rray OM HO n(tan o—¢) = 21. ee NB Cone le fae, 01 62) Ayr—15 Agr? + 945A57°—... + iat a bane Oe 2: 105A ee 1 W here T= ntan > > If £ is written for cot? d, then to five places of decimals, ig 1, A, =0°12500 + 0°20838 &, A,= 0023444 0°13368 £+ 0°11140 F’, Az,=0°00488 + 0:05941 £+ 0:12310 k? + 0°06839 £°,. A,=0:00107 + 0:02252 k+0-08371 2 +. 0°10673 28 + 0°04447 kh’, A;=0°00024+ 0:00780 £+0:04501 k?+ 0:09716 k? +0°08956 k* 4-0°02985 &°. The second term in Ag, is Bee oi? o, the incorrect value 576 agents given by Debye. being due to the omission of the factor 3 in the third term 3. (nt 1)(n+3)(n+5) cen 3! 28 | age in the expression for a,(n). * Communicated by the Author. + Math. Annalen, vol. lxvii. p. 545 (1909). 190 Dr. J.R. Airey on Numerical Caleulation of the McMahon’s formula can be derived from the above results as follows: te = as te| , equation (2) becomes tang=p+o=p+ 5 —8. ee) Also pany AG tan?@ tan oa eats and @= tand— 7 Hence ca 7 1 1 il nD To) tang 3tan'd dtan®d o Lagrange’s Theorem then gives nd 3 36° 158) 7 see ° (6) 2 ORE. OLS 33 Sntand 24ntan?d 512n?tan?d” ” Bein a 5 25 “= 8n tan ) 9 24n tan® d ~ « 384n3 tan?h (8) Therefore B=at (7) i . 5 25 g Sn?tangd ' 24n?tan?d 384n*tan?d @) and po (4p—D)a w _ Gprltin)e An 2 An : Substitute for tan d, its value in terms of 8 and we get eel 1 25 B44 gag * anti 1 s84neie and applying Lagrange’s Theorem, il 128n?—31 na 384nta® vae® ° as) te (11) (10) B=at Since pr=n( B35 — sag): ie aay Roots of Bessel Function J,(a) and its derivate J,/(#). 191 we find, on substituting the value of § in terms of «, 4An?—1 = 112n*—152n? +31 Sna 384n3a° Raeen (13) (4p—1+2n)r a Pp= na + or writing A» for na= and m for 4n?, ee m—1 4(m—1)(7m—31) Nh the same result as McMahon’s as far as the third term. Roots of J,(a). (a) When x is positive and larger than 4, the second and higher roots of J,(z) can be found with considerable accuracy from the formulee Pp =n sec @, where me (4p 1)m i I 1 5 n(tang—g) = EP= A)" 4 eee z) (15) d ae n(tan ¢4—gy) = GPP), wie 6) When po=1, n=0:00222 153 Pm il n= ipa From (16), tan ¢,;= oa +¢,, and as before 1 2 tan dee 3G? ? where | oa (4p —1+2n)7r = Substituting this value of tan ¢ in (15) we get n(tan o—9) eae “(x5 ee |- ; See nt “(sgt gq+ gg tagt-) [=e a 2 T nth — = ema where YHKt5> and i! 7 83 S00 Pi geaem Jae 2409 192 Dr. J. R. Airey on Numerical Calculation of the Hence a, 83 6949 ct oe Ane = 2 £097 ~ 13440," 7 — ++ )o(18) ae. il 5 T and om (4p—-1+2n)7 ra An () When x is large and p small, ¢ is a small angle and from (2) to a first approximation, 4 tan 6—go= be It follows from this, by putting tan? tan?d d= tand— 3 D and solving for tan @, that ot dA,” tan d= Ayt+ = 5 ea oo where ES: —_ aa 3 pa ae lara, An Therefore ee ie a With a very small error, from (2) and (3), e= tane 5 5 = Dintan® $ ~ 18Gp—Dr The second approximation, therefore, is i) n(tan@—§) = CPT 4 ae - (20) 29 tan P=A+ = ee atl oa taal. (21) Finally, Bee iy (1 ae ae am an(tr eS) ey Roots of Bessel Function J,,(a) and its derivate Jn'(a). 193 For the first three roots of J,(x)=0, py =n +1°8569n3 + ee) po= n+ 3°2447n3 + OL DS4n see «ol eo) p3=nt+4°3817ns + 5:7598n—%... ) From (15) and (16), it can be shown that p, is given by py=nt+ 1°85576n3 + 1:03315n— 3 — 0:00403n—?! —0°09083n meno 0t48n7 Bio i vd) 2/4) All the formule hold whether n be an integer or not. (c) When n is negative, X is dependent upon p, the number of the root, and n, the order of the function. For roots larger than n, _ [36 4p—4n—1)r D ne OF og E. 2 *18(p—4n— 1m 2) and for the first root, p is the least integer making 4y—4n—1 positive. J'hen p=n( one, + to): The following simple expressions give approximately the first three roots of J_,(z) when n=a—4, « being a positive integer pi=n+0°951n8+40:271n-%... po=n+2-59Gmeer 2 022n— 2... >... (26) ps=n + 3°834n3 +.4°410n7 3... If n=a—« and « is less than 4, the first root is equal to or less than the order of the function and the above formule, derived from the expressions when the argument is greater than the order, are not applicable. The roots of J_,(n) =0 are given approximately by 4 + “204 axe}, (,26_)f, FG) _, 1, osmee 6 \6«a—1/ * 840° T(4) CO eile (d) The function J_,(x) possesses complex roots also, which can be calculated for small values of n, from Lord Rayleigh’s expressions for the sum of the powers of the reciprocals of the roots. If r(cos@+isin @) are the complex roots of J_ s(2), 2 cos y) Gos 0 _._9-06a001 Mad ° su 5? _ _ 0.00619, tk A AE r=1°944 and 06=61° 85. On 194 Numerical Calculation of Bessel Function J,(#). When 1. Hence the 712, 193) 73; Will be found from the formule for the polar triangle, viz. : cos Bcos C+ cos A cosa= : : sin Bsin © or sie il 123 T 2 + 2731 - 3712 (2) a V1—,r?, 13 V1—,7; (ii.) From the relation 1—, R= G7) (1 — sri) = gin’ sin? C, it is clear that ,R.3 is the cosine of the perpendicular from A on BC. iii.) Let us write f— COSa==U, haa —seos A—wz, | [Ro xX. @ Fy— COS'D=0, 0 “ofsi—seos B=y;' oRy— ¥ (3) Tig COSC=W, af p= Cos C=z, ae so that u—vw LYE “eS and ———————— Oo V1—v? V1—w? V1—7 Nae in addition, let A= [1 4. 3 | =l—-w’?—v?—w?+2uvw, . (4) Tig Loge 731 793 1 and A'= LL Sitges) = 1 2 y= 2 — Zaye, eae 12 i = 5193 (9) Tae Le tem 1 So that sina/sin A= V/A/A’, A= (1—v?) (1—w?)(1—a’), A’=(1—y’)(1—2”) (1—v’). (iv.) It is well known that in normal distributions the 2 be END standard deviation of 72 is given by > “= (=r) and the nr Tl2 correlation between deviations in 73 and 7y3 1S R,,=w—uvA/2(1—u?)(1 —v’) 208 Dr. L. Isserlis on the Variation of Making the third variable constant we have immediately > A S = = 0-2) 3712 We must now show that Ry=—z—ayA'/2(1—z*)(1—y”?), . . (6) the formula suggested by the polar triangle. | sinb sin B’ o 1—w? 1—2? ey? 1-7 So that udu van bi, ada ue ydy Lee ee eee 1—y" Squaring and taking the mean for many samples, wt v?—2urR,, =a t+ y? = Jay Roy. But Tei ORE Sn) R,,=w yw) (=e) =w—ur(1—2z’) e+ay (7+ yz) ae id Vise Ta 2V71—-P J1—2 V1—2 V1—2? =[2(2+ ay) —(w+yz) (y+en)]/2 Vl—a? /1—y. Hence UV QeyRy= 2? +yP—w—e + Viz aes - [2(¢+ ay) —(@+ yz) (y + remembering that (1—w) (1-1-2) =a, Ry2= — raat )w— os) e( Lae) ~ 2(u—vw) (w—ue or met ee (w—uv) — (1—u?)z?] Uz A 20(1—e*) 210 Dr. L. Isserlis on the Variation of (vi.) To find the probable error of the multiple correlation coefticient Ri23, we use the relation 1—X?=(1—v’)(1—<’). Differentiating, XdX _ vdv A zdz 1—X?> 1-1? © 1—2?? whence, using (7), X? ee 2% Que UZ x= Cer none 2 or 22" er = ge eee =1—(1-—v’)(1—2’) =1—(1—X?’) ae so that Sik 1— X2)? | 2=( n ) CD ee 1:23 Vn (vii.) Correlation between deviations in Rj-23 and 743 OF 3r12 NX | bdo zdz 1-X? 7 p= 1-2” squaring and taking mean xX? ss ve— 2vX Rx = 2 ae X?4+ 0? —2? a eae Similarly A X74 22—v’? Bx. = Dax 2% * Pa M3 = (Rip + aia) [27,5 Hoa) ee (9) Be 2713 oa (Ris ag sli9 nae r?s)/ 2 3) 12 Ryo sine (10) (viii.) We can now find the probable error of A from the relation - A=(1—w?) (1—v?)(1—2’), the Multiple Correlation Coefficient. AVE so that aA one vav zdz "3A (ae iL 1l—v? ~ 1—2?* Squaring, summing for all samples and dividing by their number, we find on using (7) and known results that n> ss —A Sy? + y? + 224 Quy EeaticA ere) DER P eo emer 4? ei) Caaeie. and the right-hand member reduces to wu? + v2 + w? when we substitute 2=(w—uv)/\/ 1-4/1 —0?. oe 9A i ./ ae ° ay (), se aaleA ee (11) But (1—u?)(1—X*) =A. So that Hence XdX | udu __ dA ee Lh 2 ay ig and therefore Te. 2 ee ee 2uX gives the correlation between errors in 7,, and jee (ix.) Similarly from the relation A'=(1—2*)(1-y)(i—w’), and equations (6) and (7), we deduce Se Vey +2 and then using (=a) (Gee) = A’; we find yte?— xX? pee OS) (x.) To find the correlation in errors of two multiple correlation coefficients X=R,,., and Y=Rys;, we have (1— Y?) =(1—w’) (1— 2”) vs YdY _ wdw avdx ono: Baer Rex = 212 Dr. L. Isserlis on the Variation of Multiply by dX and sum for all samples as usual. Thus _ ye _ (L—X?) (X?+ w?—y’) (1— X?) (y?+2?— 9 Pi ROS ee IK w hg IXe which reduces to Soha XY OXY ° mm here paves eh) (xi.) Finally to determine the correlation between errors in V=173) and Y= P31 (15) We have 1—X?=(1—v*)(1—2), 1—Z?= (1—w’?) (1—y’”); so that Kak «, eee _ vd 1—-X? 1-2 1—” and ZdZ udu _ ydy Tee i eae ae 1—," Multiply these results and sum for all samples. Using the results previously obtained, we find ZX (v2? + y? 2 ey ite — ae + us( — =) ow — X\ Tee ee dees —uX( duX Wz or 2vyRry= X? fs Z2— 42 — w?—222, which can be put in the symmetrical form 2vy R= X?+ Y°?+ 2—w?—w?—a?—2. . (16) (xii.) We collect the above results into a single table in a form suitable for reference. Let ey A=1—r, — 73, — re A ed ee ane A’'=1- os A soe = habe (iia) (Fa NC dee) ; Then Reta = Pas" gh/20 — 733). — 743) | Bins fa, ae aes Pig )& f21— 753) (1, 1) ge) | R,.., ig GR; Hea ia 5 Tig” — 9113 — shin) 27 99 -17"93 « (16) the Multiple Correlation Coefficient. 213 ne (75) Cee yee!) a Grom arial 6 CT) Rir.,, 93 = (73, +7.— , R3,)/ Lalo 7 1 RR ea (12) R p,.,, as (; Ro, +75) — 31t9)/ eee rey) si) aR hee. a (9) Rr, 12g (931 + s?i2— 1 Roe) /2(pRoa) (1793) 5 ee (14) Rir.,, for = Res + 211 — Ta) |2GRos)(Tn) - - 2 ee (10) Rir,,..By, = ("io t sti2)/2GRos)(gRy) 6 - 6 2 ew (15) 2 Se 2A (2 172, +e... LL) 2 a. =2A' (7574 | oss. || ae rere (13) 2. = JEU ale | EEE Pe eee Maen 3) 1""23 §3 Ga.) The underlying assumption of the methods employed to obtain the results in the preceding section, a method valid over a large range of cases, is that the mean value in many samples of the statistical] constant whose probable error we are determining is equal to or only differs by small quantities of a high order from the true value, 7. e. the value in the sampled population. We have already seen in §1 that this is unlikely to be exactly true in the case of a multiple correlation coefficient, and we shall in this section find the mean value of Rj23 in samples of size n extracted out of an infinite population with normal distribution, and also the corrected value of the probable error when the position of the mean has been allowed for. We shal] consider n to be so large that 2 may be taken as negligible in our results, and will reserve for a separate paper what turns out to be a much more difficult investigation, the determination of these quantities correct to terms in 1/n?, an investigation of great importance when we come to deal with moderately small samples. The justification of the separate treatment of the easier problem lies in the result, obtained comparatively simply, that the formula ae Ri03 n remains true (if 1/n? be neglected) after being corrected for the deviation of the mean value in many samples from the value in the sampled population. 214 Dr. L. Isserlis on the Variation of i.) Let u; v, w be the mean values of 793, "31, 712 in many samples of size n out of an infinite population with normal distribution. Let X be the mean value of Rj-23 as calculated in such samples, and let wu, v, w, X be the corresponding values for the sampled population ‘itself. In any particular sample the correlation coefficients will have values that we may denote by u+du, v+dv, w+ dw, X+dX. Now suppose that u=u+«, v=v+68, w=w+y, X= X + &, Bi then a, 8, y are known correct to terms in 0) A: n u(l—u?) 3u(1l—u?)(14+3u?) a im | Now vet we +w —2urvw K=,/ { = =/(w, v, w) say,. (18) and X+dX=/(u+du, vt+dv, w+dw), or X+&+dX=f(u+atdu, v+-B+dv, wt+y+dw). (19) Expanding the eda side by Taylor’s theorem we have, after subtracting (18), &+dX= (a+ du) 2 5 + (B-+dv) 82 a a +5 [© ate cain 29 = + (B+ dye od a (y+ dup Sd +2(a+du)(B+dv) : of +2(8-+do)(y+dw)-—° + 2(y+dw)(a+du) or + terms of higher order in du, dv, dw. . . (20) Let us sum for all samples and divide by the number of samples. We shall have age Oo” +3{@+rye J+ (B+ oe) SL + ( TaiGns as a 2 (28 uy TuFvTuv) st aia 2(By ar FyO wow) Ov i +2(y2 outaten)sL b+ ete 2 ne ate * H. E. Soper, Biometrika, vol. ix. p. 105. the Multiple Correlation Coefficient. 215 Now, by (17) a, B, y are of order : , while o,, o,, oy are il of order —=. Hence correct to 2 we shall have /n n _ Lae ie a oo Su u( 2, ‘ 2 74 2 2 2 * feces $2 cia comment (aes) Since we are in this section neglecting 1/n”, we take a= —u(1—u?)|2n, gy= (1 —u?)/r/n, 1 g= w—uvh/2(1—v’?)(1—v"), where A= (1—u’?)(1—X?). Eee X=/flu, v, w)=V/ 7? $0? — 2Quvw// 1, we have easily X (Lut) SE = uX? ow, XW) =v—wu, X(1—u?) os =w—uv. 3 220° 2 X3(1—u?) ne = X*(1+ 2u?) — 2uvwX? — v*w?, 2 xe) SS =a 2 xu) SS aes X3(1—u’?)? or, =vw(v—wu) —(w—uv)X?, xe uy O = vw(v— wu) —(v — wu) X’, X3(1—u?) 2X ae = — vw. Ovow * We prove formally in a later section tha: the terms arising from the cubes and higher powers of the deviations make contributions of order 1/n’. 216 Dr. L. Isserlis on the Variation. of So that (22) becomes, correct to 1/n, X-X=é ul =u?) uX? we ol eee) | On MS) Trea _ wl wm) 2nX(l—u*?) | 4(1—u?)? X4(1 4+ 2u?) —2uvwX?— vw? = (1 — 0?) 2? eo i re + X(T = 28) aL we) _G ae — 1?) uvd oi nX3(1 —u?) (w- = $0 ey) “ vw(v — wu) — (w—uv)X? X3(1—u?)? j (1 — u?) (1 — w?) wud - n (v- 2(1—w?) (1 (oan ‘) vw(w—uv) —(v— wu) X? X3(1—u?)? (Ci) tie) vw = ae n (« 2(1 —v*) (1— w)) X81 — 02)’ or 2EX3(1—u?)n = X?{ —u?(1—u?) X? + ww(1 — uv?) — vo(v — wu) (1 — 0”) —w(w—uv)(1—w’*)} + X4(1—u?) (14+ 2u?) — 2uvw(1 —u?) X?— v?w? (1 — vu?) + w?(1—v?)? +¥?(1 —w?)? + {2w(1—v?) -— wv(1— X?) L{ow(v—wu) — (w— uv) X?} + {2v(1—w?) — wu — X”) } {vw(w — uv) — (v—wu) X?} + {2(1—v?)(1—w?)u—vw (1 — wu?) (1— X?) }(— ow) = GX*+ MX"? + H say. Collecting the coefficients of the powers of X, we have first Ga lau tu? + vw? —2uvw | = (l—u’)(l+u?+u?X?—2uvw). the Multiple Correlation Coefficient. a re Secondly, ; M= —uvw(1—u?)—o(v— wu) (1 — v?) — w(w—w)(1—w?) + uv?w(v— wu) + 2v(ww—v)(1 — w?) — 2w(w— wv) (1—v’) + uvw?(w— uv) —v?w2( 1 — w?) + wu(v— wu) + we(w—uv) =uvw(7 + u? — 2v? — 2w*—uvw) — 3v? 4+ vt — 8w? + wit 30?w? — 0? — wv? ;sW or after some reduction, M = (1 —u?)?X4 + (1 —u?) X?(Quew — uw? —3) + (1—u?) (wow + v7”). Finally, f=" +w* — vw? — v?wt + uvw(— 2 - 0? — w? + 207w? + 2uvw) = (1 — u*) X?—v?w?(1 — uv?) X? — wow(1 — wu?) X? =(1—wu’)X*(1 — v?w? — ur). So that EX#(1—wu?)2n = LX*+4+ MY +H = (1 —u?)X*(uw?X? — 2uvw + 1 +-u”) + (1—u?)X?| (1 —u?) X44 (Quvw — u? — 3) X? + vw(u + ow) | + (1 —u?)X?(1 -— v?w? —uvw). InEX =1-4-X*4+ X?{ —2uvw + 1+ u? + Quow— 3—-u*}, =1+4+X*-—2X?, ——— vale as en eae) 2. e. the mean value of Ri»s in many samples is always greater than the value in the sampled population, the principal term in the excess being the positive quantity eee 2 (1— Ry.)? 2nR 23 The form of this expression is highly significant. It is not only for small samples that the mean value differs sensibly from the value in the general population. If the value of R in the sampled population is small, the mean value deduced from many samples may considerably exceed the true value even in samples of large size. For example, consider samples of 1000 taken out of a population for which R=°01. Phil. Mag. 8. 6. Vol. 34. No. 201. Sept. 1917. Q 218 Dr. L. Isserlis on the Variation of Here (9999)? 9998 ‘= 9000C01) 90 7 or & is about 5 times greater than R. On the other hand, although this is of less importance, the mean value will be very nearly equal to the true value, even in small samples, if the multiple correlation 1 is very high. Consider samples of ten drawn from a population for which R=°9. Here (ELE Ute } eS Co 2 ee 0020 nearly, or Eis about 450 times smaller than R. st oe 2 (iii.) We can now find DRyoe * the same degree of approximation (7. e. correct to 1/n). Let us write eM?!) p=X2=- a mate =o(u, v, w) say. —u So that (]- wt) 98 = 2(ud— vw), 2 Sa ey, (Lut) 2 96 (1 + But) — Suv (1 wy SP =, (1 woe =2 (Inte OS = duy — Iw — ew, (1—u?)? o“o = 4duw — 2v— 2u’v, Ouoagw (1—w?) so = —2u. the Multiple Correlation Coefficient. 219 We have, as before, (X +dX)2=d(utatdu, ‘pbaghat ea or K2 + 29XdX 4+ dX?2= X?+(atdu)S¢ i 7 +5 nee: p +.. |. On summing for all samples and dividing by the number of samples, we have x a wo? og , _O¢ X?_-K?4 32 = a Bac i, iG eu) 5 8 ae + 2((aB + tetra oe + et Hence correct to oy writing X? for @, pee we 2 uu?) aia — ow) v1 = 08) 2to aw) 5 te + 2x Fy, 2n P= 1? 2n (1 — u?) 2 “Se oe = 7 i= Berto ore na (1—w? n(1 - u? as vi (1—u?) i Bee) (w ao =) Auv—2w—2u?w n 2 1—v? (1—u?)? ea) lw") a | Qu 2(1—v?)(1—w?) J 1-u? 6 Sol ee) (v- wu 1— E (4wu—2v— 2u?v) 1—w?2 (d—w)? , nr reducing to xx 52 X?{(1—w?)(1 + Qu?) + 2u?(v? + w?) — Buvw — vow} = ae — Fy? + Qu* — 5w? + Qwt + 2— Qu? 4+ 4v*w? n(1—u?) _ = 2Qu?(v? + w?) + uvw(10 + 4u? — dv? — 5w? + 2uvw) X?(1 —u?)(1 + 2u?) + 2u?X4(1 —u?) — 3uvwX?(1 —u?) at) 5, 0 —u2)2XK* 4 Sew (1 — v®)K2—5(1 eX? — 2u?(1—u?)X? + 2(1—u?) = —- 4 (1—wu?) (2X44 X°(1 + 2u?— 3uvw + 8uvw — 5 — Qu?) + 2) 220 Variation of the Multiple Correlation Coefficient. So that correct to 1/n, | , 2(1—X?)? n oe = (24) But to the same degree of approximation, we have already shown that <= i Lee a Di aXe oe o BEX ZZ Hence X?= X?+4+ pee correct to 1/n. Hence (1—X2) Se Ye (25) correct to L/n. We have thus established that for samples of fair size extracted from a population in which Rj», is not very small, a oe ee the standard deviation of the distribution of R about fre mean value is the mean value of R exceeds the true by It is important to observe that where the true value is not known, and R necessarily has to be determined from samples, we may replace R by R in these results within the same limits of accuracy—thus : Se (di ne and a (1 —R?) 2 Ss; =— — s 6 e ° ° ° ° yy a/c i) THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES “4 Al ray Gear Pere Ge Pe I> OCTOBER 1917. “= XXVI. On the Origin of the Line Spectrum emitted by Iron Vapour in the Explosion Region of the Air-Coal Gas Flame. By G. A. Hemsatecnu, Honorary Research Fellow in the University of Manchester *. [Plates II. & IL1.] § 1. IyrropuctTion. T has been shown f that there are two different types of iron cone lines: firstly, those forming Class II., which appear likewise, though only feebly in the mantle, but grow more intense on passing to flames of higher temperature ; and, secondly, Class III. lines, which are observed solely in the cone. The special interest attached to Class III. lines is that among them are found a number of lines which had always been regarded as due to the more violent modes of excitation, such as prevail in the are or spark. Thus Mons. de Watteville and myself{ were able to identify in the supplementary spectrum a considerable number of Duffield’s polar A and polar § lines§; further, we recog- nized three of Lockyer’s enhanced lines, namely, AA 3864, 3872, and 3936. With regard to the last-named line, which * Communicated by Sir E. Rutherford, F.R:S. + Hemsalech, Phil. Mag. ser. 6, vol. xxxiii. p. 1 (1917). t Hemsalech and de Watteville, Comptes Rendus de ? Académie des Sciences, vol. cxlvi. p. 1889 (1908). § G. W. Duffield, ‘ Astrophysical Journal,’ vol. xxvii. p. 260 (1908). Phil. Mag. 8. 6. Vol. 34. No. 202. Oct. 1917. R 222 Mr. G. A. Hemsalech on the Line Spectrum of is also emitted in the electric furnace at the highest tempera- ture, Dr. King has suggested that it might be due to barium*. But, according to Mons. de Watteville’s observations, barium does not emit a line of this wave-length in the flame fT, and, further, none of the characteristic flame and cone lines of barium, such as AA 5536, 4554, and 3994 show in our flame- spectrum of iron. It is therefore highly probable that the line \ 3936, as emitted by an iron flame, belongs to iron and not to barium. But, although the line has been classed as enhanced by Lockyer, its real character appears to be still doubtful. Its relative intensity is greater in the Bunsen cone than in the self-induction spark. The curious behaviour of the cone lines soon convinced us that their emission by the air-coal gas flame is caused by some yet unknown exciting agent, such as some special chemical actions, prevailing in the cone of this flame but absent from all the other, though hotter, flames examined. The present research was undertaken with the object of investigating more fully by means of divers experiments the physical and chemical conditions which govern the emission of the cone lines, and also of gaining a clue to the particular chemical actions which underlie their excitation. These experiments were divided into the following four groups :— 1. Effect produced on the spectrum by varying the com- position of the gas-mixture, §§ 3, 4, 7, and 11. 2. Effect of changing the atmosphere in which combustion takes place and of reversal of flames, §§ 8, 9, and 10. 3. Effect of heating and cooling the flame, § 12. 4, Influence of velocity of gas-mixture, § 13. The methods of producing and feeding the flames, as also the spectrographic appliances for examining them, were the same as those described in my first paper. § 2. RELATIVE APPEARANCE OF CONE AND FLAME LINES IN VARIOUS FLAMES. The gradual development of the various types of lines on passing from the base or origin of the flame upwards is best studied by forming an image of the flame on the spectro- graph-slit in such a way that the latter bisects both cone and mantle, as indicated by the dotted-line in fig. 1. When in- terpreting the spectrograms obtained in this manner, account * S. A. King, ‘Astrophysical Journal,’ vol. xxxvii. p. 2389 (1913). + C. de Watteville, Phil. Trans. A. vol. cciv. p. 153 (1904). D> Iron Vapour in Air-Coal Gas Flame. 223 must be taken of the fact, as illustrated by the figure, that the thickness of the stratum of mantle is a minimum near the base of the cone and a maximum just above its apex. Fig, 1. Mantle Relative positions of flame image and slit. For this reason alone one would always expect the flame lines to reach their maximum intensity just above the cone. We also know from the results obtained by many experi- menters—in particular, H. Bauer *—that the temperature of a flame is highest just above the explosion region. a. Air-coal Gas Flame. (Plate II. No. 1.) The cone Jines (Classes II. and III.) first become visible at the base of the cone; then, on passing upwards, they gradually grow brighter and reach their maximum of strength near the apex. After this there is an abrupt and enormous decrease in their intensities. Class II. lines con- tinue to be emitted as feeble lines by the flame above the cone (example: triplet at 4046), but Class III. lines are no longer emitted. With regard to Class I. lines, they behave in a similar manner to the cone lines throughout the height of the cone, but, though there is a slight reduction in their intensities as they pass from the cone into the mantle, they * E. Bauer, Théses de Doctorat, Paris, 1913. R 2 224 Mr. G. A. Hemsalech on the Line Spectrum of continue as strong lines in the latter and only slowly and gradually weaken as they approach the upper part of the flame (examples : AA 3860, 3886, 3920). As I have already pointed out in my first paper, the apparent enhancement of these lines in the cone is probably caused by the super- position of two independent emissions, one due to thermal excitation in the outer stratum of flame and the other to chemical actions in the explosion region. b. Oay-coal Gas Flame. (Plate II. No. 2.) With the gases adjusted so as to give the highest tempera- ture obtainable with this flame, the cone gives out the bands of the Swan spectrum with much greater intensity than the air-coal gas cone, and, in addition, many other carbon bands, which are absent or appear only as traces in the latter, be- come quite prominent. Class |. lines of iron, even the most intense, first become visible a little distance up from the base of the cone; they gradually increase in intensity and reach a maximum in the mantle just above the cone. There is no indication whatsoever that these lines are in anyway affected by chemical actions in the explosion region, such us they are in the Bunsen cone. After having obtained their greatest strength they die out again very gradually as the temperature decreases on approaching the summit of the flame. In faet, their intensities seem to be a function of the temperature. Class II. lines, as shown by the triplet at 4046, behave in a very similar manner as Class [. lines, with this exception, however, that near the base of the cone they show a very slight enhancement. Of Class III. lines, only traces of a very few are seen near the base of the cone, where they appear as “short” lines, reaching only a little distance up . from the base. An explanation for the appearance near the base of the cone of these lines will be given later. It is thus clear that the substitution of oxygen for air sup- presses those chemical actions to which the radiations of Classes II. and III. are particularly sensitive, and it there- fore appears that the presence of air or nitrogen in the gas- mixture is essential for the emission of the cone lines. These results further show that there is no connexion between the emission of the carbon bands and that of the particular types of iron lines under consideration. e. Air-hydrogen and Oxy-hydrogen Flames. (Plate III. Nos. 1 and 2.) Although there is a well-detined cone in the air-hydrogen flame, it does not appear that it has any other than thermal Iron Vapour in Air-Coal Gas Flame. 225 influence on the metal vapour passing through it. In fact, to Judge by the luminosity of the vapour and that of its spectrum, it seems that the region of maximum temperature is that immediately surrounding the cone. In this region nearly all the lines given by this flame, which comprise those of Classes I. and II. only, appear brightest, but their relative intensities are the same as in the flame above and, indeed, the same as in the mantles of the oxy-coal gas and oxy-hydrogen flames. After the vapour has passed through this region there is a rapid, though not abrupt, falling-off in the intensities of all the lines. It is possible that this rapid diminution of the intensities is due to the relatively small quantity of oxygen that remains over for completing the oxidation of the hydrogen, the greater portion of the former having already been consumed in the vicinity of the cone; the temperature of this flame would therefore decrease very rapidly on passing into the region above the cone. A very curious exception to the behaviour of the majority of lines is disclosed by the quartet group y (head line at 4376), which is apparently not in the least affected by the change on passing from the cone to the flame above. With regard to the oxy-hydrogen flame, its cone is very short and sharply defined, but no enhancement of any line whatever has been observed in it. All the lines reach a maximum of intensity a little distance above the cone, after which the vibrations die out gradually, as in the case of the oxy-coal gas flame. The spectrum given by iron in the oxy- hydrogen flame consists of Class I. and Class IJ. lines only. There is no trace of Class III. lines in either the air-hydrogen or oxy-hydrogen cones and mantles. The oxy-hydrogen spec- trum, indeed, constitutes, as it were, a generally enhanced type of the spectra given by the mantles of the air-coal gas and oxy-coal gas flames, such as one would expect to obtain by assuming temperature {o be the governing factor in the production of these radiations. That the temperature hypo- thesis seems also to afford a plausible explanation for the origin of the air-hydrogen spectrum, is supported by the behaviour of the temperature lines AX 3720, 3737, and 3746. These lines are apparently not affected by the chemical actions in the air-coal gas cone and they appear as feeble lines in the outer mantle of the same flame. In the vicinity of the air-hydrogen cone they become much brighter and in the oxy-hydrogen flame, the hottest of these three flames, they figure among the strongest lines of the whole spectrum. These lines are evidently sensitive only to 226 Mr. G. A. Hemsalech on the Line Spectrum of temperature changes and we may therefore conclude that their greater brightness in the cone envelope of the air- hydrogen flame is indicative of a state of temperature in this flame which is of a higher order than that prevailing in the mantle of the air-coal gas flame. d. Azr-acetylene and Oay-acetylene Flames. Owing to the difficulty of securing the necessary appliances for working these flames in conjunction with the electric sprayer or spark methods, it has not been possible to investi- gate, from the present point of view, the spectra of iron to which they give rise. In the course of a previous research conducted by the present author in conjunction with Mons. de Watteville, it was, however, found that in neither the air-acetylene nor the oxy-acetylene cone does iron vapour emit a characteristic spectrum. This seems rather astonishing, in view of the fact that chemical actions of a most violent character take place in the explosion regions of these flames and, in particular, in that of the oxy-acetylene flame. ‘The inner of the two cones of this latter flame is, indeed, so brilliant as a consequence of the chemical reaction that it is not possible to examine it with the unprotected eye. The Swan spectrum given by this cone is extremely intense, as are also the other carbon bands, all of which, in addition to their great brightness, show an extraordinary development. By reason, perhaps, of the great velocity which it is neces- sary to give to the gas mixture of the oxy-acetylene flame in order to maintain its stability, the lower part of the cone is enveloped by only a thin stratum of mantle, the full develop- ment of the latter taking place some distance above the cone. This may partly account for the fact that the greater number of the metal lines given by this flame do not originate near the base of the cone as in other flames, but in the region above the cone, and only the very strongest are observed also in the direction of the cone. Another explanation for the absence of lines in the lower part of this flame may be that, owing to the great velocity with which the material, mixed with the gases, is carried through this part of the flame, it is enabled to travel a certain distance before the effect on it by the temperature has been fully developed ™*. The spectra given by both the air- and oxy-acetylene flames present the same general characters as that emitted by the * Hemsalech and de Watteville, Comptes Rendus de l Académie des Sciences, t. cl. p. 329 (1910). Iron Vapour in Air-Coal Gas Flame. 227 oxy-hydrogen flame, but they are much more intense. Hspe- cially with the oxygen-fed flame the iron spectrum is so luminous that an exposure of only ten minutes gives a con- siderably brighter and better developed spectrum than an exposure of two hours under otherwise identical conditions with the oxy-hydrogen flame. With the oxy-acetylene flame the necessary minimum temperature seems to have been reached which is required for the excitation, by purely thermal actions, of those vibra- tions which in most other light sources are brought about by chemical or electrical actions only. Thus we found in this flame traces of a number of Class ILI. lines, amongst which the group at 4957 and the enhanced line A» 3936. Although in the air-coal gas cone these lines are relatively prominent, the very high temperature of the oxy-acetylene flame (nearly 3000° C.) has only been able to excite them very feebiy indeed, as compared with Class I. and II. lines. As already stated, Class III. lines have not been observed in the hottest parts of the oxy-coal gas and oxy- hydrogen flames, the temperatures of which, 2400 and 2600° C. respectively according to E. Bauer, are probably not sufficient for their thermal excitation. The results obtained with the air-hydrogen and air-acety- lene flames show clearly that nitrogen is not the only factor in bringing about the special chemical actions to which is due the emission of the characteristic cone spectrum observed in the Bunsen flame. The observations recorded in this paragraph are sum- marized in the following table, which shows at a glance the absence or presence of any one of the three types of iron lines in the various flames examined :— mare Class I. Class I. ~—Class IIL. Air- cone present present present. ‘ coal gas | mantle present traces absent. Oxygen- { cone absent traces traces. coal gas | mantle present present absent. Air- cone present present absent. hydrogen | mantle present present absent. Oxygen- f cone absent absent absent. hydrogen | mantle present present absent. Air- cone absent absent absent. acetylene | mantle present present Oxygen- f{ cone absent absent absent. acetylene | mantle present present traces. 228 Mr. G. A. Hemsalech on the Line Spectrum of § 38. EFrrect OF ADDING TRACES oF NITROGEN AND AIR TO THE GASES FEEDING AN Oxy-coaL Gas FLAME. The observations on the appearance of the cone lines given” in the preceding paragraph have rendered it very probable that nitrogen is one of the essential factors in the emission of the cone lines. It was therefore natural to expect the reappearance or enhancement of these lines in the explosion region of the oxy-coal gas flame on adding traces of nitrogen to the gas mixture. In one experiment the nitrogen, which was produced in the usual way by passing air ever red-hot copper, was added to the oxygen and, in another experiment, to the coal gas. In either case there resulted an enhance- ment of Class II. lines in the cone, not merely near its base but along its entire height; also the stronger lines of Class III. became perceptible. On the other hand, Class I. lines were not affected and remained, as before, brightest in the region just above the cone. Similar results were obtained by adding traces of air to the oxy-coal gas mixture. § 4. EFFECT OF ADDING OXYGEN TO THE GASES FEEDING AN AIR-COAL Gas FLAME. As was to be anticipated, an increase in the proportion of oxygen in the air-coal gas mixture causes a strengthening of the flame lines due toa rise in the flame temperature. Also, if, as had been concluded from the results of previous observations, nitrogen is one of the determining factors in the emission of the cone lines, the further dilution of it with oxygen should result in a diminution of the cone emission. This is, indeed, what has been observed (Pl. II. No. 33 to be compared with No.1). The principal facts derived from this experiment may be summarized as follows :— a. Class I. lines are all more or less enhanced, especially the triplets at 3720 and 3860. They show no effect due to any special actions in the cone, but gain their maximum strength just above the cone ; in this respect they behave similarly to those of the oxy-coal gas flame. 6. Class IL. lines are affected in a most striking manner. In the cone they are relatively much weakened, whereas in the flame above the cone they are con- siderably strengthened. This is well shown by the triplets at 4046 and 4384. c. Olass III. lines are considerably reduced, only traces of the brighter ones remaining visible. d. The continuous spectrum in the cone is greatly diminished, but the carbon bands, on the other hand, are markedly enhanced and more fully developed. Iron Vapour in Air-Coal Gas Flame. 229 In short, the admixture of oxygen to the gases feeding an air-coal gas flame reduces the effect of the special chemical actions in the explosion region and intensifies the thermal actions in the mantle. § 5. A PossiBLE CAUSE OF THE CONE EMIssIon. The foregoing results supply sufficient data for suggesting a preliminary hypothesis as to the nature of the chemical actions to which may be ascribed the emission of the cone lines. The particular reaction involved attains its. full development in the air-coal gas flame and, as has been shown, the presence of nitrogen is one of the determining conditions. Further, it would appear that a favourable in- fluence is exercised by one of the constituents of coal gas, perhaps methane, but probably not hydrogen. The con- ditions for the reaction may then be summarized thus :— Favourable to reaction. Opposed to reaction. nitrogen. oxygen. methane. hydrogen. acetylene. Of the two gases favourable to the reaction methane is decomposed in the cone, the carbon burning to its monoxide and part of the hydrogen to water. The former reaction is accompanied by the emission of the Swan spectrum and the latter by the so-called water-vapour bands. ‘These two spectra are emitted whether the flame coutains metal vapour or not. Hence the two elements, carbon and hydrogen, do not seem to participate, at least not to any appreciable extent, in any further reactions which might be caused by the presence of metal vapours. From this consideration we are led to conclude that nitrogen must play a paramount part in the reaction which gives rise to the cone lines. What could be the nature of this reaction? A possible answer to this question appears to be pointed out by the very inte- resting observations on the formation of metallic nitrides by Messrs. Beilby and Henderson *. These chemists have shown that many metals, when exposed to the action cf ammonia at high temperatures (varying between about 400 and 800° C.), are either converted into nitrides or else pro- foundly changed in their physical properties. These latter changes are being caused by the continuous formation and decomposition of unstable nitrides. When hydrogen is present in excess, decomposition of the nitrides takes place. * G. T. Beilby and G. G. Henderson, Journ. Chem. Soc. vol. lxxiv. p. 1245 (1901). 230 Mr. G. A. Hemsalech on the Line Spectrum of Let us now examine whether the conditions prevailing in the explosion region of the air-coal gas flame are favourable to the formation of nitrides. The gas mixture, containing hydrogen and nitrogen (the constituents of ammonia) in the free state, enters the explosion region at about room-tempera- ture—say, 15° C. On passing through the thin wall of the cone its temperature is raised considerably and near the tip of the cone it reaches over 1000° C. ‘Thus during one stage of its passage through the explosion region the gas mixture has passed through that range of temperature recognized as essential for the formation of nitrides, viz. 400—800° C. Hence the temperature conditions in the Bunsen cone are favourable to the formation of metallic nitrides. We know, however, very little about the changes in the state of the metal as it passes through the explosion region. The material enters this region generally as a minute globule of a salt solution of the metal, or as an ultra-microscopic particle of the oxide. In either case the reaction takes place with the emission of cone lines, although with the oxide, to judge by the relative faint- ness of the resulting iron spectrum, the reaction is not so intense. Is the temperature of the explosion region alone sufficient to dissociate these minute quantities of the com- pound and thus to set free the metal prior to its combination with the nitrogen, or, if not, is the chemical affinity of nitrogen for the metal so great as to liberate the metal atoms at a lower temperature than they would be set free at under the influence of thermal actions alone? ‘To these and many similar questions which naturally arise, no satisfactory answer can as yet be given. But let us suppose that the cone lines are caused by the successive formation and decomposition of a nitride of the metal, how is it then that this reaction does not take place also in the cones of the air-hydrogen and air-acetylene flames? Several reasons can be advanced in explanation of this discrepancy : Firstly, it may be that the temperatures prevailing in the explosion regions of these two flames are too high for the formation of the nitrides, 7. e. above 800° C. As we have seen (§ 2,c & d), the spectroscopic evidence seems to point to the existence in the air-hydrogen cone of a temperature higher than that prevailing in the mantle of the Bunsen flame, namely, above 1700° C. We know nothing about the tem- perature of the air-acetylene cone, but to judge by its great luminosity it must be considerably higher than that of the Bunsen cone. Iron Vapour in Air-Coal Gas Flame. 231 Secondly, there may be present in both the air-hydrogen and air-acetylene cones too great a quantity of free hydrogen and, as Messrs. Beilby and Henderson have shown, excess of hydrogen is unfavourable to the formation of nitrides. Thirdly, the formation of the nitrides may be facilitated by the presence of methane, or its reaction with oxygen,, acting as a catalytic agent. The first two of these explanations seem to me the most plausible ones, and they appear even to lend support to the hypothesis of the formation of nitrides in the air-coal gas cone. Whether and in how far the other constituents of coal gas, such as carbon monoxide, ethylene, &c., which are present only in small proportions, participate in the reaction, and also the exact réle played by methane, are problems which will require separate investigation. Until experimenta! evidence to the contrary is brought forth I wiil assume, as a working hypothesis, that the emis- sion of the characteristic cone spectrum of iron, composed of lines of Classes II. and III., is the result of a reaction be- tween nitrogen and the metal, consisting in the formation and, perhaps also, subsequent decomposition of a nitride of iron. That the nitride formed is probably an unstable one may be conjectured from the fact that the material, almost as soon as formed, passes into regions of rapidly increasing temperatures. There are no observations which might pro- vide an indication as to whether the decomposition of the nitride takes place within the explosion region or only after the material has passed into the mantle. It will be in- teresting to know this, for if decomposition took place only in the mantle this would prove that the decomposition of the nitride, like that of the chloride, oxide, sulphate, &c., is not accompanied by the cone emission, but only by that of the temperature lines. § 6. REASON FOR THE APPEARANCE OF CONE LINES NEAR THE BASE OF THE Oxy-coaL Gas FLAME. It is now possible to offer a satisfactory explanation for the appearance at the base of the oxy-coal gas cone of traces of Class IIT. lines and also for the slight enhancement of Class II. lines, referred to above (§ 2,0). Since the flame burns in air, some of the latter is naturally drawn in at the base and the nitrogen of this air is thus brought into contact with the iron compound within the explosion region. Its restriction to the base may be accounted for by assuming 232 Mr. G. A. Hemsalech on the Line Spectrum of that the temperature in the cone of this flame, on passing upwards, soon reaches the critical value beyond which the formation of nitrides will cease. With cobalt and nickel, however, the effect of the reaction in these same conditions is considerably more conspicuous and is observed to extend to a much greater height from the base of the cone. It seems, indeed, as if the limiting value of the temperature up to which the formation of nitrides of these two metals will take place were greater than for iron. § 7. EFFECT OF ADDING AMMONIA TO Oxy-coAL GAS AND AIR-COAL GAS FLAMES. Without entering into details, it is sufficient to state that the experiment was arranged so that a greater or smaller amount of ammonia could be fed into the Hame by passing the whole or only a portion of the coal gas through a flask containing strong ammonia solution. With a certain amount of ammonia the cone of the oxy-coal gas flame becomes bril- hiantly coloured yellow. Spectroscopic examination of the mantle and cone shows that the iron spectrum given by the former is the same as when no ammonia is present, namely, all the lines reach their maximum intensity in the region Just above the cone. The cone spectrum, on the other hand, and contrary to expectation, shows no trace of Class III. lines, nor is there the least sign of a strengthening of Class II. lines. Thus the addition of ammonia suppresses the slight reaction which exists at the base of the cone when this gas is absent. With regard to the Swan spectrum, its bands appear nearly as well developed as usual. But in addition to this spectrum there stand out very prominently the bands of cyanogen at 3884 and the group in the violet at 4216. ‘This most interesting result seems to indicate that ammonia is broken up in the explosion region and that the nitrogen combines with the carbon to form cyanogen. Some experiments by Grotrian and Runge appear, however, to show that these bands are net due to cyanogen but to nitrogen. In that case their emission in the flame would probably be caused by the decomposition of the ammonia and should be independent of the presence of carbon. ~ When ammonia is fed into the air-coal gas flame the cyanogen bands again show in the cone, though they appear much less pronounced. Also the bands of the Swan spec- trum are in this case appreciably weakened. ‘The influence on the various types of iron lines is very marked, the cone lines especially being affected. Thus, Class III. group at Iron Vapour in Air-Coal Gas Flame. Bas 4997, which in ordinary circumstances is plainly visible and more intense than Class I. group y, shows only traces with ammonia and is relatively feebler than group ¥. The Class II. triplets at 4046 and 4384 are considerably reduced in intensity, the latter relatively much more so than the adjacent group y. Class I. lines, although appreciably weakened, are so much less than those of Classes II. and III. Hence the addition of ammonia to the air-coal gas mixture reduces to a considerable extent the particular reaction between nitrogen and iron, and further, it affects also the temperature of the flame, as is evidenced by the weakening of the temperature (Class I.) lines of iron. It is not possible in this case to account for the weakening of the cone emission by lack of nitrogen alone, for, in ad- dition to the nitrogen set free from the ammonia, there is the usual amount available from the air which is mixed with the coal gas. But we must remember that, in addition to nitrogen, a considerable amount, about three times as much, of hydrogen is set free by the decomposition of ammonia. This, added to the hydrogen derived from the coal gas, repre- sents a large quantity, and it may be safely assumed that, even after a great portion of it has been burned to water, an appreciable amount will remain over. It may be this excess of hydrogen which restrains the reaction between the nitrogen and iron, such as may be expected to happen in accordance with the hypothesis suggested. This same cause may also account for the suppression of the cone emission in the oxy-coal gas flame. It would have been interesting to know if the so-called water-vapour spectrum is in any way influenced by the presence of ammonia. The characteristic bands of this spectrum are, however, situated in the ultra-violet and were therefore not within reach of my spectrograph. § 8. SPECTRUM OF [RON GIVEN BY AN AIR FLAME BURNING IN AN ATMOSPHERE OF CoAL GAS. The burner used consists of one row of six holes each having a diameter of 2 millimetres. Its construction is other- wise similar to that of burner No. 2, previously described * The flame is inverted in the well-known manner by sur- rounding the burner with an ordinary paraffin-lamp chimney. The air, before entering the burner, flows through a sprayer in which it can be charged with the spray from a perchloride of iron solution. A continuous stream of coal gas is passed * Hemsalech, /. ¢. p. 6. 234 Mr. G. A. Hemsalech on the Line Spectrum of through the space enclosed by the lamp chimney and the flame is started in the usual way. When the air is not charged the resulting flame is of pale blue colour and presents the shape of a semicircular are passing parallel to and over the row of holes in the burner. This are seems to constitute the feeble explosion region. If now the air is charged with iron spray the are becomes luminous and gives out a continuous spectrum ; further, from each hole of the burner a luminous streamer (though less luminous than the arc) proceeds upwards through the arc. These streamers, which seem to represent the mantle, remain absolutely isolated and do not meet anywhere. The spectroscopic observations were made end on, 2. e. in a direction parallel to the row of holes in the burner, and that part of the lamp chimney through which the light passed to the spectrograph was kept hot by means of a Bunsen flame in order to prevent the deposit of moisture on the inside. The spectrum shows, in spite of an exposure of two hours, only very few lines and a feeble continuous ground in the visible part. The bands of the Swan spectrum are absent. With regard to the iron lines it is interesting to note that only the stronger: ones of Class I. are brought out, namely, AA 3856, 3860, 3879, 3886, 3923, 3928, 3930, and traces of a few others. But there is no trace of Class II. triplets at 4046 and 4384. Hence there is under these conditions com- plete absence of any reaction between nitrogen and iron. Further, the restricted development of the iron spectrum as emitted by this flame seems tc be indicative of a very low temperature. § 9. Spectrum oF [RON GIVEN BY AN AIR-COAL GAS FLAME BURNING IN AN ATMOSPHERE OF CoAL GaAs. The general appearance of the flame is very similar to that — given by air burning in coal gas, but it is both larger and brighter. The spectrum, which was again obtained with a two hours’ exposure, is of a similar character as that of the air flame, only a little more intense. In addition to the iron lines of Class I. already recorded with the air flame (§ 8), there is also present the quartet at 4376 (group y). This fact is rather remarkable, because this group is not a particularly prominent one under ordinary conditions. A further interesting feature of this spectrum is the presence of Class Ii. line 4384. It is, however, considerably fainter Iron Vapour in Air-Coal Gas Flame. 235 than its neighbour 4376 of the quartet, and the relative appearance of these two lines is very similar to that ob- served in the mantle of the air-coal gas flame burning in air. For this reason it seems to me probable that the presence of the line 4384 does not, in this case, point to the formation of a nitride in this flame, but rather to thermal actions slightly more energetic than those prevailing in the air flame burning in coal gas. This conclusion is in harmony also with the slight increase in the number of Class I. lines observed. § 10. Spectrum or [RON EMITTED BY AN AIR-COAL Gas FLAME BURNING IN AN ATMOSPHERE OF OXYGEN. The general arrangement for producing this flame was the same as described in § 8, but only two 2 millimetre holes of the burner were used anda constant stream of oxygen was passed through the lamp chimney. The combustion of the gases is accompanied in this case by a singing flame which, in a revolving mirror, can be split up into a great number of individual streamers. In fact, the flame can be regulated by the pitch of the sound-note given out, and kept constant with the help of it. The general aspect of this flame is rather curious, in that the sharp outline of the Bunsen cone is absent and its charac- teristic blue coloration is visible only near the base. When charged with iron vapour the flame is considerably brighter than one burning in air, but the visible spectrum of the luminous flame seems to be chiefly continuous. The in- tensified chemical actions due to the excess of oxygen were rendered apparent by a thick deposit of red oxide of iron on the inner wall of the lamp-ehimney at the end of an experiment. The spectrum obtained with a one hour’s exposure shows only three lines of Class I., namely, AA 3860, 3886, and 3930. On the other hand, the Class II. triplets at 4046 and 4384 are brought out completely, the head lines being quite as intense as the line 3886. ‘Thus, it seems that the formation of nitrides will take place under these conditions. This may be accounted for either by the absence of hydrogen in the surrounding atmosphere or also by the presence of increased chemical actions of a different nature due to the excess of oxygen. But the reaction between the nitrogen and iron can only be of a very restricted nature in this case because no Class III. line is emitted. The thermal actions, to judge by the imperfect development of the temperature spectrum, 236 Mr. G. A. Hemsalech on the Line Spectrum of have not been appreciably increased by the extra oxygen. There is, indeed, no trace of the quartet at 4376. The enhanced chemical activity is further demonstrated by the appearance of the Swan spectrum. i The results of the observations made on the relative be- haviour of various groups of iron lines in the last three experiments ($$ 8, 9, 10) are summarized in the following table :— | Air Air-coal Gas Air-coal Gas in in in Coal Gas. Coal Gas. Oxygen. Cl I Quartet at 4376 absent present absent. ass | Triplet at 3860 present present partly present. { Triplet at 4046 absent absent present. Class II. ... « Triplet at 4384 absent trace of head _— present. | line only Swan bands absent absent present. From the results of these experiments, we may infer that the presence of oxygen in the medium surrounding an air- coal gas flame is essential for bringing about the formation of iron nitride in the explosion region. § 11. ExpERiments witH 4 NITROGEN-coAL Gas FLAME. A stable flame was obtained with a burner consisting of a brass tube of 2-inch bore. The flame is composed of a large non-luminous pale blue cone, enveloped by a pale green mantle. The latter passes at the summit into a bundle of pale red streamers. The green coloration of the mantle is perhaps due to copper vapour carried over by the nitrogen, which was produced by passing air over red-hot copper. When the nitrogen is charged with perchioride of iron spray the mantle becomes feebly luminous, but no change is observed in the appearance of the cone. The luminosity of the mantle was, however, so feeble that no photograph of its spectrum could be secured. This experiment is, however, instructive in demonstrating the absolute necessity of the admixture of oxygen to the combustible gas mixture, in addition to its presence in the surrounding atmosphere, in order to start chemical activity in the cone. § 12. Errect or HEATING AND CooLING AN AIR-COAL Gas Fuame. A graphite rod (fig. 2) of semicircular section, 3 milli- metres in diameter and 15 centimetres in length, is placed in the mantle just above the explosion region of a flame Iron Vapour in Air-Coal Gas Flame. 237 given by burner No. 1*. The round side of the rod faces the burner, and on the flat top is fixed a strip of asbestos in order to reduce loss of heat by radiation. The spectro- graphie observations were made in the direction of the explosion region and as near to the lower surface of the Fig. 2. | -Mantle | Asbestos Graphite Fied fBurner Heating of air-coal gas flame. graphite rod as practicable. A continuous current of 25 amps., which passed through the graphite rod, kept it at a bright red heat. ea? e—, o—O- o + (41) Hence for this case we have the special solution tae V7 (1+ 8)u mt Ti Te Fey (42) a ae V(1+6)u . mt a 97 Sn Mb mee cmaerrg eos) - (43) 2 256 Prof. Barton and Miss Browning on Coupled The ratios of the amplitudes of the quick vibrations to those of the slow ones are seen to be 1:+/%(1+8).. Ses in the y and z vibrations respectively. (ii.) Single Displacement.—Now let one bob be pulled horizontally aside while the other is held in the zero position, both constraints ceasing at the same instant. We may thus write dz dy _o — =(), for ¢=0.. . (45) oc 2=0, i Ta These conditions inserted in (36)-(39) yield equations satishied by (46) And these values in (36) and (37) give, for this mode of starting, the special solution y= —_— 5.008 mt +- 5° an F 6 6 (47) ga 5 cos mt + “00s ERE ; - (48) Thus the ratios of amplitudes of quick and slow vibrations in the y and ¢ traces are respectively Fle ere That is, the amplitudes of the superposed vibrations are numerically equal for any values of the coupling. The symmetry of the equations shows that the motions will interchange simply if the other pendulums be struck or displaced. Gii.) Double Displacement.—Let one bob be drawn hori- zontally aside, the other hanging motionless in its equilibrium but slightly-displaced position. Thus if z=b it follows that y= ean the other conditions being Ve eee ei) | dy _ dim | Ti =( and oh = |), } Vibrations elucidated by Simple Eaperiments. 257 Putting these in (36)-(39) we obtain equations which are satisfied by em nig ee 2438” e= 9) O58 71: (51) Hence the corresponding special solution may be written Pe b | 1+ 8 met x) y=— _-_ wae (52) | b +2 mt z= ya gost + 5 eo 4/1 +B) By ee (53) Accordingly the ratio of amplitudes of quick and slow vibrations in the y and z traces are respectively TO). 0) ED) (iv.) Cases of Single Frequency.—It may be seen imme- diately that if the phases are opposite and amplitudes equal the strut CC’ and the bridles ACA and A’C’A’ will all remain at rest. We should accordingly have the quick vibrations alone, which are proportional to cosmt. Again, if the phases are alike and the amplitudes equal we should have both pendulums swinging in unison, each bridle and its lower thread forming one plane. The vibrations are accordingly of the slower type of period ,/(1+()-times the other just noticed. V. THEORY OF CoRD AND LATH PENDULUMS. Description—The model here called the cord and lath pendulum consists essentially of two pendulums PR and QS, one suspended by cords from a movable point R on the lath of the other as shown in fig. 3. The bobs are shown by P and Q, the movable and fixed points of suspension by Rand 8. The cord pendulum is shown by PR. When the apparatus is used for giving double sand-traces a system of four cords and three stretchers is used so as to clear the upper board which takes the trace from the bob Q. Each cord is pro- vided with a tightener for adjusting its length. Of the two upper stretchers the back one is shown in the figure a little shorter and lower so as to make it and its cords visible. In the actual model the cords and stretchers are all symmetrical. The point of suspension R is adjustable on the lath of the pendulum QS by a small sliding metal sleeve with studs to support the cords. This sleeve is set where desired and then fixed in position by a screw-clamp. The suspension S of the 258 Prof. Barton and Miss Browning on Coupled lath pendulum consists of two screw points resting in a hole and slot in a metal plate. The two bobs P and Q are the same as those used in the double-cord pendulum, viz., metal rings with glass funnels for sand. The boards shown just Fig. 3. Cord and Lath Pendulum. beneath the funnels are fixed on the same wooden carriage and are capable of simultaneous slow motions on horizontal rails perpendicular to the plane of the diagram. This motion is effected by the rotation of a wheel whose axle winds a cord attached to the carriage. The whole arrangement is shown by the photographic reproduction fig. 5 of Plate IV. Equations of Motion and Coupling.—For the cord and lath peudulums let the masses of the bobs be P and Q, the lengths of the simple pendulums equivalent to PR and QS be respectively r and s and at time ¢ let their angular displacements be 6 and 7, the linear displacements of their bobs being y and z. Further, let as denote the distance RS between the movable and fixed points of suspension. Then for small oscillations we may assimilate sines to angles and regards the total tensions of the cords as always Vibrations elucidated by Simple Experiments. 259 equal to the weights of the bobs simply. The equations of motion may accordingly be written as follows :— d? P—5+Py0=0 and QE + Qg=Py(O—Wa: or Pot 4 Phy =PLac, ot aa aaa eee) @: (P Pa? QS + (F8 A \ge=Play. . (66) s The coefficient of coupling y is given by Psa? ~ Cast = Q)r+] + Psa” In the special case where P=Q, and r=s, which will henceforth be adhered to, if we write m? for g/, the above become (97) oY my = ie cee NEI Men ae eset fo) = + (l Patel 2=miay, ... >. (59) 2 and eo 60) lt+eat+a? So for very small values of «, y=« nearly. Solution and Frequencies for Equal Bobs and Lengths.— To solve these equations try in (58) esis.) |) s, -\e8 Gall aera COE) This gives 3 w+m 2 ea eae (62) Then (61) and (62) in (59) give the auxiliary equation in 2, w+ a(2+a+a?)m*+614+a)m*=0. . . (63) If we write this in the form aa peer pg? =0, |. 3 ao (64) we see that Gp G2... s ie fot ays too) 260 Prof. Barton and Miss Browning on Coupled Hence, on inserting the four arbitrary constants, we may write the general tion and its first derivatives as follows :— yaBiin pet) Fin a4, ee 6 (G3) 1 Pee eee vr 67 Ze in Cott eye sin (qg¢+¢), . (67) CL = pE cos ( (ee anPone4 by es Dias —— PT ein yee 1’ GF cos (qt+$). (69) Returning to the comparison of (63) and (64) we see that p+g=(2+4a+a7)m?=m? § say, Y and 0302 et a) ne eave yh oe) whence (p+¢q) = m(6+2n), | | @1 2 Gon, = 5 1/8 +20) +4/(S—2n) }, l e (72) : ee | and . p_Je+m+yG—% : > a/(O4 29)=\/(8— 29) An alternative method is to eliminate g? between the equations (70), thus obtaining the quadratic in p’, pr—(2+ a+ a*)m*p? + (L+a)mi=0. 93 29) Thus, calling the larger root of this p? and the smaller gq’, we have piami(14 5 + 5) + Ve +2a+5), aH (0D) ge=m?(1+ 5 5)- me Ha +2e+-5), | whence f2tatearta(e’?+2e+5)2 (76) q= l 2+a+ta?—a(a?+2a+5)2 which agrees with (73). Thus by (72) or (75) we see that the superposed vibrations Vibrations elucidated by Simple Experiments. 261 of the coupled system have frequencies which differ from each other and from those of the separate systems even when they are alike. Hence, in these broad features this system of the cord and lath pendulum is in agreement with the electrical system of coupled circuits. But in the closer details ditferences show oo as may be noted by comparison of (75) with 18 Imtial Conditions.—Since for this cord and lath pendulum the equations are not symmetrical, the phenomena may depend upon which bob receives a blow or displacement when starting. We accordingly treat each in turn. GQ.) Upper Bob Struck.—We may here write as follows :— dy _» &% pe, Ti Gr = 10%) b=O. ce (77) These conditions in (66)—(69) give equations satisfied by —mav mee PEEP PF) So, inserting these values in (66) and (67), we have for the special solution e=0, =0, (78) 2 a ane + — sin gt aCe a Pai 1 79 (p?—m?)v . (m?—Q*)v . ae = agp Pt ag ih HF If the amplitudes of the quick and slow 7—_, F= 45... (82 (p'— 9") p (p?—9")q (82) 262 Prof. Barton and Miss Browning on Coupled And these values put in (66) and (67) give the special solution rane 1" sin pe t+ roy sin at 33) Gar @Fcan. oe ss — tap Sin ett Saag agg singe So | sp m* —q")q pane a (=a and fp i aaa Note the contrast of (84) with (80). (iii.) Upper Bob Displaced : Lower Free.—This case may be represented by dy dz esi, 0) a —=(0)) =, =0, for t=0l-2 eo These put in (66) to (69) give equations satisfied by Po mins led ye CT Oe $= 5: p—a Tease ; (36) These values in (66) and (67) give the special solution 2 Z y= ae Toe COSMO Ua gen 3 COS gt, pa Gk: (87) (pami)gb (m= me (p?— @?)m? © Spit (p Ey 7.008 gf, So Hoe oq & SP ee FTP Note (iv.) Lower Bob Displaced: Upper Free.—lLet the dis- placement (a) of the lower bob P be produced by a horizontal torce. Then the corresponding value of the displacement (z) of Q when at rest can be found statically We thus obtain ad dy . dz ee de ye ae These conditions inserted in (66) to (69) give equations satisfied by =O, 2= =(), for ¢= O05 589) + (1ta)m?—(1+a+a2)q? =) py eae ee oo Gta | gay ye ne (Lta+a%)p'—UA+a)m? | | a ee) Vibrations elucidated by Simple Experiments. 263 These values put in (66) and (67) give the special solution (l + a)m?—(1l+a+a’)q’ (L+a+a*)(p?—q’) (Lta+ta’)—(1+a)m? (+ eee" ) (p?—m?)$ (1+ a)m’?—(l+ae+a’)q’ ; a COs pt Ba (It) a COS Gt, a COS pt (m?—q?)§(1+a+2")p?— (L+a)m?? 2 n’a(l+ata?) (p?—g¢’) COR ale So the ratios of the amplitudes of the quick and slow vibrations in the y and z motions are given respectively by EK (l+a)m*= (1+ 2+ 2*)q? FU (1l4+a+a?)p?—(1+a)m” (93) and Ho (m?—@){(Lta+a)p?—(L+a)mt Note the contrast of (93) and (94) with (88). G — (p?—m?){(1+2)m?—(1+a+a*)q?} (94) VI. Comparison OF THE Two TyPEs oF CoUPLED- PENDULUMS. Referring to equations (27) and (28), and (55) and (56), we see that the two types of model under examination have equations of motion of the form d? ae ee a) 2 Qe). |.) Ge) And in each case we have for the coupling y the relation RB ve AC’ dua) aioe see oe ee (97) These may be compared with equations (1)-(3) for the electrical case. It is there seen that the coupling involves the inductances L and N but is independent of the capacities. Whereas in (95) and (96) the A and C involve g/l, g/r, and g/s (which are comparable to the reciprocals of the capacities) as well as the masses P and Q (comparable to Land N). m?a(1 + a+ a?) (p*—Qq’) \ (92) 264 Prof. Barton and Miss Browning on Coupled Though both the types of pendulum are broadly alike and fall equally under the equations (95) and (96), their individual details, as dependent en the values of A, B, and C, are some- what different as already shown in the separate examinations. It is specially noticeable that in the double-cord pendulum, with equal pendulum lengths and masses, everything is inter- changeable and, of the two superposed vibrations when ~ coupled, one has the unaltered period of the pendulums if separated. In the cord and lath pendulum there is no such interchangeability, and both the vibrations superposed when coupled differ in period from those which would occur if the pendulums were separated. This is more like the electrical case. VII. Forcep VIBRATIONS : SPECIAL CASE OF COUPLED VIBRATIONS. It is interesting to note how the case of coupled vibrations reduces to that of forced vibrations when the coupling is small and the driving mass is much greater than the driven mass whose vibrations are forced. Thus in equations (95) and (96) let B be small but so as to be appreciable with respect to the very small mass P but inappreciable with respect to the much larger mass Q. Then (96) reduces to 2 Q& +02=0, OS ee giving as a solution z= K sin ntygay. cq ee Then, this used in the right side of (95) gives dy : ! : Pie +Ay=BK sin nt, 1 2 eee which is one form of the equation of motion for forced vibrations. By hanging a simple pendulum with bob of very small mass near A of the double-cord pendulum, we could imitate the experiment in which Dr. Fleming’s Cymometer detects by resonance the two superposed vibrations in a pair of closely- coupled inductive circuits. The pendulum imitating the cymometer would show maximum responses for two lengths corresponding to the quick and slow vibrations. For cases where P is much less than Q and B small but not quite negligible in comparison with either, the equations (27)-(29) for the double-cord pendulum and (55)-(57) for Vibrations elucidated by Simple Experiments. 265 the cord and lath pendulum assume simpler forms. It is particularly noticeable that the couplings in the two cases are then given by emer.) 008 aie. 2 0/ CLO) and ea es sys eaeitesy +), (LOZ) thus reducing to the same form for each type of pendulum. VIII. ExpermentaL MetHops AnD RESULTS. Double-Cord Pendulum.—This pendulum was arranged with its bobs near the floor, along which a black board on wheels was slowly pulled by cords so as to receive sand- traces. The board with the sand-traces was then placed on the floor at the foot of a rising stand carrying a camera by which the photographs were taken. The relations were calculated from the theory already given so as to obtain any desired values of the couplings. The total height BD (see fig. 1) was 229 cm. The results are shown in Table I. TABLE I1.—Double-Cord Pendulum. ; : Amplitude C ; Eeatio 2 ‘Actual Droop JSG Meg Ratio when oupling | CD: BC, or ce BD or Ap* for Ratio Baus SEE z mae wl 32 .3eie BD=229 cm. Dye H x P= T-y +8 ity ep ele eee aare: Per cent, cm. 5 2/19 0:095 21°81 1:051 0°952 10 2/9 0182 41°63 1:105 0-905 15 / 6/17 0°261 59°77 1:163 0-860 20 | 1/2 0°333 76°33 1°225 | 0816 25 / 2/3 0-400 91°60 1:291 | 0-775 335 1 0:500 114°5 1:414 0:707 40 4/3 0571 130°86 1527 | 0°654 S00” | 2 | 0667 152°7 1-732 0:577 GO 4: | 3 ) On 171-5 2 | 0-500 Length Found to give >60 | 173 2 | | * For Laboratory work it would be convenient to have a lath of length BD with the positions of C marked on it for each desired coupling. Plate V. shows sixteen reproductions of the double traces obtained from this pendulum. The couplings used are in- dicated on each figure as a percentage. ‘The letters F or W indicate that one pendulum-bob was started by a blow from the jinger or from a wood-block. A small circle and stroke show that one pendulum-bob was drawn aside and let go 266 Prof. Barton and Miss Browning on Coupled from rest. A small circle against the other trace indicates that it was held in the zero position and both bobs let go together. Arrows along the traces show the sense in which they were described. In some experiments the board was drawn along before the bob was let go. These traces accordingly show their own initial conditions. Looking at figs. 1-9 of Plate V. we may trace the gradual change from a 5 per cent. coupling to one of 60 per cent. The contrast between the first and last is very striking, and at first glance it would seem impossible to bridge the gulf. The first figure with loose (or moderate) coupling exhibits the phenomena of beats and the slow surging of the energy to and fro between the bobs. The vibrations appear to be practically simple harmonic throughout but of slowly- changing amplitude. This is the natural consequence of the superposition of vibrations of only slightly different periods. (See Table I.) Moreover, while these beats are recognizable the numbers of vibrations from node to node are in accord- ance with the theory. Thus, for fig. 1 the first line of Table I. gives the ratio of frequencies as 1:05 or 21: 20, and the traces show the correct number of vibrations between the nodes. So with the others. It is perhaps worth men- tioning that some early traces for the 5 per cent. coupling were found to give about 27 waves from node to node. This was disconcerting, as the following couplings were right. It was thought that the connector CC’ between the two cords might allow a little lost motion or back lash. It was examined and found not to fit quite tightly, though possibly friction might prevent any shake. Of course, if there were lost motion, a theoretical 5 per cent. might be reduced to an actual 4 per cent. (or less), and the number of waves in- creased to 25 (or more). The connector was accordingly made quite tight and traces again taken. They still showed about 27 waves from node to node. But it was soon dis- covered that an electric lead crossing the room for another research had sagged and touched the bridle cords, and thus very slightly vitiated their motion. This disturbance was removed and the trace taken which appears as fig. 1, showing the correct 20 waves. Turning now to fig. 9 we see that it shows nearly, but not quite, the compound harmonic curve characteristic of a tone and its octave. If the 2:1 relation were exact the kink would not wander on the main curve, but would reappear each time in the same position. Now this case corresponds with the ninth line of Table I., and was calculated to give a frequency ratio of 2:1, but on the supposition that the Vibrations elucidated by Simple Experiments. 267 masses of the bridle cords ACA, A’C'A’, and connector CC’ were negligible. Now for the higher values of the couplings where the droops are very great this can hardly be the case. Hence it is not surprising to find that the droop had to be made (as shown in the Table) 173 cm. instead of the cal- culated 171°5 cm. to give the exact 2:1 ratio. This arrange- ment furnished fig. 10, in which the kink scarcely wanders perceptibly. The board was also turned through a right angle and left stationary while the pendulums were used as Blackburn’s pendulums. They thus gave simultaneously the patterns in the right-hand top corner of the figure and retraced them almost perfectly about ten times. The gradual change from the case of beats in fig. 1 to the exact tone and octave of fig. 10 is very instructive. It also shows the advantage of taking traces, for in watching the motions of the intermediate cases the eye fails to realize exactly what is happening, though the extremes are easy to recognize. Several of the remaining traces for this pendulum are for the same coupling, but are varied only by different initial conditions: Thus fig. 11 has the bob held at zero while the other is pulled aside, then both let go together. (See equations (51)-(54).) In fig. 1 one is allowed to hang freely at rest (but somewhat displaced), while the other is held aside and then let go. (See equations (47)-(49).) Figs. 13 and 14 each show simple vibrations instead of two superposed. Fig. 13 was obtained by starting with unlike equal displacements due to a connecting thread, which was burnt when all was still. Fig. 14 was obtained by swinging with the hands till the amplitudes were equal and phases alike instead of opposite as in the previous case. Pains were taken to draw the board at the same uniform rate by scale and stop-watch in figs. 13 and 14. Thus the relation of periods 1:2 is correctly exhibited along with the phases. Figs. 15 and 16 show for two important couplings the initial conditions of double displacement, one bob pulled aside the other hanging freely at rest as in fig. 12, but here the board was moved before the bob was let go, as seen by the traces. In every case in which we tested them the traces were found to give a satisfactory agreement with the theory both qualitatively and quantitatively. These tests were carried out by plotting on squared paper the theoretical curves to be expected and then comparing them with the sand-traces obtained. 268 Prof. Barton and Miss Browning on Coupled Cord and Lath Pendulum.—In the present work this arrangement was used with the lengths, r and s, of the pen- dulums equal and the masses, P and Q, of the bobs equal. Accordingly the equations (58)-(60) apply and also those based upon them. Calculating from these we derive the values given in Table II. The length of the simple pen- dulum equivalent to one of the lath pendulums used alone was 112 cm. The lengths SR (=112.«) were accordingly calculated and are shown in the Table. The positions of R for the various desired couplings were then marked on the lath to facilitate setting. For completeness’ sake several values of coupling are included in the table, though they have not been used with the present model. For example, it is difficult to set the coupling for values less than 15 per cent. or greater than 55 per cent. TasLE [1.—Cord and Lath Pendulums. Coupling Ratio RS: QS Length SR Y. =f, =a X112 em. Per cent. cm. 0051331 5°75 10 0105680 11°84 15 0°163660 18°33 20 0:226020 25°31 25 0°293675 32°89 30 0°367800 41:19 35 0449612 50°36 40 0°541945 60°70 45 0°646610 72°42 50 0°767591 85:97 55 0910181 101-94 577 1 112 Plate VI. presents fourteen sets of traces obtained with various couplings or under different initial conditions. Of these the first ten, figs. 17 to 26, show single traces from the lower bob when the upper bob was struck; the couplings vary from 15 to 55 per cent. In fig. 24 the coupling was about 48 per cent., the adjustment was made by trial to give 2:1 as the ratio of frequencies. To show that the exact ratio was practically attained two lines of traces were taken, 12 waves occurring between them. It is seen that the kink shifts its position but slightly on the main wave after the lapse of 23 of the long periods. To trace the slow shift of the kink because the 2:1 relation is not fulfilled the figures for the 5C and 55 per cent. are taken in duplicate or Vibrations elucidated by Simple Experiments. 269 triplicate, the board being again passed under while the penduium was still swinging unchecked. Figs. 27-30 on the bottom row of Plate VI. show double traces obtained simultaneously from the upper and lower bobs, the two boards being placed together in right relation for photography. ‘The coupling is the same in each, viz., of the order 48 per cent. as to give approximately 2:1 as the ratio of frequencies. Figs. 27 and 28 show very plainly that the bob which is not struck executes the compound harmonic motion whether it is lower or upper, but that the bob which is struck has a very different motion according as it is upper or lower. See equations (77) to (84). Figs. 29 and 30 show the results of pulling one bob aside, the other being at rest in its equilibrium position as dealt with in the theory. See equations (85)-(94). The pulling aside was effected by a horizontal thread which was burnt when all was steady. LX. SuMMARY. 1. The paper describes two types of coupled pendulums which are considered useful both for lecture demonstration and for quantitative work in the laboratory, especially as they illustrate many important points in the phenomena of inductively-coupled electrical circuits. 2. One of these, called the doubie-cord pendulum, is like a pair of Blackburn’s pendulums in parallel planes connected by a stiff tube at the droop of the bridles. Its vibrations occur perpendicularly to these parallel planes. It may be used with couplings gradually varied from very loose to very tight, say from 1 to 60 percent. It exhibits, by double sand- traces simultaneously formed on a moving board, the gradual change of the vibrations from the phenomena of slow beats to those of compound harmonic motion. Most of the curves, however, show the superposition of two simple vibrations of incommensurable frequencies. The pendulum shows also the effects of different initial conditions. This form is specially suitable for laboratory work, as it may be set up from the simplest materials and yet give results of distinct and quantitative value. If the lengths are equal and the masses also, then the vibrations of this pendulum are quite interchangeable. Photographic reproductions are given of sixteen double traces obtained with this pendulum. 3. The other form of apparatus, or cord and lath pen- dulum, is yet easier to use for simple lecture illustration, but Phil. Mag. 8. 6. Vol. 34. No. 202. Oct. 1917. U 270 Prof. Barkla and Miss White on the requires more careful installation for the best work in the laboratory. It consists of a simple pendulum, with a light lath for its rod, from a movable stud on which hangs the cord pendulum. Even if the lengths are made equal, and also the masses, the vibrations of this pendulum are not interchangeable. The effects of various initial conditions are accordingly more striking. It may be used for couplings varying from about 10 to 55 per cent. Photographic reproductions are given of ten single and four double traces obtained with this pendulum. 4, The mathematical theories of both pendulums are deve- loped and compared with each other and with the theory of the electrical case it is sought to represent. The experiments are in satisfactory agreement with theory, though this is not always immediately obvious. 5. In the experiments and most of the theory the lengths of the pendulums have been equal and also the masses of the bobs. A large field lies ready for exploration in the cases where the quantities of both classes are unequal and varied at will. These cases are reserved for later papers. University College, Nottingham, July 18, 1917. XXIX. Notes on the Absorption and Scattering of X-Rays» and the Characteristic Radiations of J Series. By C. G. Barkua, /.R.S., and Margaret P. Warts, M.A., B.Sc., University of Edinburgh™. Introduction. A ieee experimental investigation of the absorption of X-rays of short wave-length is of importance in a study both of the theories of electromagnetic radiation and of atomic structure. The experimental work on this subject, the results of which are given in this paper, was suggested—indeed forced upon us—by the discovery that the observed intensity of corpuscular radiation from air was not what was to be expected on the simple theory of emission of fluorescent and corpuscular radiations as given in the Bakerian Lecture for 19167. Anirregularity appeared in the results which could be explained on the assumption that the light elements emitted under suitable stimulus a hitherto unobserved characteristic X-radiation—(a J radiation). Such * Communicated by the Authors. + Proc. Roy. Soc. A. xcii. May 25, 1916; Phil. Trans. A. 217. pp. 315— 360. Preliminary notes in ‘ Nature,’ Feb. 18 and March 4, 1915. Absorption and Scattering of X-Rays. 271 a radiation was looked for and found, as already briefly announced. The evidence obtained from absorption expe- riments is given below*. But in addition, there are in these results a number of other features which are of considerable theoretical interest, and to which attention shoald be drawn. EHapervments. Of the experiments little need be said. The precautions essential to the accurate determination of an absorption coefiicient have frequently been stated{. The principal danger is that of allowing secondary rays of any type from the absorbing substance to enter the ionization chamber in which the intensity of transmitted radiation is measured. The absorption coefficient ~ of a homogeneous radiation is defined by the equation 1=I,e~“*, where I is the intensity of the primary radiation of initial intensity I, still proceeding as primary radiation after traversing a thickness « of absorbing material f. In these experiments the radiation employed was almost invariably a primary radiation from which the more easily absorbed constituents had been previously eliminated. In the case of the most penetrating radiations, the remaining radiation upon which the experiments were made was remarkably homo- geneous. In order to reduce any error due to heterogeneity to a minimum, the thickness of the absorbing sheet was adjusted to absorb approximately the same fraction (50 per cent.) of the incident radiation. A few experiments were also made with characteristic radiations (K series) as primary radiations. The results obtained agreed almost perfectly with those obtained with the primary radiations direct from an X-ray tube §. * Indications of the existence of such a radiation had been obtained by one of us many years before. Some of these were recorded; others not. But the evidence was insufficient to show either that the radiation emitted was really a characteristic radiation or that it could be in any way identified or classified. Indeed the probable significance of these results had been lost sight of until associated irregularities became a serious hindrance to the development of the theory. It seems highly probable that these J radiations account in part for some of the discre- pancies between the results obtained by some experimenters and those of the writer on the scattered radiation. By adjustment of the primary radiation care had been taken by the writer to keep the scattered radiation free from admixture with such a characteristic radiation. + Barkla and Sadler, Phil. Mag. May 1909. + » isa measure of the rate of loss of intensity of the transmitted beam, whether by scattering or any other process. § Except at a “discontinuity” where absorptions are exceedingly sensitive to small variations of wave-length. In this case the discrepancy was a regular one. U 2 272 Prof. Barkla and Miss White on the As absorbing substances, copper, aluminium, water, paraffin- wax, and filter-paper were used. The results are given in : Table I. TABLE I. Wave-length Wave-length (u/e) (u/o) (4/0 ) (wu : a (Hull & Rice), (Siegbahn). Copper. | Aluminium. Wit ee Pe ax} a 09 TORS em OOD | ON Maines URE Ve eter “547 i SOL Sale 18:10 en *O894 Hl) eae ‘317 499, Ze oO AV aeaeceteee “524 Beso vas 16:4 NIEayy MMM Deane cer i ‘307 Olay 15:8 Cee Gee ee "302 VELL ois LAY -479 1 468 —,, 45> ta lednenieece 473 ‘467 so, 14:40) Wivbncesest ay Pan eo ere ‘479 POO ainees OTE | Momameeeces 436 Oy ee 13:95 AO ee ae eee 396 Loar yyy 13°10 1:28 pea Mer ive ee ‘279 ‘4440 | 12°20 a ‘267 ASO) hy 11°6 1:15 ‘400 -400 2638 ASB ae Te eRe ian eas ‘388 426s, TODS 1 Gale Sei aes 351 “AEs, 10:40 106 9.5) ceecea eae ‘252 “4108 10 9:98 | hake’ 371 ATS lane 9:60 1:03 | ALOU Wis O31 b\w ocgn ol ae 266 409, 6:99, ee *349 | 403, 7d a PE ebb nace soci: 353 ADO: 8:55 Ey eee -329 396) iy Bh bo caceges MON ie eer cil un | 258 "BOA m2 8-04 847 | 390) ” "7 "82 Aathode 302 i 389, iret Ta oe 398 q Peet Males 761 "853 pepe lmuialbine dk | 263 4 Bear (i Ay. |) ie enn on meee 389 iq BBW uk 715 815 Hl oS Ini el wei G94 nee ec SS ee 954. r 376. Wee G90 Wee 335 : 368, 6°38 707 SG te 5:08" |/ sual intel ae aeee 361 ‘3601078 em.| -360__,, 5:92 766 i 359 ” 399 ” Beit ecccee 311 j 258 N 258 i Sol BRUSH Renamer e (tlunoroao el Nh occ e "232 i. SOD MMs ati IAs BBS lo. eivabagthcvce, Mulaeeete a ne 298 E41, 302) ee 5:40 COMER AR hth ves 330 346, EEO SUT Ween 314 | ‘B45. ‘B45 1) 5:00 713 yi B48.) sy PMR A ah pecan 499°) Oe a 232 b OEM aM NOL RMN ante usc: 4°71 643 | Sy MRR DIR EMC ke CBS ALE UN) 4:70 CT EC ee a 212 S20) Ue at Mee eae Ae 5A | Wioenstae 270 P ‘307 G1 il. Ue Ueselianetavetstete laters 4:02 PEN hele cllbdy, GOQ0 ON 309 4 £210 RIB SUD MR MI h oh SHrly/ 519 tS] Ne aM RO ee 0 : Sor ayn eae il POU Masher shui is sete S268. se) dd Mea eee ‘280 VS DA RRONA ME ay) PERE ee 1 Ber die MME ie Ihe mbOae s 217 SO MMag Yea ili, | aaeesemele MESON hail i judoncee ost Wave-length (Hull & Rice). 277 < 10-8 em. | -276 266 | +262 Absorption and Scattering of X-Rays. 2738 Wave-length (p/p) (z/¢) (u/o) | (n/p) (Siegbahn) Copper. | Aluminium ater. | Paper. a eee 3°08 yay toh: bees re IS SAR oe ‘261 Rene Oy au. | ORG 391 226 Berks 2S 1 eiNee “ate Nee |, 28S), 382 BMD I). i sacs enc PABST Raa "229 MR oh) so. -+--2- | ee "Hae eta NEG i aa y CO ee aaa 2 Rane 245 1 2 ional ss | oO ie HERR "252 eee PeSOr nite cos: ‘203 POMS UL) ory win ose UST. am “214 2 ae ey |, ane 204 a ECORI s.o 0 fos weasel We aeaee PMID yh) inet. 154 "263 en TD) | Se ae 297 i Mr eee 1-29 OS eyed ce oe i PE ns peices 1.0) ae HOG cr oehena 6 Sennen TOM... IRL ATeOR MCE en esse Osman 210 | : esa TOA Weis: lapels 186 a ee ‘950 BGP on i es. Omni 153 For various reasons the absorbability of X-rays has usually been measured in aluminium. Apart from practical reasons, aluminium was one of the most suitable substances to use because no characteristic radiation from aluminium had been observed within the range of wave-lengths of X-rays in common use. In other words, the wave-length was a simple single-valued function of the absorption in aluminium. Our experiments, however, have shown that aluminium emits a characteristic radiation of wave-length about °37 x 107° cm.; it is therefore not a suitable substance in which to measure the general absorbability of radiation through a range of wave-lengths such as that used in these experiments. The same objection does not hold in the case of copper, because its K spectral lines are fairly distant on the longer wave-length side and its J spectral lines (assuming they exist) must be somewhere on the shorter wave-length side of the J radiations of the other substances dealt with. We have, therefore, plotted absorptions in aluminium, water, filter-paper, and paraffin-wax as ordinates against absorptions in copper as abscissee—thus taking the last-named as the standard of general absorbability. The results are given in fig. 1. The scale of ordinates for filter-paper has been made 6/5 times as great as for the other three. This was necessary as the water and paper (n/p) Paraffin- Wax 218 195 192 Ht | Absorption (Yp). — un Oat 274 Prof. Barkla and Miss White on the absorptions were too nearly alike to show clearly on the same scale. The interesting features of these curves are described below. Fig. 1. Showing absorption curves in the neighbourhood of J spectral lines. Absorption in Copper (%). Characteristic Radiations of J Series. The simplicity of the form of each curve is broken by a feature which indicates the existence of an X-radiation characteristic of each substance. Similar curves were first shown by Barkla and Sadler* in association with the K-radiations, and by Barkla and Collier? in association with the L-radiations. No suchirregularity has ever been observed except in association with a characteristic X-radiation. The three “ discontinuities,” as we shall call them, being due to the elements aluminium, oxygen (in both water and paper), and carbon (in paraffin-wax), we see that the higher the atomic weight the more penetrating is the radiation at which the discontinuity occurs ; in other words, the higher the atomic weight of the element the higher is the frequency of its characteristic radiation. This is in harmony with the results obtained tor K and L radiations. [Any discontinuity in the absorption curve for hydrogen would be inappreciable and could not be detected in the curves for paper, water, and paraffin-wax, of which it isa constituent. That due to carbon is clearly marked in the * Phil. Mag. May 1909. + Phil. Mag. June 1912. Absorption and Scattering of X-Rays. 275 curve for paraftin-wax ; it must also occur in the curve for paper, but the “step” is only about half as big as in paraffin- wax, and the fractional change is smaller still. There appears to be indication of this step (see fig. 1), but it is within the limits of experimental error in this case, and has therefore not been shown in the curve. It would indeed be barely appreciable in the figure. | The wave-lengths obtained from the known relation between wave-length and absorption in aluminium are approximately”: Ate ce |, Absorption, Scattering, Fluorescence. One of us has previously pointed out that an X-radiation loses energy by two independent processes. One is the process which we may briefly call scattering; the other is that which is associated with the emission of corpuscular (electronic) radiations and their accompanying fluorescent (characteristic) X-radiations. The former absorption varies slowly with the wave-length —over certain regions it is approximately independent of wave-length; the latter varies very rapidly—over certain ranges it is approximately as the (wave-length)?. The former—produced by equal masses of different ele- ments—varies little with the atomic weight of the absorbing element; the latter considerably. * These values are somewhat lower than those previously given. There is, however, some uncertainty as to the absolute values,—though not so much as to relative values—owing to the fact that neither the cha- racteristic radiations nor the primary radiations used in these expe- riments were homogeneous. The values previously given as ‘56 x 107° cm. for nitrogen and ‘5x 107° for sulphur were for the constituents of longest wave-length (a lines), on the assumption that J radiations were of constitution similar to the K radiations. There are indications that the J radiations are more homogeneous than K radiations, and that the wave-lengths of the corre- sponding £ lines which are approximately ‘5 x 107* and 44x 107° cm. respectively, are more correctly the values to be assigned to nitrogen and sulphur by the ionization method. There is thus a small discrepancy between the values obtained by the ionization and absorption methods, but the variation of Ay with atomic weight is about the same in both. In the ionization experiments K characteristic radiations were used; in the absorption experiments, primary radiations. Equal average penetrating powers in these two radiations do not correspond to exactly the same effective wave-lengths. Greater accuracy will be obtained by the use of more homogeneous radiations, or by the interference method. 276 Prot. Barkla and Miss White on the As a consequence when the wave-length is small, scattering is the predominant cause of loss of energy except in the heaviest elements. In general, the lower the atomic weight the greater is the proportion of loss by scattering to the total loss of energy. Scattering and Absorption. In paraffin-wax, water, and paper scattering accounts for almost the whole “absorption” within the range of the shortest wave-lengths here dealt with. This is shown by the following facts :— (1) The total absorption coefficient in each substance varies little with the wave-length of the radiation absorbed ; (2) the total absorption coefficient is almost independent of the nature of the light-absorbing substance; (3) there is close agreement between the magnitude of the coefficient and that directly calculated from measurements of the energy of radiation scattered. The small variation of absorption in light elements with a change in wave-length is illustrated by a comparison of the absorptions in paraffin-wax and in copper—the former in- creasing only by about 20 per cent., while the absorption in copper increases by 400 per cent. The absorbability in light substances as paraffin-wax, paper, water, or even aluminium is thus very unsuitable as a measure of the character (wave-length) of such high- frequency radiations even though the effect of scattering is not lost sight of, as it unfortunately has been both in the derivation and in the application of formule connecting wave-lengths and absorptions. We see, too, that the absorption coefficients (u/p) in paraffin- wax, paper, and water all become only slightly less than °2 for X-radiation of very short wave-length. This agrees exceedingly well with the value -2 obtained many years ago* from measurement of the energy of the X-radiation of longer wave-length scattered in light substances (with the exception of hydrogen). Owing to the constituent hydrogen the scattering coefficients for these three substances as then determined for somewhat longer waves are slightly greater than °2. Further, the value ‘2 agrees almost perfectly with that obtained by calculation, on the assumption that the number of electrons per atom is equal to the atomic number. Or, putting it another way, the value -2 when applied to Sir J. J. Thomson’s calculated expression for the energy * Barkla, Phil. Mag. May 1904. Absorption and Scattering of X-Rays. 277 lost by scattering, gives with remarkable precision tue number of electrons per atom now accepted. This method was in fact the first to lead to the present conclusion*. All these results verify not only the general laws of scattering which one of us arrived at many years ago, but they indicate also the accuracy of the energy measurement made at that time. It will, however, be noticed that the coefficient of absorption in aluminium sinks quite appreciably below °2 to a value °15, and that there is a tendency to lower values still. Hqually low values have been obtained by Rutherford, Richardson, and Barnes ft, and by Hull and Rice f in experiments with the Coolidge tube. In addition the values of the coefficients of absorption of y rays obtained by Soddy and Russell, Ishino §, and others indicate that the simple law of scattering (approximate inde- pendence of wave-length) must break down for very short waves, for values of u/p as low as ‘04 have been obtained. We have already shown|| that it does not hold for much longer waves, and that this is to be expected theoretically; that although in light elements the scattering varies little with wave-length, yet there is a slight increase with an increase in wave-length of the radiation; that this increase becomes decidedly marked when the wave-length becomes comparable with the size of an atom ; that in heavy atoms the variation is most pronounced, the scattering in- creasing many fold; but that with shorter waves the mass scattering even in the heavier elements approaches equality with that in the light. ! In view of the facts stated above, it certainly looks as though we have in the absorption by aluminium, evidence of the beginning of a second marked deviation from the simple laws—that is, evidence that a diminution of scattering and absorption is setting in when the wave-length becomes small. Thisappears to be the link connecting the absorption et X-rays with the absorption of y rays. In this connexion it may be observed that the absorption of X-rays in aluminium appears to cross that in paraffin, paper, water, &c., and to reach a lower value. This we should expect it to do as the * As the values of e/m and e were obtained more accurately, re- determinations of the number of electrons per atom were made. See Barkla, Phil. Mag. May 1904; Jahrbuch der Radioaktivitit und Elektronik, April 1908; Phil. Mag. May 1911. + Phil. Mag. Sept. 1915. ft Phys. Review, Sept. 1916. § Phil. Mae. Jan. 1917. || Barkla and Dunlop, Phil. Mag. March 1916. 278 Prof. Barkla and Miss White on the other three substances through their constituent hydrogen contain slightly more electrons per unit mass, and so scatter to a greater extent mass for mass. Such a lower value appa- rently persists throughout the results of all experiments on the absorption of y rays (see Table II.). TaBueE II. Coefficients of Absorption of y Rays. (Soddy & Russell.) Dyeeubins Shane Thorium D, | Radium C. | Mesothorium 2.| Uranium X. L/p. lt/p. b/p. 1/0. Meremryeee acne eae "O472.2 4 2 aera "0612 bead bas aaa 0405 0438 0544 0636 dN RAR eA AR 0326 ‘0388 0421 0470 Zine Asean outs 033 | 03893 0424 0465 Coppetyccsce-eeee 0334 ‘0398 0423 0472 Brass: Niece eee "0325 0389 0425 ‘0470 Tron ty.8; aah hoe 0328 ‘0399 0415 0472 Sulphures seers ‘0369 0438 0465 0516 Aluminium ...... 0324 0406 0421 0469 Slater ima ce sire 0337 "OF Fle Og nee reeenee 0469 Glassen. ce tke Meee 0352 0416 "0448 0484 Magnesia Brick... CBO. 9 is ners .0469 0478 Paraffin-Wax ... 0361 0464 0580 0502 Absorption and Scattering of y Rays. The explanation of the approximate equality of the mass coefficients of absorption in different substances of X-rays of short wave-length thus leads to the explanation also of the approximate equality in various substances of the mass coefficient of absorption of y rays. The results with these short waves bridge the gap between the very different absorptions in various substances of X-rays of longer wave- length and the approximately equal absorptions of y rays in these substances. Evidently the absorption of y rays in substances of low atomic weight is almost entirely absorp- tion by scattering, which is proportional to the number of electrons in the substance traversed*. Thus, as we see from Table II., u/p varies little with the atomic weight of the absorbing substance except in the case of the heaviest elements, in which the value may be ag much as 40 per cent. greater. It is probable, however, that the “‘ fluorescence absorption ” * Mr. Ishino thinks there is a large true absorption. This is probably true of mercury, the substance upon which he made further experiments. With regard to the lighter elements, all evidence appears to point the other way. Absorption and Scattering of X-Rays. 279 is not negligible in these heavier substances, as it is not even in copper when high-frequency X-rays are employed (see Table I.). A point worthy of notice is that as the wave-length of the X-radiation diminishes, after the mass absorption coefficients in various substances become approximately equal—owing to the “fluorescence absorption”’ becoming very small—the absorptions diminish in different substances at approximately the same rate. This indicates that the scattering diminishes at approximately the same rate in different substances with a diminution of wave-length. On the other hand, with longer waves the scattering increases with the wave-length at very different rates in different substances. This latter variation, as we have ex- plained, is probably due to the electrons acting in groups rather than individually. The variations observed—(or rather inferred from ab- sorption experiments, for it has not been observed directly) —when the waves are short, must have another explanation. It may be connected with the fact that when the wave-length is short, the deviation of the observed distribution of X-rays scattered around the substance traversed from that given by the simple theory becomes very marked. It seems certain that scattering loses its simple character. The scattering of y and X-rays may thus be considered in three stages. With very short waves (y rays) the scattering increases with the wave-length in all substances at approximately the same rate, until the mass scattering coefficient approaches °2. Secondly, there is little or no variation of scattering with wave-length over a long range of wave-lengths in light elements, over a shorter range of wave-lengths in heavier elements. Finally, the scattering increases with wave-length, slightly in light elements, more rapidly in heavier elements. The second and third stages have already been explained. The first stage needs further investigation. Absorption associated with X-ray Fluorescence. Absorption associated with the phenomenon of X-ray fluorescence may be analysed into several distinct and evidently independent absorptions connected with various sets of electrons within the atom. ‘Thus there is the K absorption which is that associated with the emission of a particular group of electrons which we call the K electrons because their emission is associated also with the emission 280 Prof. Barkla and Miss White on the of the K fluorescent X-radiation ; similarly there are the J, L, M,...absorptions associated with the emission of AP De eae electrons, and J, L, M,... fluorescent (cha- racteristic) radiations * ‘The total mass absorption coetlicient of X-rays associated with fluorescence in any element may thus be written ee pHINP. Pp where Tx/p is the absorption coefficient definitely associated with the emission of the K characteristic radiation, Ti/p 18. the absorption coefficient associated with the emission of L radiation, and so on for each of the terms. Hach of the terms Tj, Tx, Ti, &c. is zero when the wave- length of the primary radiation is greater than that of the constituent of shortest wave-length in the corresponding characteristic radiation ; as the wave-length of the primary radiation becomes shorter, the term 7x or 7, suddenly rises and soon begins to diminish again with the wave-length, continuing to do so without limit. As all the & absorption curves are similar in form, 7x/p 1s a function of A/Ag, thus for any element tx/p=/x(A/Ax), the constant k depending on the particular element. Similarly, t,/p=//.(A/Ax). Pre- sumably a similar relation may be found for ts/p. The functions /5, fs, ft,...though similar in features do not appear to be identical. Corresponding functions, however, are the same in different elements, and the relative values of the coefficients 7, &, 1, ... appear to be the same for different elements. It should, however, be pointed out that the simi- larity of corresponding functions and equality of ratios of the corresponding coefficients have not been observed for substances differing widely in atomic weight. Jt may be found that these observed laws are not perfectly general. The particular form of these functions of A/Ax cannot be simply expressed; indeed, the exact form is not known in the regions where J is slightly less than Ax, owing to lack of perfect homogeneity in the radiations upon which expe- riments have been made. It has been shown, however, to be the same for each element, for the form has been found to be the same when instead of perfectly homogeneous radiations similar beams of radiations of neighbouring wave- lengths have been employed. * For more complete account see Bakerian Lecture 1916, Phil. Trans. A, 217. pp. 815-860. Absorption and Scattering of X-Rays. 281 Absorption Formule. Various attempts have been made to express the absorption simply in terms of A, but without mentioning these in par- ticular it may be said that for the most part they give only rough averages over very limited regions ; they are respon- sible for many inaccuracies. As we have indicated, Total absorption = scattering absorption + fluorescence absorption ai is = g ae i ae p p 2 —2 Sy aa (2) 9 Pleats Ie. 2p | But a/p is a function both of the atomic weight (w) of the absorbing substance and of the wave-length (A) of the radiation. Thus (3) ; =F (w,) + [.. .jfr(r/ra) + hfs (d/Ax) + f(A) Aa) «J As the relative values of the constants j, 4, 1, &., seem to be the same for different substances, a single constant n depending on the particular atomic weight or atomic number may be placed outside the bracket, giving, within probable limits, (4) ‘i =F(o,r)+n[...fr'(A/rs) +f! (AfAw) + fl (A/An) +...) F'(@, X), however, varies little with the particular absorbing substance except when A is great. It is also approximately independent of » over a certain range of wave-lengths. The general form has already been indicated. Consequently within certain limits equation (4) may be written simply (5) F=C +kf(r), where k only depends on the particular P element *. Huli and Rice, from the experiments on the penetrating radiations from a Coolidge tube, have recently pointed out that except when A of the absorbed radiation is near the ‘(ujo—C)a * This we have previously written in the form ui Ole =k, aconstant for any two substances for any radiation of much more penetrating type than any characteristic radiation excited in either substance. 282 Prof. Barkla and Miss White on the wave-length of a characteristic radiation on the shorter wave-length side of the absorbing substance, /(A) becomes 2 simply. This is certainly very near to the truth when A is much less than Ax the wave-length of the K radiation of the absorbing substance. Aluminium is probably the best substance with which to test this relation, as the deter- minations of the absorption coefficient are probably more accurate than in any other substance. We have, therefore, plotted values of log (u/p—°2) in aluminium against logr over a long range of wave-lengths (39 to 2°28 x 107° cm.) greater than A; for aluminium, using Barkla and Sadler’s and our values for u/p and Siegbahn’s for X. The relation is not exactly a linear one and shows that if we write (w/p—°2) =Car”, nm is not a constant, but varies even over this range from about 2°7 to 3. (No importance can be attached to the exact value -2 as it is usually very small in comparison with p/p, and is only introduced here as indicating a physical fact of which account must be taken over other ranges.) ) We have shown, too, that the absorptions in other sub- stances are proportional to those in aluminium for radiations of wave-length considerably shorter than those of the nearest spectral line. Thus, if it were true for aluminium it would be true approximately for other substances. 7 Tt should be remembered, however, that such a simple relation as is obtained by giving n a constant value has not been found to hold accurately for wave-lengths anywhere really near to that of a K spectral line on its shorter wave- length side. In fact, if we can regard the absorption experiments as having given results approximating to those which would have been obtained with perfectly homogeneous radiations, the relation cannot hold even to a first approxi- mation when the wave-length is only slightly shorter than that of a K spectral line say. But further experiments are needed. Hull and Rice, from their experiments on the absorptions in aluminium, copper, and lead, showed that p/p—:12=kr? with a fair degree of accuracy and concluded that the scattering coefficient (o/p)=°12. It is probable, however, that a slight variation of (c/p) with X makes the seattering appear somewhat less than it actually is—at any rate with the longest waves used. It is highly improbable that the “ fluorescence absorption” in this region is so strictly proportional to the cube of the wave-length as to enable us to use this as a method of determining o/p with any accuracy. Absorption and Scattering of X-Rays. 283 The relation between /p and 2? is, however, so simple and convenient that we have plotted these two quantities for all the substances we have experimented upon. The wave-lengths of short waves have been obtained from Hull and Rice’s relation experimentally determined between wave-length and absorption in aluminium ; for longer waves Siegbahn’s wave-lengths (K,.; lines) corresponding to Barkla and Sadler’s and our absorption coefficients for the K charac- teristic radiations have been used. The two sets overlap through only a very short range of wave-lengths, and this range is a particularly critical range, but the agreement is remarkably close and by the graphic metnod the results may be made perfectly continuous*. The wave-lengths obtained from these relations are given in Table I. columns 1 and 2 respectively. The two sets of wave-length determinations are distinguished by dots and circles in figure 2, in which absorptions in copper, aluminium, water, and paraffin-wax are plotted against the (wave-length)?. The discontinuities in the aluminium, water, and paraffin curves indicate again the positions of the J spectral lines for aluminium, oxygen, and carbon. There is also a slight break in the copper curve suggesting a copper J radiation. But there may be some doubt as to this interpretation for the change is not so pronounced as we should have expected ; it appears somewhat near to the J lines for aluminium ; and the fact that the two sets of values are confined to the two sides of the discontinuity leads us to treat the result with some caution. On the other hand, there are features in the relative absorptions in copper and other substances which suggest a change in the copper absorption in the neighbourhood of this particular wave-length (see aluminium absorption curve, figure 1); and we have no reason, beyond those stated, to question the accuracy of the result. We thus see that over a limited range of wave-lengths the equation eae): (where & is a constant for a given substance) fairly accurately represents the experimentally observed relation—very much more accurately indeed than any simple formula of the type u/p=hxr”. It, however, does not express the fact of the variation of o with X such as has been described, nor does ‘A’ appear to hold at all accurately * Tt is unfortunate that Hull and Rice continued their experiments only up to the point where the discontinuity occurred both in aluminium and in copper. 284 Absorption and Scattering of X-Rays. except for wave-lengths far removed from those of the characteristic radiations excited in the absorbing substance. Fig. 2. 2:5 Absorption (40). Ws fe) Qe pen.0 rat tin Wax. Oe : a mea Ton (Wave-length)® There were one or two minor features in our experiments which we may just mention, though we have not yet had time to investigate them fully. In the absorption curve for paraffin, for instance, there appeared to be a consistent slight irregularity at certain wave-lengths. When the absorption (u/p) in copper was about 3:5, or A=*29 x 10~* cm., certain rather high values were obtained for the absorption in paraffin, values say 10 per cent. above the normal. There seems no doubt that these irregularities really exist, but a permanent increase in the absorption for wave-lengths ik ea a a Spectra of Helium, &§e. in the Ultra- Violet. 285 shorter than a certain value has not been observed. This, however, can be accounted for if corresponding radiations in other elements are of neighbouring wave-length. Similar irregularities were found in the absorptions by paper and water, both within the same region. {t seems just possible that these are connected with a spectral line or spectral lines of still higher frequency— possibly of an I series. This, however, is a point which would require careful investigation by another method. It should also be pointed out that though we have drawn smooth curves—for the most part straight lines—through observed points, this does not indicate that certain slight deviations are not real. We know certain of them to be so. but they are only small, as the figures show quite clearly. For the present they are of only secondary importance. We wish to acknowledge our indebtedness to Miss J, Dunlop for some preliminary work on the absorption of high-frequency X-rays. XXX. The Limiting Frequency in the Spectra of Helium, Hydrogen, and Mercury in the Extreme Ultra-Violet. By O. W. Ricuarpson, #.2.S., Wheatstone Professor of Physics in the University of London, and Lieut. C. B. Bazzont, General Staff A.HF., formerly Harrison Re- search Fellow of the University of Pennsylvania *. Hee investigations described in this paper were under- taken for the purpose of detecting and measuring the frequency of the shortest vibrations emitted from the various gas atoms under electron impacts. A preliminary notice of some of the results obtained has already been published in ‘Nature’ t. From the quantum relation eV=hy a certain frequency v can be calculated for each gas from its ionization potential V which should be the maximum frequency ob- tainable from that particular gas when subject to the same type of ionization. This calculation gives a probable minimum wave-length for hydrogen of 909 x 10-8 cm. and for helium of 422 x1i07° cm. assuming Bohr’s ionization potentials (13°6 volts for hydrogen and 29°3 volts for helium) to be correct. In making these computations the value of eis taken to be 4°77 x 107° E.S. unit and of h 6°55 x 10727. Taking the experimentally determined values of 10-4 volts * Communicated by the Authors. + Richardson and Bazzoni, ‘ Nature,’ vol. xcviii. p. 5 (1916). Phil. Mag. 8. 6. Vol. 34. No. 202. Oct. 1917. xX lb a OO 286 Prof. Richardson and Lieut. Bazzoni on the and 20 volts* respectively to be correct, the limiting wave- lengths come out as 1188x107* cm. and 618x 107° cm. When these experiments were begun in 1915 no “ ultra- violet” radiation shorter than 9000x1078 cm. had been detected. On the other hand, various measurements with X-rays had shown the maximum known , Wavelength of these to be in the neighbourhood of 1x10-° cm. It seemed from the considerations given above that the investiga- tion of the radiation from helium promised to fill a large part of this gap in the radiation spectrum. To carry on work in the neighbourhood of 2400 it was, however, necessary to devise a new method of experimentation. us is to be observed that difficulties arise not only because such short waves appear to be absorbed completely in all solid media, but also because the radiation corresponding to the ionization potential may be expected to be highly absorbable by the gas giving rise to it. To detect such radiation it is therefore necessary not only to eliminate all lenses, prisms, and other absorbing bodies but it is also desirable to avoid the long path through the gas demanded by vacuum grating spectrographs. The gas must be worked with at a low pres- sure, and preferably with low and definitely known applied potentials. It is further necessary in dealing with helium to have an experimental arrangement in which the gas can be maintained of extremely high and dependable purity. The previous investigations in the region of very short waves have been made by Schumann’, who extended the spec- trum from 41850 to 11230 by the use of a vacuum fluorite camera-spectrograph, and by Lymant, who in 1914 by the use of a vacuum grating camera-spectrograph extended the spectrum to 900. Since the experiments dealt with in this paper were begun Lyman has published§ the records of work carried on in helium in the apparatus above referred to which give definite lines down to about 1600. This spectro- graph is constructed of brass with greased joints. The radiation path is something over 2 metres in gas at pressures ranging from 1 to 3mm. A disruptive discharge is used. Under these conditions it is extremely difficult, if not im- possible, to keep the helium free from impurities, particularly * The data and curves given by Bazzoni in Phil. Mag. vol. xxxii. p. 566 (1916) make it improbable that the ionization potential of helium in this neighbourhood exceeds 20 volts. However, Franck and Hertz found a value 205 volts. If this is used the corresponding wave- length is 603 x 10-8 em. ¢ ‘The Spectroscopy of the Extreme Ultra-Violet, by T. Lyman p. 75 (1914), t Op. cit. p. 78. § Astrophysical Journal, xliii, No. 2, March 1916. aa ee he pe ys Mo Spectra of Helium, &c. in the Ultra- Violet. 287 from hydrogen. It is especially hard to anticipate what may be the effect of traces of hydrogen on the helium spectrum. It is further to be expected that all radiation at all absorbable in helium will be eliminated before arriving at the photo- graphic plate. On the other hand, the radiation detected occurs in definite lines, the wave-length of which can be determined with considerable accuracy. In the method which we have used the radiation is allowed to fall on a metallic target, and the velocity of the photo- electric electrons emitted is measured. It is thus possible to determine the frequency of the impinging radiation, and from the distribution of velocities amongst the emitted electrons something of the distribution of frequencies in cases where the radiation received is complex. In making these determinations itis necessary to assume the correctness of the equation 4mv?=eV=h(v—v) and its validity in the new region of very short waves. This equation has, however, been established beyond any question on both sides of this region. In the ultra-violet the researches of Richardson and ‘Compton * and of Hughes fT have shown that it applies to a very large number of very varied materials, and Millikan tf and his pupils have carried out tests with a few carefully selected substances which show that it holds with very high accuracy. The researches of Whiddington and others have shown, though not so directly, that it also holds in the X-ray region, and there is no reason for doubting its validity in the intermediate part of the spectrum. Determinations of this kind offer advantages over spectroscopic methods for this particular problem in that they can be carried on in an apparatus of small dimensions without windows or lenses, and of such a character that it can be madeand kept free from occluded gases. An obvious method for measuring the maximum velocity of the electrons lies in the use of a back potential to check the: emission. We tried this method with an apparatus we have described in another connexion §, but the phenomena were complicated by a large photoelectric emission which took place from the walls of the collecting sphere surrounding the target. Owing to this the results obtained by this method were indecisive and consequently were not published. We then determined to sort out the electrons by bending their paths into circles by a magnetic field and calculating the velocity of emission from the value of the field necessary to * ‘Science,’ May 17, 1912; Phil. Mag. vol. xxiv. p. 575 (1912). + Phil. Trans. A. vol. 212. p. 205 (1912). { Phys. Rev. vol. vii. p. 355 (1916). § Phil. Mag. vol. xxxii. Oct. eo 288 Prof. Richardson and Lieut. Bazzoni on the throw the electron into a circle of known radius. By a continuous variation of the magnetic field one is able to get a spectrogram of a complex radiation similar to that obtained for infra-red radiation with, for example, a rock-salt prism and a thermopile, but in this case the spectrum obtained is not a pure one; so that it is not in general particularly suitable for locating individual lines. The method, in fact, is similar to one used by Ramsauer™* in investigating the velocity distribution of photoelectrically emitted electrons. Fig. 1. Horizontal cross-section of apparatus. Drawn to scale. The apparatus finally developed is shown in fig. 1. The tube is made entirely of transparent quartz. The metallic parts are all of copper without solder, excepting the fila- ment which is of tungsten and the sealing-in wires which are of molybdenum. The leading-out arms are all 10 to 11 cm. long and are sealed at the ends with the usual lead- molybdenum seals made by the Silica Syndicate. The tungsten filament, which is 1 cm. long and is used as a thermionic source of electrons for exciting the radiation, is shown at F. It is supported from a side tube in such a way * Sits. der Heidelberger Ak. d. Wiss. 223 July, 1914. Spectra of Helium, &c. in the Ultra- Violet. 289 that it can, if necessary, be readily cut off, repaired, and sealed on again. At A, is a copper plate anode *5 cm. wide and 2 em. long, at A, a copper wire anode 1 cm. long. The anode A, was lage in the position which it occupies in order that the radiation which might come from the copper itself under high velocity impacts could be investigated. The extra anode is in any case a valuable adjunct since, when charged to a small negative potential, it permits the highest vacua to be measured and continuously followed if changing by the fluctuations through a galvanometer of the positive ion current * collected by it. These parts are con- tained in a vertica! cylinder 8 cm. high, the cross-section of which is shown in the figure. At P are two copper plates "8 cm. by ‘5 em., fitted into a rectangular neck connecting this first cylinder with the rounded quartz box at the left. These plates are used to prevent the passage of any ions or electrons from the discharge-chamber into the quartz box. In this box is a copper cylinder 2 em. high by 2 cm. in diameter, provided with a bottom and lid and divided internally into four compartments as shown. At the front, facing the aper- ture between the plates P, is a slit °8 cm. high by ‘2 cm. across. At T is a vertical copper target 1:2 cm. high by 2 em. wide. At §, and 8S, are bevelled chtaan copper partitions. These slits are -*7 cm. high by °16 cm. wide and are arranged to lie exactly on the circumference of a circle the radius of which, taken at the centres of target and slits, is exactly °‘5cm. Beyond §, is a curved copper plate or cup contained in an insulator, I, moulded of one piece of quartz and provided with a leading-out tube which projects through a hole in the side of the copper cylinder. B is a quartz prop or strut to keep the cylinder in plaee. G is a leading-out wire connected to the cylinder. M is the outlet tube for attaching to the pump and auxiliary apparatus. The quartz apparatus was made by Mr. Reynolds of the Silica Syndicate. The apparatus was supported inside by a double coil of the Helmholtz type, specially designed to receive it, in such a way that the copper cylinder lay at the centre of the coil system. These coils were 8:25 cm. in radius and 8°25 cm. apart, and so wound that fields up to 100 lines could be maintained in the space between them. The dimensions were such that one ampere gave a field of 20 gauss. The calculated values of the field were carefully checked with a fluxmeter and found correct within the limits of error of the fluxmeter. The field was studied with a special small test- coil on the fluxmeter and found remarkably uniform both * Cf. Buckley, Proc. Nat. Acad. of Sci. yol. ii. p. 683 (1916). 290 Prof. Richardson and Lieut. Bazzoni on the vertically and horizontally almost up to the rims of the coils, so that small displacements of the cylinder out of the centre can have no influence on the results obtained when the ap- paratusisin use. Since the discharge-cylinder was contained within the coils, it was found necessary to compensate the field. inside the cylinder in order to prevent a shifting of the dis- charge when the magnetic field was altered. This was done by winding around the cylinder a two-layered solenoid the dimensions of which were calculated to give a zero resultant field inside for all values of the main field. Tests with the fluxmeter showed this condition to be exactly attained and also that no disturbances of the main field outside the sole- noid resulted at distances greater than -2 or ‘3 em. from its. walls. The connexions of the main coil and this compensating solenoid were so arranged that the proper adjustment between them was obtainable at all times by a single setting of the series resistances. This circuit was taken from a 100 volt main and carried up to six amperes. The apparatus was supported on asbestos props, so that (after the magnetizing coils had been withdrawn) it could be maintained at a bright red heat with a blowpipe flame or otherwise as long as might be thought desirable. In the preliminary treatments the whole of the quartz excepting the seals was repeatedly maintained at a bright red heat for half-hour periods, while a liquid air and charcoal vacuum existed inside. In this way and by repeated electronic bombardments of the electrodes under potentials up to 800 volts and currents up to 30 milliamperes, the metal parts were finally entirely freed from gas, at any rate as far as the temperature ranges used in the experiments are concerned. This end was, however, not easily accomplished. Various extraordinary and in some cases interesting results were obtained during tests made on the electrodes before they were completely gas free. It was observed that as long as any part of the apparatus was immersed in liquid air any hydrogen which may be present is very rapidly eliminated by the action of the glowing filament and thrown down in an apparently modified state on the cold glass*. The arrangement was entirely airtight, and after suitable heat treatment held the highest vacua for indefinite periods. Running out vertically below the copper cylinder was a quartz tube closed at the bottom which could be immersed in liquid air. Just beyond M, which was expanded at the end so as to receive within it a long glass cone and sealed round with a small quantity of hard wax, was a glass liquid * Cf. Langmuir, Trans. Amer. Chem. Soc. vol. xxxiv. p. 1810 (1912). Spectra of Helium, &c. in the Ultra- Violet. 291 air-trap, then a discharge-tube for examining the spectrum of the gas, then a mercury cut-off valve, then a second liquid air-trap containing charcoal. There was next another mercury cut-off, beyond which was a bulb with electrodes for the purification of the gas by sparking with oxygen over P,O;, then two tubes, one containing cleveite and the other potassium permanganate, with suitable stopcock. This part of the apparatus was also provided with a palladium tube for the introduction of hydrogen and with a McLeod gauge reading to ‘0001 mm., and was finally connected with a Gaede mercury-pump. The helium was prepared by heating the cleveite, then mixed with oxygen obtained by heating the permanganate, then sparked over the pentoxide for about one hour, then passed into the tube containing charcoal in liquid air, and finally admitted into the apparatus proper. The gas thus obtained was entirely free from hydrogen and from all other impurities that could be de- tected spectroscopically or otherwise. When hydrogen was used it was introduced by heating the palladium tube with a bunsen burner. The electrical connexions (excluding those to the coils for producing the magnetic field) used in the majority of the experiments are shown in fig. 2. A, isa milliammeter or some- times a unlpivot microammeter with or without a shunt as Fig. 2. as 400 V. + ij -=- ee DY: Hp }i===-f]k—— \ LIVI bt | Te ) Vee | ir A) } wd t Fed a “ ws k ——. ‘ —— oy ; ae Ta Tele 9 ay eee We pe--7 Leen ie “a aR Val eqilifitifi--- haf jOO V. may be required. G is a galvanometer sensitive to 10~° ampere or in some cases a suspended-coil microammeter. Wi is a Kelvin electrostatic voltmeter reading from 200 to 1000 volts. For lower voltages a Weston voltmeter of suit- able range was used in place of the electrostatic instrument. 292 Prof. Richardson and Lieut. Bazzoni on the A is an ammeter reading to 1/30 ampere, T a Dolezalek electrometer sensitive to 600 divisions per volt. The currents in the field coils were read from an Elliot standard ammeter to 1/20 ampere, and checked against the readings of a second ammeter in the circuit of the compensating coil. The 400 volt battery was made up of small storage-cells. The other batteries, as well as an auxiliary battery giving up to 1000 volts, were composed of dry cells. — We shall now consider briefly the action of the apparatus. When in use electrons are accelerated across from the fila- ment to one or other or both of the anodes under a known potential. The electrons strike the gas atoms and, if of sufficient energy, directly or indirectly cause the liberation of radiation from them which will, speaking generally, con- stitute an entire spectrum. This heterogeneous radiation passes between the guard plates and strikes on all parts of the target in the copper cylinder, liberating electrons from it with a great variety of velocities and in all directions. The electrons which come out normally will travel in circles under the influence of the magnetic field, and all of those moving in circles the radii of which end in the slits will arrive in the terminal cup and cause a deflexion of the electrometer. Itis seen then that for every setting of the magnetic field the presence of electrons between certain velocity limits will be recorded by the electrometer. The finite size of the slits exerts a further influence in that electrons which come out of the target in directions inclined a certain amount to the normal both vertically and horizont- ally, will also pass through the slits in quantities depending on their place of origin on the target and on their angle of emission. This effect is a maximum for the central part of the target. The distribution of energy received at the cup for any particular field as determined by graphical methods * is shown in fig. 3 A for a plane across the slit and perpendi- cular to it. In the calculations for these figures the slit width is taken for the sake of convenience to be 2mm. The distribution has also been contoured in the other perpendicular plane due to the height of the slit admitting screw-motion electrons. Fig. 3B represents the final distribution taking account of both of these effects. The horizontal coordinate in both of these curves is the radius of the electron path— the radius at the centre of the slits being 5mm. This curve shows that for any particular setting the main part of the * Cf. Ramsauer, Sitz. der Heidelberger Akad. d. Wiss. 19 Abh. p. 8 (1914). | Spectra of Helium, Se. in the Ultra- Violet. 293 effect observed is due to radiation corresponding to radil near the centre of the target, provided that the radiation is completely heterogeneous between the limits corresponding to the edges of the slits. It is to be observed that no Fig. 3. :- 20} PACA om electrons can enter the cup on radii shorter than the inside radius of the slits, so that when the field is raised to the elimination of any particular spectrum, we are concerned only with this inner radius in calculating the limiting fre- quency. Since we are chiefly considering in this paper the determination of maximum frequencies, the energy distribu- tion across the slit for any one setting is not of immediate interest for our present purposes. The limiting frequencies received in the cup at successive values of the field for the slit width (16 cm.) used in the apparatus are shown in fig. 4. The solid line given for each field value shows the total range of frequencies received. The dotted lines represent the same limits expressed in wave-lengths. The slit was purposely made very wide in this apparatus. The limits can consequently be very con- siderably narrowed by using a narrower slit. The calculations are made in the following way. At any given setting of the magnetic field the velocity v of the photoelectrons travelling on any given radius 7 can be deter- mined from the relation é — =H , m 294. Prof. Richardson and Lieut. Bazzoni on the The minimum velocity with which an electron can enter the cup is given from the calculations for the innermost radius of the slits—the maximum velocity from the calculations for the outermost radius—slightly corrected to allow for the Fig. 4. Dotted lines (@) 400 800 1200 i600 2000 2400. 2800 3200 x10" cm. SEE Eee eee Geass A a SERRE NS a we 2 ess a R-4 S| i gs y¥-0 12 93°4 5 @ 7) 6 9 10 W i2° Nata VS0e ener sd Solid lines fact that electrons can enter on radii somewhat larger than this outside radius. This correction does not affect the further calculations or deductions used in this paper since we are, as pointed out above, particularly concerned only with the inside radius. With the slit widths used we have (neglecting the correction referred to) calls: x77 x 10°'x B= '743x x ae = 080 X 177 x 10’x H=1-026 x Hx 107% The impinging frequency corresponding to any particular velocity is determined from the relation tmy?=eV= oes, Vo), whence V=Vo+ =) or, putting m='898x 1077 and A=6°55X10-”, v="0686 v? + pv. Spectra of Helium, Sc. in the Ultra- Violet. 295- In this formula ») is the threshold frequency for the cepper target, that is to say the frequency at which photo- electric emission begins from copper. For this we take a weighted mean of tha values determined by Richardson and Compton *. This mean is °38x10". We have therefore, finally, vy =°0686 v? +°38 x 10”. Putting in the values of v,,,.. and Ue. LOK each magnetic field, we get directly the values of vy. and v,,, These values are tabulated below. Fig. 4 is a graphical represen- tation of this table. The value of the threshold frequency exerts a very large effect on the determinations at low frequencies, but at high frequencies the proportional effect is o£f course much less. H. Vmax. Amax, Ymin. Amin = 1:096 2730 1:04 2880 8 1°445 2069 1°224 2450 12 2°024 1485 1529 1960 16 2°836 1057 1:977 1515 20 3°880 773 2°512 1195 24 5°155 582 3'166 948 28 6°663 450 3°964 Tou 32 8:403 307 4-910 611 36 10°35 289 5°900 508 40 12°578 238 7-060 424 cas 15°02 199 8°358 308 48 17°68 170 9-774 307 If now the radiation is not a continuous spectrum but consists of lines, we might expect to locate the individual lines directly with greater or less definition depending on the width of the slits, but, asa matter of fact, the known character of the velocity distribution amongst the photo- electrons liberated by monochromatic light renders this impossible. This distribution as determined by Richardson and Compton has ihe form shown in the curves at the bottom of fig.5. For purposes of illustration the total effect obtained from seven lines of equal intensity falling simul- taneously on the target can be determined by summing up the seven corresponding curves as shown in this figure. If we take a series of observations of the electrometer deflexions corresponding to a series of values of the magnetic field from zero field up to the field necessary to give zero deflexion of the electrometer, we get, therefore, a curve like the resultant * Phil. Mag., Oct. 1912. 296 Prof. Richardson and Lieut. Bazzoni on the eurve of fig. 5 which is obviously difficult to analyse into its constituent curves. We do not attempt to make any such analysis in this paper. It is, however, clear that the curve furnishes a direct indication of the maximum frequency occurring in the spectrum. Fig. 5. ae bea i a aed Ee aed ie ee ieee oon JK Specimens of the results obtained when the apparatus was filled with pure helium are shown in fig. 6, which gives the negative electrometer deflexion as a function of the field. The figure also gives the frequencies corresponding to the inside radius for each setting and the related wave-lengths. Spectra of Helium, &c. in the Ultra- Violet. 297 The dotted curves show the results obtained with the field reversed, and serve as zero curves for eliminating the effects Fig. 6.-—Radiation in Helium. Mm.in 30 sec. < 60 50 SS: Se eS anne 2 abe aS ERE 0 +2 -4 +6 -B 10 12 4 16 I-68 2:0 2-2 2-4 2-6 AMPERES. A = 2880 2450 1960 5 1195 948 787 610 508 424 358 307 Angstrom Units V = 1-04 1-22 153 1-98 2-51 3-17 3°96 4°91 5-90 7-06 8:36 9-78 x10'>seé! due to stray radiation. In fig. 6 A we have the same curves corrected for this stray effect. These curves indicate that the helium spectrum contains no detectable radiation shorter than 425x10-§ cm. Since there is a distinct effect at 298 Prof. Richardson and Lieut. Bazzoni on the 1°8 ampere and zero effect at 2:0 amperes, radiation must be present between d 425 and d 500, as reference to fig. 4 will show. Asa matter of fact definite effects were obtained at 1:9 ampere, and irregular traces of an effect at 2 amperes, Fig. 6A. pay meres Scale for A,,;, and v In min, as on fig. 6. which localizes the limiting frequency close to 425. The limiting frequency, the shape of the curves, and the position of the maximum are seen not to be functions of the driving potentials on the electrons nor, with certain restrictions, of Spectra of Helium, §c. in the Ultra- Violet. 299 the pressure of the gas. The pressure must be below that at which the mean free path of the electron becomes of the same order as the distance around from the target to the cup, and it must be above a certain value where the collisions of the bombarding electrons become so few that there is not enough radiation to produce an effect that can be detected on the electrometer. If the mean free path is less than the distance around, disturbances arise from the deflexions of the slow moving electrons by the gas molecules. In the curves shown the pressure was about ‘(06 mm. That these curves are due to radiation from the gas and not to radiation from the filament or from the metal parts of the apparatus, is evident from the following considerations. Some of the curves were taken with the anode A, at a low negative potential, the discharge then passing to the anode Ay, which is out of sight from the target. These curves do not differ in any way from the others. Before admitting the helium, while the apparatus contained a liquid air vacuum, careful observations were made with the purpose of locating any radiation which might be given off from the copper electrodes. Anode A; was used and voltages up to 900 with electron currents up to 10 milhamperes were applied, but no radiation effect was discovered. Curve A, fig. 6, which coincides with the horizontal axis, was obtained at this stage. This was in some ways a disappointing result, as it was thought that radiation from the metal might be expected at these potentials. In these observations the apparatus was, however, not set to indicate radiations shorter than about 60 Angstrom units. Under these conditions the entrance of the slightest trace of gas would produce radia- tion at a point corresponding to the maximum of the curve given at higher pressures. It is proposed to make further experiments on the emission of radiation from solids under similar conditions. The lowest driving potential shown in fig. 6 is 200 volts, but similar curves were taken at 100 volts and at 80 volts. The intensity of the radiation at these voltages had fallen to such an extent that the spectrum was shortened so as to end > at 1:8 ampere. The apparatus, of course, records only a very small fraction of the total effect due to the radiation, and the total effect itself is extremely small at the limiting frequency. Itis proposed to use a more sensitive electro- meter so as to be able to take spectra with driving potentials in the neighbourhood of 30 volts and of 20 volts. It is significant that the limiting frequency here deter- mined corresponds very closely, if not exactly, with the 300 Prof. Richardson and Lieut. Bazzoni on the value for 29°3 volt impacts as determined from the eV=hy relation. The formula gives a frequency of 7:12x 10-, and a wave-length of 422x10°% cm. Our experimental results show that the limiting wave-length lies between r 425 and A 462 (the value for 1:9 ampere), but the trace of an effect at 2 amperes makes it probable that there is radiation close to X 425; and since the shape of the distribu- tion curve (fig. 3) is such that no appreciable effect is likely to come from values corresponding to the extreme edges of the slit, no marked effect at 2 amperes could be expected from A 422 if it were present. This is an indication that the ionization potential of helium is 29°3 volts as given by Bohr’s calculations, but the evidence of most other experi- ments* strongly indicates that the potential is actually 20 volts. If the maximum energy corresponded to 20 volt impacts, a radiation of frequency about 5x10” would be liberated, and the spectrum in these curves ought to ter- minate at about 1°6 amperes. This suggests again the probability that we have put forward heretofore f, that gas ionizations are not simple impact phenomena but relatively complex processes in which radiation plays a part. It may be that a 30 volt impact is necessary to ionize a normal helium atom, but that 20 volt impacts only are required on atoms thrown into an abnormal state through the absorption of 20 volt radiation. This is a point which we have not been able to settle to our complete satisfaction. Whatever may be the theoretical significance of the observation, the fact remains that this radiation is shorter than has ever hereto- fore been located in this region. Concerning the distribution of energy in this spectrum, it is possible to say that since the maximum at °6 ampere is considerably greater than the effects at -3 and -4 ampere, there must be a length of spectrum beyond the ordinary ultra-violet in which there are very few or no lines, followed by a region in which there are several lines or at any rate considerable energy. The smallness of the effect at °3 and ‘4 ampere indicates, in fact, a scarcity of lines with fre- quencies between 271300 and 21800, assuming that the maximum of the photoelectric curve corresponds to a velocity one-half of the maximum velocity due to the exciting frequency. In the same way, the maximum at * Franck & Hertz, Verh. der Deutsche Phys. Ges. 1913—positive ion method; Bazzoni, Phil. Mag. Nov. 1916—negative ion method. But see Aston, Proc. Roy. Soc. June 14, 1907, who gets 30 volts by an in- dependent method. + ‘Nature,’ Sept. 7, 1916. Spectra of Helium, &c. in the Ultra- Violet. 301 ‘6 ampere is probably due to lines lying about 1800. Since ‘2 ampere corresponds to frequencies at or little above the threshold frequency for the target, the sharp drop in the curve at that value is easily explained. The increase in the effect at -4 ampere seen in the curves for the higher driving potentials is probably due to an increase in the intensity of the radiations lying around 1800. The position of the maximum makes it further evident that there are some lines between 2800 and 271200. These conclusions are in accord with the observations of Lyman *, who could detect no lines in the helium spectrum between 12000 and 1250, although he found some 27 lines between 11250 and 600 with relatively strong ones about AA1215, 1175, 1040, 977, 833, and 720. It is, however, possible that any of these lines may be due to either hydrogen or helium, since most if not all of them above 2900 occur in the hydrogen spectrum with as great or greater intensity than in helium. The principal lines occurring in helium only are AA833, 720, and 600. Summing up the curves of these lines with the intensities given in Lyman’s paper together with a curve for a line about 1400 and one or two between 21000 and 41200, one gets a resultant curve very similar to that obtained in these experiments. A similar resultant could, however, be obtained from a quite dissimilar set of curves if the intensities were arbitrarily fixed at suitable values. After the helium had been finally pumped out and the apparatus properly washed with air and subjected to an extended heat treatment, it was refilled with hydrogen. Curves obtained with this gas are shown in fig. 7. Here again the form and limits of the curves are very little altered at different driving potentials and different pressures between the limits already referred to. The maximum of these curves lies well to the left of the helium maximum, and the limiting frequencies are seen to correspond to wave-lengths of 830 units to 948 units, with a probable value of 2» 900. This radiation would be liberated by impacts with an energy between 13 and 14 volts, which is in close accord with Bohr’s calculated ionization potential of 13°6 volts, from which the limiting wave-length would be X910. All of the experi- mental values for the ionization potential of hydrogen, how- ever, lie around 10:4 volts, which would give a limiting wave-length of 1190 units. Lyman has obtained no hydrogen lines below X900, but he has found a number of * “Spectroscopy of Extreme Ultra-Violet’ 1914, and Astrophysical _ Journal, March 1916. Phil. Mag. S. 6. Vol. 34. No. 202. Oct. 1917. Y 302 Prof. Richardson and Lieut. Bazzoni on the lines in this gas between 1.900 and 11200. Our results with hydrogen are thus in substantial agreement with Lyman’s, although, as we have already pointed out, in Lyman’s method Fig. 7.—Radiation in Hydrogen. My ae i ; U Sec. ren eb | 4 a ANS | lo 40 10 Ts | p-060t0-027 \ te a pe th A ‘S 0 +2 -4¢ © -8 $0 Fe t-4 1-6 Amperes Scale for \ and v as on fig. 6. it,is to be expected that all radiation highly absorbable in the gas experimented with will be lost, so that it would not be Spectra of Helium, &¢. in the Ultra- Violet. 303 surprising if our results do not check exactly with his. These observations also support the hypothesis stated above, that the absorption of radiation facilitates impact ionization. In order to keep mercury vapour out of the hydrogen it was necessary to keep the trap adjacent to the apparatus immersed in liquid air throughout the experiments. In consequence, the glowing filament cut down the pressure of the gas steadily and very rapidly during the course of the observations. ‘The resulting changes in pressure are shown on fig. 7. The character of the effect of this on the shape of the curves is seen by comparing curves A and B. The observations on A were taken in the usual way, going from low values of the field to high values—those on B were taken in the reverse sense. After the hydrogen had been thoroughly cleaned out of the apparatus, a short glass tube containing mercury was sealed on near M and an attempt was made to examine the mercury spectrum. The vapour was produced by heating the short tube in a bath. It is not possible under these conditions to get a steady or definitely measurable pressure of vapour. In order to do satisfactory work with mercury the apparatus will need to be redesigned in certain parti- culars. The results obtained strongly indicate that the mercury spectrum ceases at somewhat longer wave-lengths than does that of hydrogen. The curves shown in fig. 8 are sufficiently definite on this point, but uncertainty arises from the suspicion that the shortening of the spectrum may result from an insufficient pressure of vapour in the discharge chamber. Pressure A was in the neighbourhood of 001 mm. If the limit shown (1 ampere) is the proper one, the limit of radiation is indicated to lie between X1000 and 21200, which checks either with Bohr’s calculated ionization potential (10°5 volts) which gives radiation at 11180, or with the limiting value calculated from the eV =hy relation and the observed secondary ionization potential (10°25 volts). It is proposed to repeat these measurements under more satisfactory conditions. The extension to 1:2 amperes in curves C and D may possibly be due either to the mercury pressure being higher than the admissible value or to the presence of some hydrogen contamination. It has already been stated that the reflexion disturbances arising from too high gas pressures cause the spectrum to spread out. The effect then becomes a function of the driving potential on the electrons. It is easy to tell when one enters the region of too high pressure, since the readings x2 304 Prof. Richardson and Lieut. Bazzoni on the with the field reversed then become of the same order of magnitude as the direct readings. These facts are well Tio. 8.—Madiation in Mercury. < Amperes Scale for A and p as for tig. 6. shown by the curves of fig. 9. Curves A, B, and C were taken in a mixture of hydrogen and mercury vapour which had a pressure of roughly -1 mm. The mean free path under these conditions is about 10 mm., while the distance around from target to cup is about 15 mm. Curves of the type of D were obtained with pressures above *5 mm. The ’ reverse readings are generally about equal to the direct ones under these conditions, so that the curves have no signi- ficance. Spectra of Helium, &e. in the Ultra- Violet. 305 In concluding we wish to point out that this method can be developed to exhibit considerably greater definition, so that it promises to be useful in a variety of connexions. By narrowiug the slits to *3 mm. the spectrum can be explored Fig. 9.—Mercury and Hydrogen High Pressures. 5 ro) tN) p Cop) @ ro) nm py ron) Fe) \ oe ty PO | N | S < Amperes Seale for \ and » as for fig. 6. in the region between 600 and 2100 in steps of the order of 25 wave-lengths (or better, depending on the field current adjustment). The use of an electrometer with a sensibility of 25,000 volts per division ought to make the use of such narrow slits possible. The present experiments are thought to be accurate and dependable between the limits indicated 306 Prof. Richardson and Lieut. Bazzoni on the in the body of the article, but on account of the width of the slits are meant to be merely preliminary. It is of obvious importance to increase the sensitivity of the arrangement to such a degree that the minimum voltages required to excite these new short radiations may be accurately determined. The statements contained in our note in ‘Nature’ already referred to, are of some interest in this connexion. We at that time stated that we had secured the many-lined spectrum of helium at 22°5 volts, and that somewhat higher voltages were required to bring out certain of the higher frequency lines. The apparatus was, however, contaminated with mercury, so that we could not get the helium spectrum without previously exciting the mercury spectrum, which made it possible to suspect that the presence of mercury may have had something to do with the low potential required to produce the helium lines. Shortly after publish- ing these results we tried the experiment in pure helium free from mercury and in a mercury-free apparatus. There was no difficulty in maintaining the many-lined helium spectrum at 22 volts to 22°5 volts under these conditions. ‘The same phenomena were observed as before with regard to the successive appearance of certain of the lines according to their frequencies as well as to the series in which they belonged. There is no reason to suppose that with conditions properly intensified it would not be possible to get this spectrum at potentials lower than 22 volts. We have here an analogy to the phenomena observed in mercury, where Hebb has recently * obtained the many-lined spectrum down to or below 5 volts, while a secondary ionization potential seems, from the work of Franck and Hertz and others, to be established at 4:9 volts and a primary ionization potential at about 10:25 volts. By primary ionization potential we mean the potential at which heavy ionization sets in. These facts all strengthen the supposition that the impact ionization potential is lowered by absorption of radiation. It should be added that tests were frequently made with the different gases experimented with to see if there was any radiation of appreciably shorter wave-length than the limits given above, by applying larger magnetic fields. These tests were carried up to fields corresponding to about 60 A.U., but a complete record of the tests has not been made as they were uniformly negative. In the case of helium we have records of such tests under various conditions made at 3:0, 3°5, and 4:0 amps. or H=60, 70, and 80 lines. * Physical Review, May 1917. + Spectra of Helium, &c. in the Ultra- Violet. 307 These correspond to wave-lengths roughly in the neighbour- hood of 200, 150, and 100 A.U. respectively. Nothing was detectable with these fields, although the effects of the line near 420 were yuite marked in the same discharge. Again, with a clean copper target freed from occluded gases and in a liquid-air and charcoal vacuum, it is recorded that nothing was found at H=60, 80, or 102 lines. The last magnetic field corresponds to about 70 A.U. and was made with 625 volts driving the exciting electron current. Summary. The most important results of this investigation may be summarized as follows :— The high frequency limits of the spectra of helium, hydrogen, and mercury when stimulated by large electron currents under potentials up to about 800 volts have been determined. ‘The measurements are believed to be most accurate for helium and least accurate for mercury. The helium spectrum extends to a Jimit which is certainly between 470 and 420 and probably near to the latter value. The hydrogen spectrum terminates at a wave-length between 830 and 950 Angstrom units and probably close to 900. The mercury spectrum terminates at a wave-length between 1000 and 1200 Angstrom units. * So far as we are able to ascertain, the observed terminal frequencies are identical with the frequencies calculated from Bohr’s theoretical values of the ionization potentials of the respective gases, and they exhibit no obvious relationship to the ionization potentials which have been determined experimentally by Franck and Hertz and others. The ultra-violet spectrum has been extended to a wave- length which is certainly shorter than 470 and probably close to 420 Angstrém units. The high frequency limits of the spectra referred to are independent of the applied potential up to about 800 volts, provided this potential exceeds a lower limit which has not been determined with any accuracy. Wheatstone Laboratory, King’s College, Strand, W.C. o08 AXXI. On Evaporation fron a Circular Water Surface. By Nesta THomas, B.Sc., Assistant-Lecturer in Botany in the Royal Holloway College, and AtuaN Fercuson, M.A., D.Sc., Asststant-Lecturer in Physics in the University College of North W ales, Bangor ™. Pie laws of evaporation from a circular liquid surface have received a certain amount of attention, both on the theoretical and experimental side, but the results of the work done are by no means so well known as they might be. Kiven now, it is not at all uncommon to find, in text-books and original papers, the statement that the rate of evapo- ration from such a surface is directly proportional to its area, in spite of the fact that, so long ago as 1881, Stefan had established from theoretical considerations that the rate of evaporation was proportional, under certain specified external conditions, to the radius and not to the area of the surface. And even where this fact is recognized, the con- ditions under which the linear law does hold are not perhaps clearly appreciated. Further, in many books and papers which touch upon this point there exists a certain amount of “ dimensional ” con- fusion in the application of the theoretical equations involved ; in particular, there is, in the application of the fundamental formule, some confusion between mass and volume—partly traceable to the use of the equivocal term “amount” (of— liquid evaporated per unit time) or its corresponding German equivalent ‘“‘ Menge”—which makes the checking of calculations a matter of some difficulty. The existence of these sources of uncertainty is the more surprising as an excellent annotated bibliography of literature on evaporation has for some years been accessible + ; the importance of an exact knowledge of the laws—involving, as they do, considerations equally valuable to the physicist, botanist, and meteorologist—is beyond cavil. We therefore propose to give in this paper a résumé of the work already done, with an account of such points in the treatment of the subject as appear to us to be either obscure or erroneous, and, finally, to describe some experiments which we have made on the evaporation from circular water * Communicated by Prof. E. Taylor Jones, D.Sc. y+ Mrs. Grace J. Livingston, ‘An Annotated Bibliography of Evaporation.” Reprinted from ‘Monthly Weather Review (U.S.A.),’ June, Sept., and Noy. 1908, and Feb., March, April, May, and June 1909. Evaporation from a Circular Water Surface. 309 surfaces under “ every-day ”’ conditions—a point which so far as we know, has received very little notice. That aspect of the question which is of botanical interest we have already discussed * ; here we confine ourselves to a treatment of the matter purely from the physical side. Apart from the tendency to assume & priori that the evaporation from a liquid surface is proportional to the area exposed, it is probable that the definite statement of Pouillet f had great influence in fixing ideas on the matter ; and it was not till 1881 that Stefan f, on theoretical grounds, advanced the true law of evaporation. A comparison of the equations of diffusion with those of electrostatics shows that the amount of evaporation per unit time from a circular surface of radius a is given by P—p yee, where & denotes the coefficient of diffusion, P the atmo- spheric pressure, and p; and po the pressure of the vapour at the surface and very far away from it respectively. If po and p, be small with respect to P this becomes § _ tka(pi—po) i P Stefan further shows, by an extension of the argument to elliptical surfaces, that ‘tin einem ziemlich weiten Inter- valle die Capacitit einer elliptischen Platte von jener einer gleich grossen kreisformigen nur wenig verschieden ist.” He also obtains expressions for the evaporation from a definite section of the liquid surface, and gives an approxi- mation which shows how the evaporation is affected by the * “Annals of Botany,’ April 1917, p. 241. t “Kléments de physique expérimentale et de météorologie, 1837 ” (abstracted in Mrs. Livingston’s bibliography, p. 25). t Wied. Ann. xvii. p. 550 (1882). § This equation is misquoted as in Preston’s ‘ Heat’ (p, 291, 1894 edition) and in Brown and Escombe’s paper “On the Static Diffusion of Gases and Liquids .... in Plants” (Phil. Trans., B, 1900, p. 251). In the revised edition of Preston (1904, note p. 357) the equation is corrected, but another error remains to which reference will be made later. 310 Miss N. Thomas and Dr. A. Ferguson on lowering of the level of the liquid below the rim of the containing vessel. But it is of primary importance to note exactly the con- ditions under which these formule are applicable. As we have said above, the common assumption is that evaporation is proportional to area, and even where the linear Jaw is assumed, it is often supposed that the law applies only to surfaces of small area. If, however, the conditions specified by Stefan be fulfilled, the law of evaporation is the same, whatever be the dimensions of the evaporating surface. These conditions are given by Stefan in the following words :— “In einer unendlichen Ebene, welche keinen Dampf aussendet, auch keinen absorbirt oder durchlasst, befindet sich eine Vertiefung, welche mit einer Fliissigkeit derart gefiillt ist, dass das Niveau der Fliissigkeit mit dieser Ebene zusammenfallt. Die Fliissigkeit verdampft in die oberhalb der Ebene befindliche unbegrenzte Luft.” It is clear that these conditions are not usually fulfilled in ordinary every-day cases of evaporation, and that a better approximation to the conditions is afforded by a number of small apertures pierced in a plate than by a ten centimetre erystallizing dish full of water ; and it is this fact that has probably given support to the idea that Stefan’s formule are approximations applicable to surfaces of the order of magnitude of a few square millimetres. Moreover, Stefan’s equations are based on the fact that the lines of flow and of equal vapour pressure are similar to the lines of force and the equipotential lines of the analogous electrostatic problem ; this requires a homogeneous and steady atmosphere over distances from the aperture comparable with its dimensions, and, as Brown and Escombe have pointed out, such con- ditions are more likely to hold good in the case of small surfaces than of large ones. Various experiments have been made to test the validity of these results. Srenewsky * pointed out that the evapo- ration from drops varied as their linear dimensions, but the earliest experiments designed actually to test Stefan’s formulz were made by Winkelmann +, who observed the rate of evaporation from small vertical capillary tubes con- taining benzo], which were closed at their lower ends and immersed in a circular basin filled with the same liquid. It was found that the rate of evaporation from such tubes * Beibl. d. Phys. vii. p. 888 (1883). + Wied. Ann. xxxv. p. 401 (1888). Evaporation from a Circular Water Surface. att was a function of their position, being greater for those nearer the periphery than for those in the neighbourhood of the centre of the dish. This is in accordance with Stefan’s theory, but the quantitative agreement is by no means exact, a result which is rather to be explained by non-fulfilment of the theoretical conditions than by any deficiency in the theory. Later, v. Pallich * instituted experiments with a similar object and obtained similar results, finding, for example, that the curves of equal vapour-pressure (which, according to Stefan, are ellipses) are indeed approximately elliptical, but have an eccentricity of about twice the value given by the theory. In a lengthy monograph on the physics of transpiration phenomena, Renner f has given the results of some experi- ments on the evaporation from free water surfaces. He assumes that the rate of loss of vapour in grams per second from such a surface is given by M=4kpa= Ka (where K=4kp), where & is the coefficient of diffusion of water vapour into air, a the radius of the surface, and p a quantity to which he gives the name of “ potential difference,” and defines as the difference between the weight of a c.c. of vapour saturated at the given temperature and the weight of a c.c. of vapour in the surrounding atmosphere. After finding experi- mentally the rates of loss from surfaces of different radii, he proceeds to compare the observed values with those given by the equations i Ka, Ni eeoeeand M=K ima, finding that the formule applicable are functions of the radius of the surface. Why the equation M=K.7a should have been employed is not very clear, as it merely involves a substitution of the semi-circumference for the radius; possibly it is intended as a tribute to the occult qualities which are sometimes associated with 7, but in any case such comparisons have no important physical significance— results deduced therefrom are merely expressions of the not very recondite mathematical fact that two quite unrelated curves may approximately coincide over a portion of their lengths. * Berl. Akad. Sitzber. 106. p. 384 (1897), and Sci. Abs. i. p. 203 (1898). + Flora, 100. p. 474 (1910). 312 Miss N. Thomas and Dr. A. Ferguson on If the whole of his results be plotted logarithmically, it can be seen that they may be represented with mederate accuracy over the whole range of the experiments by M= Kal? in fair agreement with our experimental results, which we shall presently proceed to discuss. In a later paper * Renner gives the results of experiments made on the evaporation from moistened pieces of bibulous paper and, wnéer alia, gives a qualitative confirmation of the relation established by Stefan between the evaporation capacity of a circular surface and that of an ellipse of equal area. The discussion of previous work might ‘in judicious hands Extend from here to Mesopotamy.” but theretis no need to particularize further. Mrs. Livingston’s bibliography provides full references up to 1909, and the courtesy of the Director of the Meteorological Office has enabled us to give one or two later titles, which are appended below fT. We turn now to the discussion of several errors which, it appears to us, have somewhat obscured the treatment of this part of the subject. They have probably arisen from the fact that we can employ either of two differential equations in defining k. Thus, if we employ the equation ot Oa?’ we ubtain, following Maxwell f, a definition of & as “the apm ol gas, reduced to the unit of pressure, which passes in unit of time through the unit of area when the pressure is uniform and equal to p, and the pressure of either gas increases or diminishes by unity in unit distance.” OP1 oa ep P1 * Ber, Deutsch. Bot. Gresellsch. xxix. p, 125 (1911). + F. H. Bigelow, ‘The Laws of Evaporation, &c.’ Buenos Aires, 1911. B. F.E. Keeling, ‘ Evaporation in Egypt and the Sudan.’ Survey Dept., paper 15: Cairo, 1909. R. Strachan, ‘Basis of vaporation.’ London, LTO: J. R. Sutton, ‘ Evaporation in a Current of Air.” Trans. Roy. Soc. Africa, i. p. 417 (1910). t Phil. Mag. xxxv. p. 201 (1868). Se > : Evaporation from a Cireular Water Surface. aks On the other hand, if we empley the equation Opt _. en 0 Pi OES x OE y) we are led to a definition of k such as the following :— If layers of equal density are horizontal planes and p i the density of a gas A at a height x above a fixed horizontal plane, then in unit time the mass of A which passes down- wards through unit area of a horizontal plane at a height x is proportional to the density gradient and is equal to kOp/da, where & is the coefficient of interdiffusion of the gases wand, Bb. Whichever definition be employed, the dimensions of k are, of course, the same and are those of [Surface ] [ ‘Time | As we have already mentioned, the equivocal nature of such terms as “amount.” has also had considerable influence in introducing confusion. Now, considering Stefan’s equation for the evaporation from a circular surface, viz., V =4ka log, ae. : examination of the dimensions shows at once that V must stand for the volume evaporated per unit time; and, although Stefan consistently uses the ambiguous term “4 Menge,” the symbol V shows clearly that the quantity under discussion is volume and not mass. If, indeed, we interpret V as meaning evaporation in grams per unit time, an application of the above formula gives results of quite the wrong order of magnitude, as the following calculation shows. In one of our experiments a basin of radius 2:08 cm. lost 1:727 oms. of water by evaporation in 19° 41™ under the conditions immediately following :— Ht. of barometer (mean), 76°7 cm. Mean relative humidity, 56 per cent. Mean temperature, 15° C. or 288° A. * See, for example, Poynting and Thomson, ‘ Properties of Matter,’ 6th ed. 1918, p. 196. 314 Miss N. Thomas and Dr. A. Ferguson on At 0° C. and 76 cm. pressure Winkelmann gives h, as 0-216. Hence we have La (6 i mm ._ = 2) k bor) 0236. Also p; = maximum pressure of aqueous vapour at 15° C. = 1:28 cm., and therefore from the humidity data p> =°72 em. Substituting these values in Stefan’s equation we have V ='0143 as the amount (Menge) evaporated per second. The observed loss in grams per second is yall Ad POSES UC and it is seen that the two quantities V and EH which, if V stands for the evaporation loss in grams per second, should at least be of the same order of magnitude, stand in the ratio of about 600: 1. If, however, it be assumed that V stands for volume lost per second, the results, though not showing any quantitative agreement, are of the same order of magnitude. Nevertheless, in both editions of Preston’s ‘ Heat’ * the symbol V is exchanged for M, and it is definitely asserted that in the equation E -= 0000244, M=4ha log. Po ; ay M stands for the mass of liquid evaporated per unit time — a result which is dimensionally impossible. The same error appears to have been made in Waitz’s article on “ Diffusion”’ in Winkelmann’s Handbuch f, although the use of the term ‘‘ Menge” makes the results difficult to interpret. Just as in Preston, Stefan’s equation is here given with “ M”’ substituted for “ V,” and as in the general discussion ‘‘ Menge” is used where mass is undoubtedly meant f, it is probable that the same holds good in its use in Stefan’s equation. * 1894, p. 291 note, and 1904, p. 357 note. +t Handbuch d. Phystk, i. pt. 2, p. 1430. { Z.g.: “ Betrachtet mau ein Gasvolumen zwischen zwei um dz voneinander abstehenden Querschnitten Q, so geht durch den unteren Querschnitt desselbe in der Zeit dt von dem in Richtung der wach- senden v7... .sich bewegenden Gase die Menge Ai = —1Q 961 a” Evaporation from a Circular Water Surface. 315 In Brown and Escombe’s well-known paper on diffusion * there is in places an obscurity of terminology which renders the account needlessly difficult to follow. Thus (p. 228, note) we read of a density gradient of one atmosphere per metre; on p. 251 we have Stefan’s formula wrongly approximated, with the substitution of ‘“‘mass” for “volume” already discussed in connexion with the same mistake in Preston’s ‘ Heat’ ; on the same page we have an equation for the rate of absorption (Q) of atmospheric CO, by an absorbing disk of diameter D, viz., Q=2kpD, where p is the density of atmospheric CO,. Q should there- fore, as dimensional considerations show, be given in grams per unit time, whilst in the experimental discussion of the formula the amounts absorbed are given in cubic centimetres per hour. Again (p. 263, note), the volume (Qu,) of CO, diffusing per hour down a tube of cross-section A is given at 0° and 760 mm. by 2: ee Quo= Ta T 760 where L+z2 is the “corrected” length of the tube, & the coefficient of diffusion, and p the density of the CO, in the atmosphere. ‘This formula again is dimensionally impossible, as the right-hand side has dimensions MT’, whilst those of the left-hand side are L?T~1. It is on turning to p. 232, where we find density and partial pressure treated as con- vertible terms, that the reason for the discrepancy begins to be apparent ; and on p. 239 the matter is cleared up. Here the symbols used in the equation just mentioned are defined, and p,, the density of the carbon dioxide in the atmosphere, is defined as the volume of CO, contained in unit volume of air. This arbitrary use of the word density also explains the apparent anomaly in the handling of the equation Q=2kpD, but it appears to us to be a quite indefensible use of so well defined a term, and one which ean only make for confusion. x 3600, We proceed now to give a brief account of some experi- ments which we have made on the evaporation from circular * Phil. Trans., B, p. 223 (1900). 316 Miss N. Thomas and Dr. A. Ferguson on water-surfaces under different external conditions. The ex- — perimental methods employed were quite simple. -A series of circular crystallizing dishes of radii varying from two to ten centimetres was exposed under certain fairly definite conditions, each dish in any given experiment being filled to a definite depth d below its rim. The vessels after being weighed were placed on a table, separated from each other by distances several times greater than their own diameter, and were left exposed for a definite period. They were then weighed again, the resulting rate of loss (HK) by evaporation being expressed in grams per hour. Assuming the rate of evaporation to be given by EK= Ka’, by plotting log HE against loga, K and n can be graphically determined. In all cases it was found that the curve so obtained was linear to a very fair degree of accuracy, so that the values of k and n could be determined without appreciable ambiguity. The barometric height, hygrometric state of the air, and maximum and minimum temperature experienced during any given experiment were also recorded. The experiments were carried out under the following conditions :—(A) in a dark room, having blackened walls and a floor space of about 400 square feet. This room was chosen for its steady temperature qualities, the temperature over a period of 24 hours never varying more than one or two degrees ; (B) in a large room used as a general Jabora- tory, well lighted from above, and having a floor space of about 600 square feet ; and (C) in the open air. The greater portion of the experiments were made in the summer of 1916 in the University College of North Wales. Some readings were taken during the cold spell of last winter at the Royal Holloway College, but these readings are not so numerous as we could wish ; we hope, should circumstances permit, to continue the experiments, using much larger surfaces, as we believe that the results, apart from their purely physical interest, may be of service to meteorologists. It is not necessary to give full details of all the figures obtained in the various experiments. In Table I. below is shown the result of an experiment made under condition (A). This shows the accuracy obtainable in one given experiment and will serve asa sample of the rest of our observations. The first column gives the radii (a) of the dishes used, the sixth column gives the observed rate of evaporation in grams per hour, and the seventh column shows the rate of Evaporation from a Circular Water Surface. 317 evaporation as calculated from the equation H=O010la'® ; the constants of which equation were obtained graphically, as described above. The last column gives the percentage errors, and it will be seen that the average error neglecting sign is less than 1°5 percent. In this Table, and in all cases, d stands for the depth of the liquid surface below the rim of the containing vessel. TABLE I. Mean height of barometer = 29°6 ins. (steady). Max. temp. =18°8 C. Min. temp. =17™7 C. Relative humidity =63 per cent. d==()'s em. K=6;0101. N=) ee Loss in | o Loga. | Time. | weight Log E Hi 2 FOr cent, (cm.). (grams). (obs.). | (obs.). | (calc.). | error. —— |_| ee 1017 | 10073 | 43" 4m] 22:10 | 1-7103 | 05188 | 05064 | —1:34 | 826 | 091701 43 0 | 15:15 | 1-5470 | 03524 | 0:3579 | 41-56 | 703 | 08470 | 42 58 | 11:90 | 1-4424'| 0:2770 | 02727 -| —1°55 | | 584 | 07664 | 42 54) 860 | 1:3021 | 0-2004 , 0-1989 | -0-'75 | 500 | 06090 | 42 53] 6-45 | 11774 | 01504 0:1530 | 41-73 438 06415 | 42 54 | 5:25 | 10878 | 01224 0-1224 | +0-00 B91 05922 | 42 58| 4373 | 29977 | 0-0995 O-1011 ; +161 | 330 | 05185 | 42 59] 3286 | 38835 | 00765 00758 | —091 | | 284 | 04533 | 42 59| 2-422 | 27510 | 0-0564 | 00589 | +4°48 “| 908 | 03181 | 42 58| 1-484 | 20884 00345 00348 -+0'87 _ Average error regardless of sign = 1°47 per cent. The second table gives a condensed view of all the experi- mental work carried out in the summer. Experiments under the heading A were made under dark-room conditions; those labelled B and C were made in the general laboratory and in the open air respectively. (In the experiments labelled A; and A, a maximum and minimum thermometer was not used, and the mean is the result of several readings during the course of the experiment). The values of K and n given in the last two columns were obtained, as explained, by graphical calculation. Phil. Mag. 8. 6. Vol. 34. No. 202. Oct. 1917. Z 318 Miss N. Thomas and Dr. A. Ferguson on TABLE II. a ( Rn Temperatures in © O. Rel. : mi { (Mean)! ay ES em : in inches. Wee ti ian.) Mean. (Mean). S é Culneixcenita aie A, 30°19 Toe a7 EO 56 0-0 | 0:0507 | 1:48 ANG 30 14 ls7 | 15:0 | 1435 is 03 | 0:0126 | 1:60 Ay 30°20 a4 wee 1G*5S 73 0-5 | 00187 | 1:49 AN, 30°78 mt: Bea (OES ay Ga 05 | 0:0168 | 1:50 ING 30°98 22°5 | 18:7 | 20°60 69 05 | 0:0158 | 1:46 A, 29°94 16-1 | 15:3 | 15°70 74 05 | 00109 | 162 AY 29:60 PS Li. \ Alsi 70 O7 O-O10! | 1:69 Bhs 29:77 184 | 17-8 | 18:10 We 1:0 | 0:0079 | 1:78 A, 29°60 NSS) 17-7 | 1825 70 1:2 | 00063 | 1:86 Ay 29°49 18°8 | 18:1 | 18:45 76 15 | 0:0053 | 1:82 A 30°13 Poy 15:0) ors 73 15 | 0:0048 | 1:88 ANS 29°95 18-7 | 174 |18:05 67 2:0 | 0:0053 | 1:99 Aes 29°69 NOS ele.7 wo ie 2°5 | 0:0050 | 1-99 Ae 29°57 Zia! 19:2 9) L975 71 25 | 0:0056 | 1:97 AGEs 29°83 ISO) nee aye || serie 71 30 | 0:0049 | 1:97 Bie: 30°24 DOT NIIP WISE) 62 03 | 0:0291 | 1:58 B, 30:23 21'7 | 18-1 | 19°90 60 0-6 | 00268 | 1:57 Be 30°21 21-3 | 18:3 | 19:80 66 0-7 0:0206 | 1°65 Bis 29°55 16°8 | 14:7 |15°76 90 2°5 | 00031 | 2-00 B, 29°61 ete 9) lor 84 2° | 0:0047 | 1:98 C, 30°21 21°83 | 189 | 20:35 68 05 | 00607 | 1:67 C, 29 95 WB 7 eye a eT 78 27 0°0379 | 1-77 OR. A Ta 2°4 1:4 19 3°0 00097 | 2:06 The table brings out clearly several points of interest. First, it is to be noticed that under no circumstances does the value of n fall so low as unity, which Stefan’s theory demands. For vessels brimful of water, in which no “ rim- influence” is therefore apparent, the value of » is approxi- mately 1°5—about halfway between the value given by Stefan’s theory and the value which holds if it be assumed that evaporation is proportional to area. As the depth of the liquid below the rim increases, the value of n also increases, rapidly at first and then more slowly, until when the liquid has reached a depth of about three centimetres below the rim, n has become practically constant and equal to 2, so that the evaporation is now proportional to the area. A reverse series of changes takes place in the value of K. Large when the vessel is brimful, the values become smaller as d increases, finally, at a depth of about three centi- metres, reaching a limiting value, which is a function of the external variables—temperature, pressure, humidity, and wind-velocity. | Evaporation from a Circular Water Surface. 319 At constant depth changes in the external circumstances in the direction of greater atmospheric disturbance do not make so large a change in the value of n as might be expected from the distortion of the lines of flow which must necessarily ensue. Thus, for d=0°5 cm. the mean value Gieweis about 1:52 in series A,.1°58 in B, and 1:67 in C. During this last experiment, which was performed in the open air, the liquid surfaces exposed were continually ruffled by the wind, but the increase in n, though quite distinct, is relatively small. Series A contains sufficient observations to show quantita- tively how n and K vary with d. Considering K first, if we assume an equation of the type KRepitgqe™, we find that the results are fairly well satisfied by K=:005 +'025¢e~-7. Exact correspondence is not to be expected, as the values of Pp, q and s, which are independent of d and of a, the radius of the surface, are affected by the external variables men- tioned above. In series A, however, the external conditions were sufficiently constant. to justify taking mean values where more than one series of readings was taken for a given value of d. The graph of the above equation was sketched on squared paper, and the agreement between the observed and calculated values of K is shown below. Tarn vet We Ve ite ae d(em.) ...|0°0 ane OD JO 7iOmgets? 1:5. | 2-08 ear 3:0 | | | | ae | | K pe O19 ‘014/011 0080 0070 | 0060 0054 ‘0051 | 0050 | Bee | K (obs.) ... 081 013 015 -010 0079 0063-0051 -0053 |-0058 | 0049 With the exception of the value for d=0°3 em., it will be secn that tle agreement is as close as could be expected. In the same series the value of n is fairly well represented by an equation of the type n=p—qe if we assume for the constants values which give nu—2-0—-60e7*. —sd 5) ZL 2 320 Evaporation from a Circular Water Surfuce. Table IV. shows the agreement between the observed and calculated values of n. TABLE LY. ir Maver | _ eae d(om.) .....{00. 1038 |05 [07 {10 |12 Fed 20 125 130 n (cale.) ...... 1-40 | 1:56 | 1-64 [1°70 | 1-78 | 1-82 | 1:87 | 1-92 | 1:95 | 1-97 M(ODS.) seen 1:43 | 1:60 | 1°52 | 1:69 |1°78 |.1°86 | 1-85 | 1:99 | 1:98 | 1:97 The value for d=0°5 cm. excepted, the agreement is fairly good, when it is remembered that the temperature, pressure, and humidity, all of which affect the constants of the above equations, must necessarily vary to a greater or less degree during the progress of any given experiment. We may say then that the evaporation from a circular surface of radius a, at a depth d below the rim of the con- taining vessel, is given in a steady atmosphere by an equation of the form sd Bi (it Gil tae ae 3 where 1, 91, 51, ~, g, and s are independent of a and d, but vary with temperature, pressure, humidity, and wind- velocity. The dependence of these quantities on the above variables is a matter for future investigation, but it may safely be assumed that, in all ordinary conditions, for sur- faces from 2 to 10 cm. in radius, when the value of d is greater than about 3 cm., the value of the exponent of a in the above equation is constant and equal to 2. One further point may be noticed. It is customary to give evaporation results in linear measure—inches or centi- metres, as the case may be. ‘The above results show that in such cases the amount of evaporation recorded will be a function of the radius of the vessel used. Tor, if d be the depth evaporated per unit time, the mass evaporated will be H=7a’pd, and if the law of evaporation be Haas then d a SS Ope. mp and is only independent of a when n=2, If we assume Astronomy and Electrical Theory of Matter. 321 n=1°5 for dishes nearly brimful, then the linear evaporation will vary inversely as the square root of the radius of the vessel used. This is possibly an important factor in ex- plaining the irregularities observed in the linear evaporation from comparatively small surfaces. Our thanks are tendered to Professor E. Taylor Jones of the University College of North Wales in whose laboratory most of the experimental work was carried out, and to Professor V. H. Blackman of the Imperial College of Science and Technology to whom we are indebted for several references to the literature of the subject. August 1917, XXXII. Astronomical Consequences of the Hlectrical Theory of Matter. Note on Sir Oliver Lodge’s suggestions, LI. By Prof. A. 8. Epprneton, W.A., F.RS., Plumian Professor of Astronomy in the University of Cambridge”. 1. WIR OLIVER LODGH?’s theory, given in the August eS) number of this Magazine, makes use of the well- known equation of particle dynamics, du F/m Je: @ = jee - 5 - e e ° . (1) In the course of correspondence between us, it has appeared that this equation requires amendment when m is taken as variable. If we recall the steps by which it is obtained in text-books, we find that m has been assumed constant ; and it is therefore necessary to examine the effect of the terms in dm/dt which have been left out. Since Lodge’s theory is a non-r cavity theory, it is essential to refer all such quantities as momentum to axes fixed in the ether, and not to axes travelling with the sun. But I think it will be clearer if I divide the work into two stages, using first momenta relative to the sun, and after- wards introducing the further terms which appear when we refer (as we ought) to fixed axes. The radial and transverse components of momentum are then h=mr, h= mr, where m is the inertia, which varies with the velocity of the planet relative to the ether. The rates of change of momenta * Communicated by Sir Oliver Lodge. 322 Prof. A. 8. Eddington on Astronomical in the instantaneous radial and transverse directions are given by the usual formule for rotating axes, viz. “ at Bho, Be + Oh. Equating these to the corresponding forces (—F’, 0), we have ) Clie i. hm) —mrf? =—F, | oad . (2) ay 6rd) + mr == 0 \ The second of these is equivalent to ld a Bi (Giwec)) == Ws so that mr’? = constant = Mh, say, where M is the mass at rest. Then | Mher Mhdr du 1 mr = agi adm — Mh, («=;) Also mré = Mhu. Hence, dividing the first equation of (2) by @, d’u . — Mh aa — Mhu = —F/0 | — - mr? ee aa Whenee, finally, d*u ee) ie 72° Me {, ih . 6 . e (3) Vhis differs from (1) by having the factor m/M instead of M/m. It is easily seen that the change will just reverse the sign of all the perturbations predicted ; but this correction makes no essential difference in the application to astronomy, since we have only to make a corresponding reversal of direction of the sun’s motion through the ether. There is another way of looking at the correction. We may try to think in terms of mass instead of momentum ; but in that case we have to distinguish between the longi- tudinal and transverse mass: the latter is the same as our m. When a transverse force acts on a particle, the magnitude of the velocity does not change, and hence dm/dt is zero ; but it is not zero for a longitudinal force. The transverse mass therefore corresponds to dm/dt=0 ; and the neglect of dm/dt Consequences of the Electrical Theory of Matter. 323 in forming (1) is equivalent to neglecting the difference of longitudinal and transverse masses and using the latter throughout. At first sight it might seem that for a nearly circular orbit the force is nearly transverse, so that the use of the transverse mass is justifiable. But we are here con- cerned with the true path of the planet through the ether, which is a spiral, and the force is not really transverse. By basing the analysis on momentum we obtain a simpler treatment, which avoids the introduction of two kinds of mass. 2. Turning now to the question of absolute instead of relative momentum, it is found that yet another correction will be needed. If V is the velocity of the sun through the ether, u the orbital velocity of the planet, the absolute momentum is m(V +); and the rate of change of momentum is du _dm dm ee he re hey, since 7, ==) In Lodge’s original analysis only the first term was taken into account. We have introduced the correction repre- sented by the second term. ‘There remains the third term to consider. If the inertia of the planet increases during any part of its orbital motion, an additional impulse will be needed merely to enable it to keep up with the sun’s motion through space ; otherwise the translational velocity decreases as the inertia increases, in accordance with the conservation of momentum. Let V be the sun’s velocity through the ether, and let the longitude @ of the planet be measured from the direction of V as zero-point; accordingly, the radial and transverse com- ponents of V are V cos@ and —Vsin@. Then the momenta of the planet relative to axes fixed in the eether are hy = m(V cos 0+7), h, = m(—V sin 6+76), in the instantaneous radial and transverse directions re- spectively. Forming the equations corresponding to (2), we have 2 (mr) —mr6?+ V cos @ =—F, d (rd) font vine 0 : ‘ dt cae 324 Prof. A. 8. Eddington on Astronomical We have, therefore, to consider the additional perturba- tions caused by the terms containing V. If w is the orbital velocity of the planet, and v its total velocity, m = M(1+v?/2c?) = M(1+(V?+w—2Vusin )/2c?). Hence dm EN cos 0.6. es We have here made the approximation of treating the orbit as circular for calculating the small perturbations ; that is to say, we neglect e in the terms which have the large denominator c?. This is the same approximation as Jn the previous papers. By (5) the term V cos gm has the variable factor cos? 0, and therefore goes through its period twice in one revolution of the planet. It follows from Lodge’s discussion (pp. 85-86) that it can only give rise to periodic perturbations which would be insensible to observation. Secular perturbations, which we are seeking, can only arise from terms having the same period as the planet, which therefore give rise by resonance to continually increasing effects. Similarly, the term V sin@ ae gives only periodic perturbations. In determining the secular perturbations, the terms in (4) which contain V can accordingly be dropped, and the equations become identical with (2). We have seen that these lead to GU ee Age pee and the conclusions of § 1 are valid. 3. We have neglected the eccentricity in expressions having c? in the denominator. This means that if eda is expanded in powers of e, thus— eda = do tayet age?+..., our approximation gives only a, It is easily seen that there must be a term a,; the term v’/2c? in Lodge’s equation (2), p. 84, contains a periodic part with e in the coefficient. The treatment of orbits of high eccentricity is much simplified by the aid of a geometrical theorem. It is well known that the orbital velocity can be resolved - Consequences of the Electrical Theory 0) Matter. 325 rigorously into two components of constant magnitude *, WZ: (1) A constant speed uy perpendicular to the radius vector ; (2) A uniform translation we parallel to the minor axis, in the direction 0=a+90°. The approximation hitherlo made by Lodge and the writer consists in treating the planet’s orbital velocity as constant and perpendicular to the radius vector; e. g., in obtaining equation (5). This condition is rigorously ful- filled by the first component uy. The work will therefore become exact if we take separate account of the second component, woe. This second component simply combines with the general motion through the ether, —V sin a, in the same direction ; and we have therefore to write we—V sina for —Vsine in equation (3), p. 165, of my previous paper. Accordingly, iM de= 53 .V cos a, SAGES UO EUg” (6) 2 i a - el, eda = 92 Vsino+ ¢ 0. The signs have been amended in accordance with § 1. The strict value of uy for an eccentric orbit is = or 27a T(1 —e)* The results (6) are inaccurate in one particular: since @ is not strictly a constant, the terms containing V in (4) will produce secular perturbations when e is no longer neglected. The exact effects can be computed by the methods of dynamical astronomy. The variations of the elements due to radial and transverse forces S and T respectively are given by the general formule Tf de_h ws = = “18 sin (0a) + (cos (0—c)+- )t, a 0 Se ={ —S cos (oe sin (@— a) f In the present problem we have MS 2 epg) WT win 62 dt dt * I cannot find any explicit reference to this result in any of the text- books I have consulted, though it is familiar as an examination question and is really the basis of the practical calculation of stellar aberration. T See, for example, E. W. Brown, ‘ Lunar Theory,’ pp. 61-62. 326 Astronomy andElectrical Theory of Matter. Making this substitution, and dividing both sides by d@/dt, we have de eave 10s WE —cos # sin (9@—@) +sin 6 cos (@—@) 7 : a—?7 , dm Lge sin 0 tS | . e = me ; = cos 8 cos (9— a) +sin 8 sin (@—a) r : 8 Vian eat 2) sin 6 sin (@—a) (ae Instead of (5), we must now use the rigorous value of dm/d@, taking account of the component we. It is easily found that To obtain the secular terms we must pick out the non- periodic part of (7). It will be seen that the first two terms in the bracket give sin a dm/d0@ and cosadm/dé@ for the two equations—expressions which are purely periodic. We need therefore only consider the third terms. Re- membering that w=p/h, the formule reduce to de V.aa=me ) aa =, sn A(V cos 0+ we sin (9@—a)), | da Vi ee: , ; r (8) aT ae qe 0 sin (9—w)(V cos 0 + ue sin (0—a)). J We can expand r in powers of ecos(@—@), and with a little trouble the non-periodic part of (8) can be found as a series in powers of e. It will suffice here to give only the first terms of the series. de. NBA: dor i Ne aaa 3,26 sin 20, op) a ae Integrating and adding to the results already found, we have finally e cos 2a. 2 d _ MVE mpi VO hoe i 2¢? 8c? (9) U V0 ; UY 20 V70 | eda = — 322 sin o+ gone t Ba" cos 2a 5 correct to the first power of e. It will be noticed that the terms independent of e (the High Potential Batteries for small currents. 327 primary terms) involve the product u)V, whereas the new terms containing e involve V7 and wo”. Hence if V is very large or very small compared with uo, the terms containing e may be larger than the primary terms, even when e is as small as it is for the planets. It is thus theoretically desirable to carry the expansion as far as this point. But there is not much object in going further, since the terms in e’, e’, etc. involve no new combinations of the velocities V and wu), and are therefore necessarily of lower order of magnitude. The part of the motion of perihelion dw=u)0/2c? is of particular interest, since it is independent of the sun’s motion through the ether. It has therefore appeared in some relativity theories of planetary motion, given by de Sitter, Silberstein, and others. For Mercury it gives Just one-sixth of the observed discordance of perihelion ; for Venus and the Harth it is negligible compared with observational errors, on account of their small eccentricities. The other terms involving the eccentricity cannot be of great importance unless V is very much greater than uw. In that case the primary and secondary terms are both extremely large; and, though they might happen to cancel for one planet, it is scarcely possible that they should so nearly neutralize one another for all the eight elements of the four inferior planets. So far as I can see, the conclusions of my previous paper are not materially modified by this more rigorous calculation. XXXII. High Potential Batteries for supplying small currents. By Frank Horton, Sc.D., Professor of Physics in the University of London”. f Wars difficulties attending the use of a high-potential battery capable of supplying a current of a few milli- amperes are familiar to all who have experimented with the discharge of electricity through gases. The type of battery most commonly employed for this purpose is one consisting of a number of small secondary cells with lead plates, and the trouble usually experienced is the “rotting ” of the lead of the positive plate at the point where it passes through the cover of the cell. The rotting consists in the formation of a white powder which analysis shows to consist mainly of lead sulphate, and this goes on until in a few months, or even weeks, the rod is separated into two pieces. The rapidity of this action depends on the quality of the lead used in the * Communicated by the Author. 328 Prof. F. Horton on High Potential plates, for the plates supplied by some makers last much longer than those of others. When this type of small storage-cell was first made it was usually provided with an indiarubber cover, but it was afterwards thought that the contact of the rubber and the lead was the cause of rotting which occurs. The lead rods of the electrodes were therefore covered with short glass tubes to prevent this contact; and this device has generally been found to lengthen considerably the life of the cell. More recently, wooden tops well soaked with paraftin-wax have been substituted for the rubber and glass tubes; but these wooden lids have been found by the writer to be quite as bad as the old indiarubber ones. About 20 per cent. of the positives of a new battery of 320 such cells recently fitted up in this laboratory rotted through in the course of 3 months. The remaining positives, and the new ones replacing those spoilt, were therefore covered with glass tubes where they pass through the wooden covers; but though, as usual, this increased the length of service of the plates, after a few months broken positives were continually being found and the battery was never reliable. The rotting of the positive plate in this way is due to electric conduction across the lid of the cell which is wet with sulphuric acid. In the case of the wood and india- rubber covers which fit tightly round the lead, the action goes on more rapidly than when the rods from both plates pass loosely through glass tubes. The rotting may be prevented altogether by doing away with the cover. It is then necessary to adopt some other device for keeping the plates in position and to prevent the splashing of the acid when the cell is being charged. It would be convenient to have the glass cells made with ridges to keep the plates vertical, but such cells cannot be obtained at the present time. Ina long row of cells in series the connecting wires can be made to keep the plates in position, but a safer device is to cut a strip of thin celluloid of width equal to the distance apart of the plates and to bend this into a f) and place it between the plates of the cell. The top of the celluloid separator should be below the level of the acid in the cell, and a small hole should be made in the top of it to allow the gases to escape when the cell is being charged. The splashing of the acid can be prevented in the usual way by covering the surface with athin layer of oil ; theliquid petroleum sold by chemists for medicinal purposes does very well. A battery of secondary cells arranged in this way has been working satisfactorily for several months. | ee rie ees Batteries for supplying small currents. 329 The advantage of a battery which does not require periodical charging is obvious, and several types of primary cell have been used for this purpose. Primary cells are usually very satisfactory for electrometer work and for experiments where potential only, and not current (or only a very minute one) isrequired. They are often troublesome to fit up, but require no further attention if treated carefully. For currents of the order of ‘01 ampere dry cells may be used, and the writer has found those made by the British Ever-Ready Co. very convenient for this purpose. These cells have the advantage of being small, thus enabling a large number to be packed into a small space, and their electromotive force falls but slowly when currents of only a few milliamperes are taken from the battery, so that the cells can be used fora long time before it is necessary to replace them. Owing to the impossibility of obtaining a further supply 5 of lead plates for small storage cells, the writer was induced Fig. 1. —200 Volt Battery. to experiment with dry cells early in the present year These were arranged in batteries of two forms which have been found so convenient and satisfactory in use that a description of them may be useful to others. The battery illustrated in fig. 1 is for supplying potentials up to about 200 volts. The cells are contained in a wooden box 61 em. long, 18°5 cm. wide and 115 cm. high. This has an ebonite plate on the top which insulates the plug-keys by means of which the cells are arranged in series. The eells used are the Hver-Ready dry battery No. 1689 of about 4°0 volts. It is advisable not to have too many cells Prof. F. Horton on High Potential connected in series when the battery is not in use, and the box therefore contains 3 sets of 10 small batteries (each set giving about 40 volts), and 5 sets of 5 sinall batteries (each set giving about 20 volts). The sets are insulated by micanite, and they can all be connected in series by means of the plug-keys. The required potential is tapped off by inserting special plugs into holes in the insulated brass pieces con- nected to the cells. In these special plugs the insulated wire leading to the apparatus goes through a hole along the axis of the plug, and the wire is fixed into the metal part of the plug by means of a small screw, the head of which is sunk in the ebonite so that it is not touched by the fingers when handling the plug. Four of these special plugs are shown in the illustration, two in the extreme left-hand brass piece, one in the front of the fourth brass piece, and one at the extreme right hand. They are of the same size as the other plugs and fit into any of the holes in the brass pieces so that the required potential difference may be obtained. This battery has the advantage of being compact and portable ; it has been found very convenient both for research work and for lecture purposes. Fig, 2.—35 Volt Battery. The other arrangement of dry cells which has been found very useful in practice is illustrated im fig. 2. The box contains 25 separate dry cells and gives a total electromotive force of about 35 volts. The cells are connected?in series inside the box and, by turning the handle in the centre, the difference of potential between the two terminals can be increased by approximately equal steps from 0 to 35 volts. Batteries for supplying small currents. 331 A convenient feature of the battery-box shown in the illus- tration is the ease with which the cells can be removed and replaced by new ones when necessary. The cells as supplied by the makers are cylindrical in shape, the outside being of zine which is the negative pole of the cell. A small brass cap connected to the positive pole protrudes from the centre of the top of the cell. The cells are each about 5 cm. high and 1*4cm. in diameter. They are arranged in a circle between two sheets of ebonite, one of which forms the top of the box and the other is inside the box and is supported from the top by four ebonite rods. Hach cell is held in position by two copper springs arranged as represented in fig. 3. The cell A rests on the lower ebonite sheet (;%; inch thick), Fig. 3. and is gripped by the spring clip B which is supported from the brass base E screwed to the lower ebonite sheet. The brass cap of the positive pole of the cell presses against the copper spring C fixed to the brass piece D which is screwed to the under side of the upper sheet of ebonite (4 inch thick) which forms the top of the box. The upper brass piece D, connected to the positive pole of the cell, is joined by an insulated wire to the lower brass piece (corresponding to I) in connexion with the zine of the next cell. This method of connexion is continued round the circle, so that all the cells are joinedin series. Inside the box a radial arm makes a rubbing contact with the under side of D, and the position of the arm is indicated by the pointer which moves over the dial on the top of the box. This arm is connected to the left-hand terminal seen in fig. 2; the other terminal is con- nected to the zine of the first cell in the series. It has been found convenient in practice to have one position of the pointer in which there is no connexion between the ter- minals (“ off”’). This forms a simple method of breaking the battery circuit. It is also convenient for some purposes to have a position in which the terminals are connected, but Da2 Mr. J. Prescott on the with no difference of potential between them (“0”). The next position (“1”) puts in the first cell, and thereafter the potential difference between the terminals rises b approximately equal steps of about 1:4 volts as the handle is rotated, until all the cells are included in the circuit. The outside dimensions of the box shown in fig. 2 are 22 em. square by 9°5 cm. high. Cells of about twice the capacity of those mentioned above can also be obtained. These larger cells are 5°5 cm. high and 1°9 cm. in diameter, and twenty of these can conveniently be arranged in a box 25 cm. square by 10°5 cm. high. ‘The larger cells are recommended as having a longer life than the small ones, but the arrangement is not quite so compact. [From the diagram (fig. 3) it will be seen that any cell can readily be slipped out of the springs which holdit. It is thus easily replaced by a new one—there is no soldering to be done or screw connexion to make. The Physical Laboratory, Royal Holloway College, Englefield Green. XXXIV. On the Motion of a Spinning Projectile. By J. Pruescort, W.A., Lecturer in Mathematics at the Manchester School of Technology™. On the Motion of a Spinning Projectile. 1. FN order to make a beginning of the problem of the motion of a projectile, we need a formula giving the resistance of the air to a body moving through it. Bash- forth found it convenient to assume that the resistance varies as the cube of the velocity, while he pointed out that the main reason for his choice of this law was that it was the easiest to work with mathematically. Actually the cubic law is very far from the true law except over two very small ranges of velocity. Bashforth’s method was to express resistance in the form K,V%, V being the velocity of the projectile, and K, a variable which is treated as a constant in his mathematical theory, the errors introduced in conse- quence of this incorrect assumption being kept small by his dividing the path into small portions and using a different K in each portion. 2. From Bashforth’s own results, however, which are based on observations, it is clear that the resistance of the air is much more nearly proportional to the square of the * Communicated by the Author. Motion of a Spinning Projectile. 333. velocity than to the cube, except over a range of velocities between 900 and 1200 feet per second. Denoting resistance by R and velocity by V, it is found, on plotting s against V, that the curve is very nearly a horizontal line for veloci- ties below 800 feet per second, then it rises rapidly till V=1300 feet per second, after which it is nearly horizontal again as far as observations go. Thus R is nearly propor- tional to V? when V is below 800 feet per second, and again when V is greater than 1300 feet per second. The steepest art of the curve is somewhere near the point where V=1080, which, it should be observed, is about the velocity of sound. It is reasonable that there should be a change in the law of resistance at the velocity of sound, for, when the velocity of the projectile is less than that of sound, the particles of air encountered by it at any instant had already been set in motion, before the projectile arrived, by the pressure which was transmitted ahead of it, this pressure being transmitted with the velocity of sound. But when the projectile is travelling with a velocity greater than that of sound no pressure waves are transmitted ahead, so that the projectile meets, and has to set in motion, stationary air particles. 3. This change in the behaviour of the air is shown in photographs of flying bullets. When the bullet is travelling faster than sound, there isa great density of air round the nose and, as the bullet travels, this leaves behind it a single wave of compression consisting of a pair of straight lines equally inclined to the direction of motion, and joined to- gether bya curve surrounding the nose of the bullet. When the velocity is less than that of sound no such compression wave is seen; but in this case also there must certainly be high density at the nose, which, however, will not be so great as in the other case, and, moreover, it will decrease gradually from the nose outwards, which will explain why photographs do not show it. The essential difference between the two cases is that, in the first case, the air is at rest at normal pressure a very short distance in front of the nose, while in the second case the air has a pressure which gradually decreases from the nose forward, and all this air under extra pressure has a forward motion. 4. Bashforth also states that his observations show that the resistance of the air is exactly proportional to the area which the shot presents to the air in its motion. Thismeans that the resistance to elongated shot, which travel nose fore- most, is proportional to the square of the diameter of the shot. Phil. Mag. 8. 6. Vol. 34. No. 202. Oct. 1917. 2A 334 Mr. J. Prescott on the 5. The curve (fig. 1) shows the relation between = and V for Bashforth’s standard shot, which is such that the number of pounds in the weight W is equal to the number of inches in the diameter d. It will be seen from the curve O0- 400 ‘800 4200 - 1600 2000 2400 2800 Curve showing the relation between resistance, R lb., the velocity, V feet per second, and diameter, d inches, for an elongated shot. that the resistance is fairly well represented by assuming that R=KV? or R=K,V? according as V is greater or less than 1060 feet per second, K and K, being a pair of constants. It will be shown in this paper that very good agreement with Bashforth’s tables, as well as with the range tables for bullets, can be obtained on these assumptions. 6. Projectile in a Low Trajectory—The trajectory is called low if the direction of motion throughout the flight always makes a small angle with the horizontal. 7. The assumptions we are making concerning the resist- ance will be used in the following forms :— 2 R= when V>1060 feet per sec. 2 and R= aan when V<1060, 1 W being the weight of the iprojectile in pounds, R the Motion of a Spinning Projectile. 3395 resistance in pounds, V the velocity in feet per second, 1 and J; a pair of constants depending on the size and weight of the shot. 8. For similarly-shaped shot with the same mean density = is proportional to the area of the section of the shot perpendicular to the axis, h being the length of the shot. Since R is also proportional to this same area, it follows that wis proportional to wr or iy and therefore that / and J, 1 are each proportional to h. Moreover, because R and W have the same dimensions as each other, it follows that lg and l,g have the same dimensions as V*, and consequently that / and /, have each the dimensions of a length. Thus for all different but similar shot / and /,; are the same pair of multiples of A. 9. In finding the equation to the trajectory the assump- tion will be made that the area presented to the air by the shot in its flight is constant, which amounts to assuming that the shot always travels nose foremost with its axis exactly in the direction of the line of flight. The extent to which this assumption is in error will appear in the latter part of the paper where the motion of the axis is investi- gated. Fig. 2. 10. Let \ denote the angle that the line of flight makes with the horizontal ¢ seconds after the shot has left the muzzle; uw and v are the horizontal and upward vertical components of the velocity V ; X and Y the displacements 2A2 336 ‘ Mr. J. Prescott on the corresponding to these component velocities, the displace- ments being measured from the muzzle of the gun. The values of quantities at the muzzle of the gun will be denoted by a suffix 0, thus Vo, m%. The letter « will be used to denote the whole angle through which the line of flight has turned in ¢ seconds, so that «=A)—A. 11. The equation for the horizontal component motion when V >1060 is ; W du W V? u Saati: CoO aaa that is, du uV When V <1060 then /, must take the place of Jin (1). Now in a low trajectory V differs very little from u, so that equation (1) can be written du we, and the integral of this is a i oe (8) Again, since du _ dudX _ du, GT dX di kes equation (2) is equivalent to the equation du U : the integral of which is log = = 0 é Nas aM . 0 (5) or HPs U=Upe ! 12. The component acceleration perpendicular to:the curve and towards the lower side is V a and the component force in the same direction is W cosx. Hence W da u ate = W cosrA=W Vv: Motion of a Spiniung Projectile. 337 Again replacing V by w, we get dag a i ae a (6) Dividing corresponding sides of equations (2) and (6) we get de, gl ee ee Consequently, 0) aak(- =) Ae AIS) that is, un EE os 2 ee 5 (is =). SS en Ga But since A is jalways small, it is approximately equal to Sees ¥ tan A, which is TX: Therefore, adY er 1 aX a tan No 2g 7 —_ tel 2x = tanAy— s,(e? -1) by (95). Integrating this, and adjusting the constant so that Y=0 when X=0, we get Y=X (tan y+ 25)— 2 a 10 = (tan ot 52 Balle Sart i bis wwe ene Ku) The horizontal range of the shot fired at elevation Xy is the value of X that satisfies the equation My) 9? (a 0=X (tan Xo + =) ~ Arg? (¢ -1). prlliie (iL) But the value of X that satisfies (11) is the value of X at the intersection of the line Y=—XtandAjy and the curve Vis gl gl? = ) Se —Il > ° ° ° ° (12) which is the curve described by a shot projected horizontally. The ordinate in (12) is negative for all positive values of X. 338 Mr. J. Prescott on the 13. Thus, if OSP is a portion of the eurve given by equation (12), then OQ is the range when the elevation of Fig. 3. the gun is X». Moreover, if we write Y' for Y in (12) to distinguish it from Y in (10), then Y=Y'+X tan Aj, so that, if ON =X, we get Y=Y’+MN =—-SN+MN =MS.° - Thus MS is the true height of the shot above the muzzle of the gun when it is fired at elevation X». Consequently, if MS were erected at N on the base OQ, then S would be a point on the curve of the real trajectory with initial elevation A,. Since the angle X, is small, it will be seen that, if the curve OSP were rotated about O until P fell on the line OQ, then M and P would nearly coincide with N and Q respectively, and MS would be nearly vertical. The curve thus rotated would differ very little from the true trajectory in which the horizontal range is OP. 2 14. Writing c for “0 the equation (12) becomes | Lae 7s Y=5,X-Z(e?-1) Ree ik The angle of elevation, X», which gives the range X, is given by x Wi (ew l tan A, = — = i,(¢ >) » « (14) If we knew the constants / and ¢, we could make a range table from the last equation. Conversely, given the range table for a particular shot, we can find the values of the constants / and c. | Motion of a Spinning Projectile. 339 15. The constant / can be calculated from Bashforth’s tables, and the constant c¢ involves only the muzzle velocity, which is known from observations. But, as I believe the range table is more accurate than any other data for a given shot, I have used these tables for the Marks VI. and VII. bullets to calculate / and c, and therefore wp. 16. So far we have only considered that part of the trajectory which is described while the velocity is greater than 1060 feet per second. In the other part of the trajec- tory we must use J; instead of / and make the two portions continuous at the junction. Indicating by a suffix 1 the values of quantities at the point where wu=1060 (except, of course, 1,, which applies when u< 1060), then starting from equations such as (2) and (6), we get, by integration, the following equations :— ie (| aA i, cy eg ee (16) Uy l; ae 1 A= — 39h, & oi 3) (17) 9 9 2X—X)) Y-Y,=(K—X,)(tan 4 $5) = gh Le : -1}. (18) 2u, du,? The last equation has the form 2X VS ReeBer eh. oe HL) For a shot fired horizontally we get the same form of equation as this last one, and we may write the equation thus: oX YS eEeee Ce ig ss Lie (20) using Y’, as before, for the ordinate of the trajectory in the particular case of horizontal firing. Equations (13) and (20) are the equations to different portions of the same trajectory, the one being applicable while uw is greater than 1060, and the other, while wu is less than 1060. 17. If our assumed laws of resistance were absolutely correct, then we should get the trajectory absolutely correct also by making the ordinate and the slope of the trajectory continuous at the point where the law changes, that is, where w= 1060 feet per sec. ; but as the assumed resistance 340 Mr. J. Prescott on the is not quite correct, that trajectory which is continuous at the point of junction will be wrong for all values of the velocity less than about 1300 feet per sec. We get much better agreement with the true trajectory over large ranges by getting two independent curves for the two laws of resistance, and not troubling to join them up at the point where wis 1060. This latter method is open to us when we have a range table from which 1o determine our constants ; but we should be obliged to make a continuous curve if we had no other data than the initial velocity and the values of l and Lie 18. By the same reasoning as that by which equation (14) was proved, it can also be shown that, when w is less than 1060 feet per sec., the angle of departure, A», which gives the range X, is given by ox Xtana,==Y’=—A—BX—Ce a) ea We have already assumed that our trajectory is so low that there is no appreciable difference between Ao and tan Aj. Consequently, there will be no loss of accuracy in putting A, for tan Ao. 19. It is known that a rifle jumps up or down on being fired, so that the angle of elevation of the rifle barrel just before firing, which is the angle observed in experiments, is not the same thing as the angle of departure of the bullet ; that is, the axis of the rifle before firing is not a true tangent to the trajectory. If) denotes the upward jump of the rifle, and y the angle of elevation of the rifle just before firing, then A\»>=(y+7), or tan Ay=(y +7) approximately. 20. If we now substitute (y+ 7) for tan Ay in equations (14) and (21), and then transfer Xj to the right-hand sides of these equations, we get a pair of equations which may be written thus : Xy=p+qX+re™* when u>1060, . . (22) Xy=p! +q'X+re* when u< 1060, . . (23) where all the quantities except X and y are constants, and wherein alsop=—vr. In these equations we may assume, for convenience, that the angle y is expressed in minutes instead of in radians, since this only introduces a constant factor all through the two equations. 21. Let us write s for Xy. Since a range table gives corresponding values of X and y, we can immediately deduce the values of s therefrom. Now we shall denote by Motion of a Spinning Projectile. 341 5, $1, Sy the values of s corresponding to ranges x X+6, X+428. Then, if all three ranges are described while u >1060, we get the following equations :— s=ptqX+re™, ‘ s=pt+q(X +8) treh Fr, ' ; (24) ssp 4 q(X+28) + rhE+28) Hence s;—s=got+ fete — De ll s9— = 98 + re" Xe" (e? — 1), pas eke (25) These last two equations give, on subtraction, sg Osean (e 1). en If we now take a similar set of three ranges in arith- metical progression with the same common difference 6, and denote the new quantities by dashed letters, X' being the smallest of the three ranges, then $y! —2s;/ +s’ = re" (e*? —1)?. pe BAR) (27) From equations (26) and (27) es! ae AG + sl k(X'—X) ae = e e ° e e 2 S3— 25,475 : ( 8) Since all the quantities in this last equation, except 4, are known from the range table, the equation determines &. Then equation (26) gives 7; next, either of equations (25) will give g; and finally, any of the three equations in (24) determines p. Thus we know how to get all the four constants p, g, 7, and k. 22. In the preceding paragraph it was assumed that all the ranges involved were described while the velocity was greater than 1060 feet per sec. An exactly similar set of equations will be true if all the ranges are described while u< 1060, but in this case we shall determine the dashed letters p', gq’, 7’, k’. Application to the Mark VII. bullet—We must first get some idea of the range when the velocity has been reduced to 1060, so that we may know at what range the law of resistance changes. According to Bashforth the resistance is given by the equation re whiny \? R=—a (x00) ae 342 ; Mr. J. Prescott on the where d is the diameter of the shot in inches and K, a variable given in his tables. But we have assumed that 2 a ee Gt By equating the two values of R we find that ae Td? Whee When V=2000 feet per sec. K, is given in Bashforth’s tables as 68°8. Also the weight of the Mark VII. bullet is — of a pound, and its diameter about 0°303 of an inch. l Therefore; 174 x 10° ™ 7000 x 0°303? x 2000 x 68°8 = 1967 feet. Taking 2000 as a rough value for J we find, from equa- tion (5), that the horizontal distance travelled by the bullet while uw drops from 2440, the given muzzle velocity, ito L060, is l 2440 1060 =1667 feet = 556 yards. Then the law does not change till the bullet has moved more than 500 yards horizontally. 24, The values of X and y in the following table are taken from the ‘Musketry Regulations’ (1909 Edition), and the values of s are calculated from them. X= 2000 log, feet, XM feet2-.- a. 600 900 1200 1500 Y diode tiene enon 8"0 11''3 15"3 20'2 Syebae tease 4800 10170 18360 30300 Now, taking X=600, X’=900, 6=3800, equation (28) gives soor__ 30300—36720+10170 _ 375 3 "* Sr psseo== 20340-4800) mse) Therefore, D) 600 a= k log, 3°75— log, 2°82 = 2105 . . (29) Motion of a Spinning Projectile. 343 Since the data from which / has been calculated are not very accurate, only three figures being available in the angles y, it will be good enough if we take the round number 2100 for 7. Then, by equation (26), 1200 600 2820 = re 2100 (¢ 2100 —1)?, = ret(e7—1)?, from which r= 14562. Next, by the first of equations (25), 10170 —4800=300g + ret(e? —1), that is, 3009 =5370 — 8527 = — 31597, g== —10°52. Lastly, by the first of equations (24), 4800 =p +6009 4-re’, = p— 6314—25786, p= —14672. This should be equal to —7, and it is near enough to give confidence in our results and theory. We shall take the value of 7 to be the mean of the results for r and —p, namely 14620 approximately. Now, recalculating g with these values of r and p, the first of equations (24) gives 4800 = 600g + 14620 (e7 —1), = 600g + 11255, = — 10°76. 25. If the angle y were expressed in radians instead of minutes, then our constants p,q, 7r would each have to be * 4. 7 : multiplied by 70800" Consequently, from equation (14), in which y is expressed in radians, ets mr 146207 Au 2 10800 10800. Therefore. Up:= 2888. Again, if there were no jump, it is clear from (14) that g would be equal to a Tt and this relation remains true 344 Mr. J. Preseott on the when ¥ is expressed in minutes. But since the term—X7 is involved on the right-hand side of (22), it follows that 9 a aM A TS = —13'-92410'76, = — 3/16. | That is, the rifle jumps downwards through an angle of about 3/2. 26. Considering the degree of accuracy of our data our constants can be regarded as only approximately correct. We shall take. in future, é i=2100, u=2890 feet per see: 2 eu) The muzzle velocity given in the ‘ Musketry Regulations ” is 2440; but this is obviously wrong, for a very brief con- sideration of the range table will show that the downward jump, which is the elevation of the rifle for zero range, is something in the neighbourhood of 3’. This can be dis- covered quickly by plotting y against the range and producing the curve backwards to the point where the range is zero. The value of the jump would make the angle of departure 5! for the 200 yards range. But a projectile fired at this angle with a velocity of 2440 feet per second would only have a range of about 180 yards if there were no air- resistance at all. The estimate of the jump given in the ‘Musketry Regulations, namely, between 4’ and 5’, makes the case a great deal worse. It is very probable then that our value of uw is much better than the value in the ‘ Musketry Regulations.’ 27. With our values of uy and J the horizontal distance traversed by the bullet before the law of resistance changes is 2890 ~ 1060 Since the value of the velocity (wm) at which the change takes place is not very definite, it will be more convenient to take X, to be exactly 700 yds., and then by equation (5), ty = 2890 e-1=1063. . | 2,2 (82) Now the equation for y, when the range X is not greater than 2100 feet, becomes X,=2100 log = 2106 feet=702 yards. 2) (G0) 2x Niy=14620 (en —1)— 10:76 X, 9 es Motion of a Spinning Projectile. 345. 28. Zhe constants for the trajectory when u<1063.— Another portion of the table given in section 24, for larger ranges however, is given below. Mv feet,..... 3000 3600 4200 4800 5400 6000 | >) oe 60"1 85'°7 1184 | 159"5 | 210"1 | 271'8 sx10-2...| 1803 | 3085-2 | 49728 | 7656 | 113454 ae Putting X=3000, X’=4800, 6=600 in an equation differing from (28) only in having &’ for k, we get 3600 21. . 1680800 — 2269080+765600 — 12732 497280—617040+180300 ~ 6054’ whence |, =4840 approximately. To get r’ it is better to take 6 as large as we conveniently can. Putting, therefore, 6=1200 and X=3600 in an equa- tion similar to (26), we get 7200 2400 1630800 — 1531200 + 308520 = r'¢* (¢ #4 — 1)2, i) ye or 408120=r'e'™ (e #11), whence 7’ = 223800. 180 60 Also 765600 —308520 =1200 g'+7'e¥! (e!—1), and therefore g' = —148°93. Lastly, p' is obtained from the equation 180 bia 308520 =p! + 36009’ + 7'e 121, which gives p =145900. The equation for y when u< 1063, that is, when the range is greater than 700 yards, is x Ky = 223800 e°—145900—-148-9'X, . . (34) the angle y being expressed in minutes. 29. Zhe constants for the Mark VI. bullet.—By applying similar methods to the results given in the range table for 346 Mr. J. Prescott on the the Mark VI. bullet, we find /= 2600, 1, = 6000, Up = 2118, X,=1800 feet (when u=1060). When X is less than 1800 feet | x Xy=41710 (¢ 1800—1)—38:77 X, < (25) and when X is greater than 1800 feet x Ky =483900 ¢ 30 — 221-8 X —423900, . (36) The jump is 2 . . I= sam x 417104 38°78 minutes, = —32':08 + 38°78, =(Oiide The value of J above was deduced directly from the range table as in the case of the Mark VII. bullet. Then J, was calculated by assuming that the ratio of J, to / is the same as for the Mark VI. bullet. This ratio is, of course, the same for all shot. Thus, for the Mark VI., . we x 2600 = 5992, which is so near 6000 that it does not matter which value we use. | 30. It is worth while to notice that, since /and /, are each proportional to the weights of shot which have the same diameters, we could have deduced their values for one bullet from those for the other bullet and the weights of the bullets. The weights of Marks VI. and VII. are given in the ‘ Musketry Regulations’ as 215 grains and 174 grains respectively. Consequently, for the Mark VI. bullet, we find l 215 l=2100x 174 77299 PANES) assuming that J and /; for Mark VII. bullet are correct. - Motion of a Spinning Projectile. 347 31. In the following table the values of y calculated from equations (33), (34), (35), and (36) are put side by side with the values from the ‘Musketry Regulations.’ Of the pair of values given in one column at 600 yards range, one is calculated from (35) and the other from (36). A similar remark explains the pair of values at 700 yards in another column. Marx VI. Marx VII. _ Range 7 Y y Y in yards. | (Musk. Regs.). | (calculated). || (Musk. Regs.). | (calculated). 500 21°5 21°5 20:2 20:16 600 30°0 30°6', 32°5! 25°9 ) Co ro) S on ow oS oe ae) io 50° He OO wh Bib oOOANOS PRD pCO mt Gx HS TDD CUE OO DDO OH 7.09 0 OCD COMNOMNSNONSCMOOUNUHSS — fey) S cm) OH en Ore B09 OS LOD DR ee et SCNWTWHO NW AOADBR AWS wns & OOo oe & 09 09 DO DOD RH ee ore to (se) (se) HOow»r WWWNN HH Ree OH Coord CmewWHRDe BD SUNT FAO OQ WO ATO FS OVNI DO bo bo OO Ol O11 ORR AT RE OD O Oru C9 09 C9 1D DOR RH St et Oo > © O1 09 Oe Ore bo ee maT OS & OO DF Sorby © ANDO POND OO bw Ie bo eo 0 OBO AND Wee Rom 32. Motion of the Axis of the Shot.—Reasoning from the behaviour of a spinning-top, we should expect that a pro- jectile with sufficient spin would keep its nose forward throughout the motion. But in the case of the top the force acting on it, namely, gravity, acts in a fixed direction, whereas the resistance to a projectile rotates with the line of flight. The precise behaviour of the axis of the shot cannot, therefore, be foreseen from that of the top. In the follow- ing investigation it is shown that the axis pursues the line of flight while lagging a Jittle behind it, the angle of lag increasing with the range. Moreover, the axis of a shot having right-handed spin is deflected a little to the right (and for left-handed spin, to the left) of the line of flight. The action of the air on the side of the shot exposed by the deflexion of the axis causes a shot with right-handed spin 348 , Mr. J. Prescott on the to veer bodily to the right, giving the deviation known as Drift. In the following investigation this drift, as well as the deviation due toa side wind, will be determined as far as possible. The effect of the wind is already known, and is given in Professor Greenhill’s ‘ Notes on Dynamics,’ but the theory of drift does not seem to be well known, for Professor Greenhill’s proposed formula for it, for which, however, he claims no theoretical basis, differs very considerably from the result arrived at in this paper. Since the drift is cal- culated in this paper from the same differential equations as the wind effect, and our formula for the latter is known to be correct, this should give confidence in the calculated drift. 33, In fig. 4, OApg is the vertical plane containing the Fig. 4. 4 N‘ axis of the gun, OA being the direction of departure of the shot. The figure is drawn in this position because the geometry seems more intelligible than it does in its true position. O is the centre of mass of the shot, H the point where the line of resistance meets the axis. Og is the direction in which O would be moving at the instant we are Motion of a Spinning Projectile. 349 considering if the resistance always acted contrary to the motion of O, which is what we have assumed in finding the equation to the trajectory. That is, AQg is the same angle a as in fig. 2. The angles w and y are small component angular deflexions of the axis, z being the lag of the axis behind the line of flight, and y the deflexion to the right as seen from the gun, the spin of the shot being assumed to be right handed. Suppose a wind is blowing with velocity w perpendicular to the plane of the trajectory, and, assuming that Og is the direction of the velocity of O relative to the earth, let OQ be the direction of its velocity relative to the air, so that tan @= —. Although Og is not the actual direction of motion of O, yet it will avoid complications to assume that it is at present, and we can correct our results afterwards. Moreover, we shall assume that the resistance acts parallel to the motion of O relative to the air, that is, parallel to QO. If the shot were spherieal, this last assumption would be quite true, but for a long shot, such as a bullet, it is certainly not true, but we shall con- sider later what modification is necessary to correct this assumption. 34. Let us denote by B the moment of inertia of the shot about its axis of symmetry OP, and by A its moment of inertia about OY, which is perpendicular to OP. The axes OP, OY, and one perpendicular to both of these, are a set of principal axes of the body. 35. We shall denote by v the number of radians through which the shot has turned, in the right-hand direction, about its own axis OP, relative to the moving plane PON. The shot has angular velocities (¢—#) about NO; ¥ about OY; and p about OP, this last being in the right-hand direction. When these angular velocities are resolved along the three principal axes OP, OY, and the third axis not shown in the figure, the components are p—(a—&) siny; ¥; (4—&) cosy; the third axis being taken on the side of the plane AOY opposite to N. Therefore the kinetic energy of the motion is U=sA{y’?+ (6 —4)? cos? y} + $Bip—(a—#) sin y}?. The force R can be resolved into components R sin B and Reos@ parallel to ON and gO respectively. The com- ponent Reosf can be resolved again into R cos cos w and R.cos § sin x parallel to PO and YO respectively. Denoting Phil. Mag. 8. 6. Vol. 34. No. 202. Oct. 1917. 2B 350 Mr. J. Prescott on the OE by ce, the component Rsin 8 hasa moment Rsin£.¢ cos y about OY in the direction in which y increases ; the com- ponent Rceos 8 cos wx has a moment Reos@cosx.csiny in the same direction about OY ; and, lastly, the component Reos@sinz has a moment R cos 8 sinw.ccosy about ON in the direction in which 2 increases, regarding Og as fixed. Thus the couples increasing x and y are respectively Re cos 8 cos y sin & and Re(cos @ sin y cos «+sin 8 cos y). 36. Now let us assume that all the angles involved in the kinetic energy and the couples, except a and v, are very small. Then the couples become approximately Rew and Re(y+ 8), while the kinetic energy becomes =3A{y + (4—8)"} 44BE-y(4-a) Hence if oe Shs — 6) + Byles oU On o2 =BUi-y(¢ -2)}, =) o= = —B(é —4) {5 —y (¢ —4)}. Using 2, v, and y as generalized OCEANS, Lagrange’s equations of motion give d 0U eel. Sy dt Ov Ov that is, al OUR dt Ov whence Bip—y(a—#)}= a constant= Bao say. Motion of a Spinning Projectile. 351 Also ae eh Re depuanen that is, £ {—A(é—@) + Bay} = Rea, or Aé+ Boy —Rev= Aa. Again, d QU OU ae OS ee Be(y+- 3); or Ay —Boé—Rey= —Boa + Ref. 37. We will now return to the modifications of these equations that are necessary In consequence of our wrong assumptions. Since the resistance does not always act in the plane AOY, it is clear that the centre of mass of the shot will deviate from this plane to some extent, so that the velocity of O, instead of being along Og as we have assumed, will make a small angle, which we shall denote by », with the plane AOY, and we shall suppose 7 to be positive in the same direction as y. Moreover, the velocity of O will make another small angle with the plane NOg, this small de- flexion being likewise the result of the fact thatthe resistance does not act along the line of the velocity of O ; this latter angle will be denoted by e, and will be reckoned positive towards Op from Og. The new assumptions amount to the same thing as saying that the line of flight is not in the position Og, but makes small component angles (#—e) and (y—n) with OP instead of w and y as we assumed earlier. The angle 8 must still be regarded as the angle between the true and the relative velocity, and is not now measured from Og, but from the new line of true velocity, so that tan B is still = 38. Furthermore, the line of resistance to an elongated shot does not act along the line of the velocity of the shot relative to the air. It acts in the plane containing the axis and the relative velocity, but for an ordinary shot it will make a greater angle with the axis than the relative velocity makes. Suppose @ is the angle between the relative velocity and the axis OP, that is, 6 is the angle which has components (y—7+ 8) and (e—e), and suppose 6! is the angle between the resistance and the line of relative velocity. The relation between 0’ and @ depends on the shape of the 2B 2 352 Mr. J. Prescott on the shot, mainly on the ratio of the length to the diameter, but partly also on the sharpness of the nose. For a shot of given form 6’ is a definite function of @ which is zero when 6 is zero. Let 0 =7(@): Expanding this in powers of 0, : , 6’ = (0) +6/'(0) + S fon ae To make 6’ zero when @ is zero /(0) must be zero. Then, since @ is always small in the actual effective flight, 6’=0f'(0) approximately. We may write this =f...) is where f is a constant greater than unity for any ordinary shot. It is conceivable that fcould be less than unity for a flat-nosed shot. 39. Our conclusion is now that the resistance makes an angle f@ with the axis of the shot. Consequently, the small component angles between the line of the resistance and the axis OP are f(~—e) and f(y—y7+ 8) instead of x and (y+) as we assumed in deducing our equations of motion. Our new conclusions change the couples from Rew and Re(y+ 8) to Ref(e—e) and Ref(y—n+ 8), but they do not affect the kinetic energy. Consequently, our cor- rected equations are Aé+Boy—Refia—e)=Aa. . . . . . (88) Aij— Bot—Ref(y—n)=—Bod+RefB. . (89) It is assumed, of course, that the angle between the axis and the line of motion does not appreciably increase the magnitude of the resistance. ) 40. The resistance makes an angle with OP which has components /(#—e) and /(y—n+ 8), while the true velocity of O makes an angle which has components («—e) and (y—mn) with the same line OP. Therefore, the resistance makes an angle which has components (f—1)(#—e) and f(y—n+8)—(y—n) with the true velocity of O; and thus the resistance has components R(f—1)(#—e) and R{f(y—n +8)—(y—n)} perpendicular to the velocity. These are the small component forces that produce the angular de- flexionseand 7. The component accelerations that accompany Motion of a Spinning Projectile. 353 these angular deflexions are Vé and V%, which are approxi- mately the same as ué and w. Hence o wé=B(f—1) (2). AAR Haass C20) We. Pi! aaah Bin aie CL} 41. Hquations (38), (39), (40), and (41), together with equation (2), namely SE oy (, ae ne and the expression for the resistance Ww W wv? = — — or — — g ! g 4 will give the values of 2, y,¢, and 7. Since the mathe- matics is just the same whether we use / or 1), we need only consider one of them. We shal! therefore use only / in the following investigation. 42. Writing T for we get R al), dil aaa) l u de) duvide ue so that aide the only difference between T and ¢ being the instant from which they are measured. Differentiations with respect to t are therefore the same as differentiations with respect to T. We shall use T as variable instead of ¢. 43. From equation (6) we have Consequently, 44. Now, rewriting equations (38) to (41) in terms of T instead of u, we get ax dy Wel; Ag Se mam pe PO ae tee 2) 354 Mr. J. Prescott on the dx Welf a2 ATs — Bo a 1 gue Ya ae gl? B; (43) de L—E | a ee (44) d = B a Mai These equations are all linear in «, y, e, 7, and these angular deflexions are the result of ¢, ¢, and 8. It is clear, then, that each produces its own effect quite independently of the others. Thus the wind, to which @ is due, produces its effect independently of the effect produced by the rota- tion (¢) and the acceleration (a) of the line of flight in the plane of the trajectory. Then we may consider the effect of the wind by itself, dropping « and & while we do it. 45. Dropping 4 and & from (38) and (39), it is clear that these two equations, as well as equation (40), are then satisfied by 2=0, c=0, y = a constant, 7—7— 6.) ae) We have to show that these values will satisfy equation (41) also; and we need not use any particular expression for R, as it will be seen that the result is independent of the law of resistance. We shall eliminate R from (41) by means of the equation W du 7 Gua ae From this last equation and (41), by division, we get =u 5) =(f—-1)(y—1) +8. The values in (46) make the right-hand side of this become , that is, - while the left-hand side becomes _, AY +8) du =—-uUu a eae dus aN which is the same as the right-hand side. At the muzzle Ww 4=0 and p=", so that the constant value of y is ——. 0 | 0 Motion of a Spinning Projectile. 300 Therefore the particular integrals corresponding to the wind velocity w are c= Oe —U, w w wt aN ag (47) : ae Uy” ca Uo and these results are independent of the particular law of resistance. But it is worth while to note that, with our particular law of resistance, dn wdu w wr de eee a ly ° . 6 (47 a) so that the side wind makes the plane of the tangent to the trajectory rotate with one or other of two constant angular velocities. 46. The result expressed by i means that the shot points its nose directly against the relative velocity, for 8 is the angle between the true and relative velocities, and (n—y) is the angle between the true velocity and the axis of the shot. And, furthermore, the result expressed by Ww ou, shows the nose of the shot points in a fixed direction. Thus the shot, by keeping its axis in a fixed direction, always faces the relative wind. 47. The solution we have just obtained represents only a part of the motion of the shot. The remaining part, which has to be superposed on this motion, will now be considered. The nose of the shot cannot, of course, suddenly face the wind as soon as the shot leaves the gun, but it will be seen that the axis begins by gyrating about the line of relative velocity as a mean position. 48. Having accounted for the effect of 8, we may now drop it from our equations, since they are all linear. Then let us put z=at+y, poet, where i=,\/—1. Now multiplying both sides of (43) by 356 Mr. J. Prescott on the and adding to the corresponding sides of (42), we get, on neglecting 8, dz ede | Welyi Aas 1Bo dT rrr gi? \© Similarly, multiplying (45) by 2 and adding to (44), again dropping B, | | AB 7 (A —~iBoT). (48) dnp a a pork a (49). We will write, for shortness, m for se and r for : ra then equation (48) gives @ne no ES 9g 72 —2im 7 (c—W) = T (L—2imT). . (50) We have to solve equations (49) and (50). 49. From equation (49) Therefore, Beds 1 Gee a fig ot a ; Be _ftidp 1 pty meme iat? 2 Foe ee Substituting for z and its differential coefficients in (50), writing ¢ for a and multiplying up by (f—1), the result- ing equation is fp ieee Ba Ve Bin) e vl (7 +2im) fb =(f-1)¥ (1—2imT). . AS) This equation being linear, the complete value of ¢ consists of two parts, one of which is the particular inte- gral corresponding to the terms on the right-hand side, and _the other is the complementary function, that is, the value of ¢ satisfying the equation obtained by dropping the terms on the right-hand side. We will first deal with the comple- mentary function. | Motion of a Spinning Projectile. 3957 50. Putting . re (8s Cienega Rial (BD in equation (51), and dropping the terms on the right, the equation for the complementary function becomes d? ; l rt poe ==) as es _ { (f= 1)em—m?T + a} Ge Next, putting f+1 Mac) *, MPA One UN Es) our equation becomes dy (f= Sime fh )a} ps: et 4 ee ( m Pay TT X=0. (54) Writing this, for the sake of shortness, 2 Bee 0,» «cDegnae en (35) dis and then making the substitution espe |. a ae eee (oe) then equation (55) becomes d’p du \? Bedpudp, |. dapiiay . mye oe + op + 20 ar aT + ip rae =0. SE Ly, Before proceeding further with the solution of this equation, it will be useful to get an idea of the magnitudes of the quantities involved in equation (54). 51. For the present we shall deal with the Mark VII. bullet, for which d (diameter) = 0-303 inch=}$ inch: | /=2100 feet, 1, =4840 feet ; | h (length) = 1:28 inches, fs (08) 1p = 289C, | w§ 21TUg BA. | O= 334 = 21800, J the value of w being obtained from the knowledge that the bullet makes one turn on its axis as it leaves the muzzle for 83 calibres of forward motion, that is, for 10 inches of forward motion. We should need very accurate details of 358 Mr. J. Prescott on the the shape and composition of the bullet to caleulate the precise moments of inertia B and A. To get approximate values we will assume something reasonably near the truth, namely, that the bullet is a homogeneous cylinder 1:2 inches long with the actual diameter of the bullet. Also, the length c, which is the distance from the centre of mass of the bullet to the point where the line of resistance meets the axis, cannot possibly be greater than 0°7 of an inch, and is very likely less than 0:4 of an inch. We will assume the value 0°4 for this at present. Of course it is a very un- certain quantity, and it is quite possible that its value is not more than 0°15 inch. However, taking the value 0:4, which is very likely too great, BO. id?o he x ie > m= yaN — 2(qbyh? + qed?) = ROL = 9952. 3 \d m? = 990200, L (59) Wel cl | sive = =p SOUL gA P+ he J All that we know about fis that it is sure to be a small number greater than unity. It might be 2 or 3, it might conceivably be 5 or 6, but it could not be much greater, nor is it likely to be less than 2. Thus rf is very uncertain on account of both ¢ and /, and reasonable guesses at its value might range between 60,000 and 600,000. 52. When u <1063 feet per second we must use instead of J, so that, writing 7, for the new 7, m= 80170 x Soo" = 184770. Also ch ae U U which is as small as 2 when w=2800 feet per second, and as large as 12 when u=403. 53. Without substituting the preceding numerical values in the expression for o, it is clear that this is a very large number. It is equally clear that the term +(f?—1), which is part of the coefficient of T-? in the expression for a, is insignificant compared with the part rf. We may, there- fore, drop altogether the former part. 54. Let us now return to equation (57). Since we re- placed one variable y by two variables p and yp, we are at Motion of a Spinning Projectile. 359 liberty to assume one relation between these variables. Suppose, then, that dd? dp d Page oF F201) de) eeet (60) dT dT Then (57) and (60) give dp du \? Pla) +op=0. ° . ° ° (61) Multiplying (60) by p the equation becomes di (yap av (Pat) => dy _ al Wea: where FE is some constant. Again, from (61), YD 1 d’p (a) =e, a 5, gl ERG whence (62) 2 Now, there is no reason ly = a should be a large quantity, and we know that ois large. Let us suppose, for the moment, that the latter is much larger than the former. Then dp? : it) =? approximately, . . . . (64) and consequently =m ‘—1)2 rf. pe) 55. We can now show that the approximation in (64) is justified, for we get from (65) log, p= log, H—4 log, o. Therefore, hd? Pea 5 (- =) > aga o al od ~~ dcdEt 16 — Now the largest term in o is m?, and this term disappears from the differential coefficient of o Consequently both 360 : Mr. J. Prescott on the terms on the right of (66) are small quantities, and all that we had to show to justify (64) was that the whole quantity on _ the right of (66) is small compared with the large quantity o. ‘Thus (64) is justified. 06. From ce. (64) d (f—1)imT pein == =+ LN er a 5) ary Pa - (67) approximately, since the fraction in the brackets is a small one. We might carry the expansion further, but nothing would be gained by doing so. We only want to discover the type of motion, not the precise amount of it. Integrating (67) we get p= + ({4/me— 2 — oe Z fame pal a {nef + Jifein V7 HYP 1) log (mT 4./nloa )} +F, (68) F being a constant. Also p=Ho* =E( Sapa aa! +a on } i to the same degree of approximation as we used in (67). Therefore, T2 1 fe meknoubes, Clue pe =E(;, a) Dm bythe i(F+s) x {1+ intl rf ) € 5 sO M=mT+ Vin! of, ) (71) = Vif sin T/T mT af § The double sign in (70) gives two separate results, and since the equation for y is linear the sum of the two results, with independent constants of integration, is a value for x. Motion of a Spinning Projectile. 361 Hence ue + x= = — a) «(f—1)mT « {1+ op) an eM Ms (72) E,, E., F,, and F, being four independent constants. 57. On retracing the transformations we find that dy amy 2 Cae pe HEY ea) x having the value given by (72). On integrating this last equation we should get two more constants (one real and one imaginary), thus giving six independent constants in the value of yr. ‘This is the full number of constants, for equation (51) is of the third order in w, which contains two variables « and 7, and we need three constants for each. 58. If ¢ is always small, then y is sure to remain small, consequently z will remain small also, as equation (49) shows. That is, we may be sure that the shot will be stable with its nose foremost if we know that ¢ is always small. Stability will be assured, then, if we know that H, and H, are small and that the terms in the expression for ¢ do not greatly increase as T increases, the functions such as eT, which have imaginary indices, being treated as unity. Now the largest term in ¢, neglecting the factors with imaginary indices, is = \'E Mes- Ym a( F4+1) m1?—rf) * ieee: é ~1\3(/-D,,--4 a (ua ) Ti Ta x “ (74) Now, MT ae and the greatest value this can have is 2m, which occurs when T=, and its least value, on the assumption that mT? > rf, ism. Thus MT™ does not greatly increase as T increases, and the other factors involving T in (74) decrease as T increases. If, then, m7I?>rf we may safely assert that o remains small if it is small at the beginning, which is all that is necessary for stability of the axis, 362 Mr. J. Prescott on the 59. We have been assuming throughout the preceding argument that m?T?>rf. If this inequality is reversed, then the quantity we have called s changes its form. Thus S =(a/ — ui ait =e a aT, = inf rf) logel — log, (v rf + V rf —m?T?)} 471M vf —mPT?. Therefore ¢“ contains a factor T”’%, which is such a large power of T that it increases very rapidly as T increases. However small ¢ may be initially, therefore, it very quickly becomes great, and consequently (zg—r) becomes great. But for a bullet travelling at a great speed, the angle w, which measures the angular deflexion of the line of flight, will not change so quickly as z, which is the cause of w. That is, ze must be large before y can become large. It follows, then, that the shot will be stable travelling nose fore- most provided m7I?>r/f, but unstable if this inequality is reversed. 60. We will consider what sort of motion is represented by equation (73) when the axis is stable. It would be a considerable feat to integrate this equation to get yw. After all we do not really need to get the precise motion. It will be sufficient to know the type of the motion. It is safe to conclude that wW will be similar to yf in that it will contain a factor e™™, and that its terms will also contain factors e* and e-**, Likewise zwill be similar to both in these respects. Consequently all the four quantities e, 7, #, y will consist of such terms as P cos (KF +mT=+s), where P is a function of T, and F is a constant. Now m is such a large number that (mT +s) increases very rapidly, making the periodic terms pass very quickly through their successive maxima and minima. Then each of the two terms in y which have coefficients H, and EH, indicates a very rapid conical motion of the axis, and a consequent helical motion of the shot in its path. The angles of the cone are not constant, and the helices do not lie on a circular cylinder. In the complete motion of the axis the two conical motions are superposed ; that is, the axis of the shot describes one cone about a line which itself describes the other cone. The angle of one of the cones, the one represented by the term containing the factor e—*, is a rapidly diminishing angle. Motion of a Spinning Projectile. 363 61. We have found that the shot is stable with its nose foremost provided mT? > rf, that is, provided B*w? iP Welf 4A? Ww? Ge? which may be put in the form [Biri S> io)! ey eh rere Ud Oi), If we write N for the number of calibres of forward motion of the shot for one turn on its axis at the muzzle of the gun, then, d being the diameter of the shot, @ Ug On = Nd’ ° ° . . . . (76) so that the condition for stability of the axis becomes Amu? s 4A fcl?W 2 = N?2d? gu : vv awe:(2) N"< 7 ® Ware et ee (77) To ensure stability throughout the motion we must make N? less than the least value of the right-hand side of (77), and this occurs when w=w. According to our earlier assumption that the shot may be treated as a cylinder for the purpose of finding its moments of inertia, the above condition for stability becomes, when u=wo, 7 4a? ed, 2 BE a i f ygh? + e@? 8c (78) We have already shown (section 8) that / is proportional to h for similar shot, and c¢ is clearly also proportional to h, whence it follows that / is proportional to c. Moreover, fis the same for all similarly shaped shot. Thus the condition for stability of the axis gives the same minimum value of N for all similar shot of different sizes. Hven if we take the accurate moments of inertia, instead of assuming the cylin- drical form, it is still true that the condition for stability gives the same minimum value of N for similar shot. Since f depends on the shape of the nose the similarity referred to here is absolute similarity of shape, and especially of the nose. 364 Mr. J. Prescott on the 62. Stabrlity of the Mark VIT. bullet.—For stability of the axis in this case De Tee L200 f Al? +3d?8 ¢ ? that is, L a 6 x 0°303? 2100 f 4(1:2)? +-3(0°303)? 8 ° or 15°38 IN ieee eal << V7e (79) Tt f were 4 and ¢ were 0:4 inch, then we should get. 15:38 V12 V16 ° - that is, ie PL Actually, for this bullet, INE) and the fact that the bullet is stable for this value of N shows that 33 In the case of the Mark VI. bullet, for which /=2600, the result corresponding to (80) is | ’ ° 2 fe< ("35") feet, or <2°6 inches. . (81) 2600 | N< 42 < AGT nae PM p00) | oo and the result corresponding to (81) is fe~< 3°22 inches. |.) 4) eae 63. Particular Integral of Equation (51).—Equation (51) can be satisfied by an infinite series of powers of T beginning with the first power. Thus, assuming that o=(f-1)2 { W,T+H,P+ HP + ae J}, equation (51) gives T{2H,+3.2H,T+4.3H,1?+....... } +(f+1—2imT)}H,+2H.1T+3H,W+...... i —f(r+2imT){H,+ H,T+H,T7+...... =1—2imT. Motion of a Spinning Projectile. 365 Hence Cf fr) et th ee ei (85) (4 + 2f—fr) lal 2Qim (f+ 1)H, ad 21m, — —2im and for the rest {n(n—1) +n(f+1) -fr} HH, = 2im(n+f—1)H,_,. (87) For an actual shot the number 7 is so large that small integers are negligible in comparison with it. Conse- quently the equations (85) and (86) give approximately the following results :— 1 ‘Qim Hi ee H,= Tr ° ° ° ° (88) After this the relation between successive coefficients is <2) 2masie— 1) (39) n fr—jfn—n n—-1? which, when ? is small compared with 7, is approximately the same as 2im(n + f—1) ry? fr nm—1° But when n? is large compared with r the relation in (89) is approximately H= (90) __ 2im nu n n—1 In a region intermediate between the two we have just considered, that is, in the region where n’ is nearly equal to yr, the denominator of the fraction in (89) is very small while. the numerator is large, and if V fr+4f?—3f happens to be exactly an integer the fraction becomes infinite when n is equal to this integer. We need not, however, consider the infinitely improbable case in which this number is exactly an integer, but it should be noticed that, when n is near this number, equation (89) shows that the coefficients go on increasing as 7 increases until we reach the point where (n?+jn—fr) is greater than 2m(n+f—1). Yet the earlier coefficients of the series, to which equation (90) applies, diminish fairly quickly, and the corresponding terms of the series diminish also because the ratio of mT to r is a very ia. Mag. 3. 6.. Vol. 34. Ro. 202. Oct. 1917. 2C 366 Mr. J. Prescott on the small fraction. Itis clear, then, that the series first converges, then diverges, and finally converges absolutely. The series will be useful if we can show that, by stopping at the point — where it first converges, we get a good value for d. This we shall now prove. 64, Let us put $=¢i+(f-1)7 { HiT +HT?+ ahihiattiowe el He (91) the series stopping at the nth power of T and the coefficients having the exact values given by equations (85), (86), and (87). Then, substituting for ¢ in equation (51), a d liegt +(f+1—2imT) an G + dim oy =2im(n+f)(f-1)F HAT". . (92) Now, equation (51) gives the value of ¢ caused by a disturbing force represented by the two terms on the right- hand side of the equation, and (92) is a similar equation, so that we may regard the quantity on the right-hand side of this equation as a disturbing force giving rise to ¢, Just as the disturbing force in (51) gives rise to ¢. If the disturb- ing force in (92) is small compared with the disturbing force in (51), then it follows that ¢, will be small compared with o, for physical considerations tell us that a small force must produce a small effect. Since mT is very large compared with unity, we need only consider the ratio of the disturbing force in (92) to the imaginary part of the disturbing force in (51). This ratio will be least, of course, when the dis- turbing force in (92) is least. If, by increasing n, we can decrease the right-hand side of (92) we are sure to be decreasing @,. Now the ratio of the disturbing force nffecting @,, when we carry the series to the nth power of T, to the disturbing force when we carry it to the (n—1)th power, 1s n+f ebay Paeg 20 Ff . nth term of series n—l+fH, 1 n—1+f ~ (n—1)th term of series’ n-+f For all but small values of n the fraction is nearly n—1+f unity. This means that, as long as the terms in the series for @ are decreasing, @; is becoming smaller as m increases, and therefore the series approaches the value of ¢. This Motion of a Spinning Projectile. 367 justifies us in assuming that a good value of ¢ is obtained by carrying the series only as far as the point where the terms are small. 65. As we are only going to use a few terms of the series, we may make use of the relation (90) between the coeffi- cients. Then the result is o=(f-1 ef-s = ne (Vases. 2 ee — arin? +(24/)73 IAN B+/ et... 4, (93) where anes 1 Bol gA TON u Wel 1 Bag u, : a “Ta ee (94) Using equation (76) to eliminate w from this, we get 1 27Bg uw _ 1 27K? uw ae fNWelaef Ned u? 97 ee. where K is written for the radius of gyration of the shot about its axis. The coefficient of = in the last expression for t is the same for all similar asi, provided N is the same for all, for ¢ is proportional to the ieee dimensions of the shot. "For the rifle-bullets, for which N is 33, _a 44 Uo ad io fa ~ fe 42 u’ te ee T= T£ we assume that c= ad, then ik aye teh = mG = Wy, jar WO) When wu has dropped down to for the Mark VII. bullet, then 3 T= 355 Se!) WA ee) 2C2 1 G Mo» about 480 feet per sec. 368 Mr. J. Prescott on the which is sure to be less than a and might reasonably be as emall as = Then our series of powers of 7 is sufficiently convergent to be useful at any effective range of the rifle- bullets. 66. For any shot similar to a rifle-bullet, but of larger size, the drop of velocity for effective firing is smaller than for the rifle-bullet ; this follows from equation (8), from which it appears that, for the same fall of velocity, the value of « is proportional to 1, which is proportional to the dimen- sions of the shot. Thus the angle through which the line of flight of a three-inch solid shot, similar to a rifle-bullet, would turn for a given fall of velocity, is about ten times the angle that the line of flight of the bullet would turn through for the same fall of velocity. For instance,* the Mark VI. bullet has an angle of departure 28'2, and an angle of arrival 36'-0, that is, “the line of flight turns through 1° 4'°2, in the 500 yards range, during which the velocity fala Grom 24100) to abont 1180, Tho lemon flight of a similar three-inch shot would turn through 10°42’ for the same fall of velocity; but as X is also the product of J and a function of the velocity, the range for the larger shot would also be about ten times as great, that is, about 5000 yards instead of 500 yards. For effective firing it is clear that a cannot be a large angle with high-velocity projectiles, and consequently the fall of velocity for the larger shot is smaller at maximum effective ranges than it is for the smaller shot. 67. It follows from the preceding reasoning she for similar shot of different sizes, 7 is less at the longest effective ranges for a large shot than for a small shot. Hence, since the series in (93) is sufficiently convergent for a rifle-bullet, it will be all the more satisfactory for any other spinning shot. 61. We have now got all the results which make up the complete expression for @ in equations (47), (73), and (93). The value of @ given by (47) is inet in=iGh =i7 ori? by (47a). (99) The complete value of ¢ is the sum of the three values in (73), (93), and (99). The four constants H,, Hy, Fy, Fs, eae ed in (73) are required to make «, y, and their rates of increase all zero at the muzzle. Four new constants are required to suit the motion after the velocity has fallen Motion of a Spinning Projectile. 369 below uw, at which the law ‘of resistance changes, and these new constants are determined by making 2, y, and their rates of increase continuous at the critical point where U=Uj. 69. Deviation due to Wind.—In equation (47) we have found the angular deflexion, 7, of the line of flight due to a wind perpendicular to the trajectory. But the useful form of the result will give the linear deviation of the shot from the plane in which it started its motion. If Z, denotes this linear deviation, then dLy = tan n=7 approximatel 7 aa n= App y> Wry mes yas Therefore, dy = | Lai 4 See U9 ‘udt = =w =w}—— — A, U Uo =u(1-=), ELIS ody 5. | 2 emer duel) Uo a result attributed to Colonel Younghusband in Greenhill’s ‘ Notes on Dynamics.’ 70. The values of ¢ and X can be obtained from Bash- forth’s tables giving T, and S,, and thence the wind devia- tion can be calculated for any range X. We may also calculate the deviation in terms of X by using the expressions for ¢ given in equations (3) and (15). By means of these equations we find Xe Zy=" {tet 1X}, ES Sb) when X< X,, and eae ens a wh 4) (102) 1 (4 a se when X>X,. In the following table for wind deviation the first two columns of values are calculated by means of Bashforth’s tables for T, and S,, and the last column by means of equations (101) and (102). The good agreement between 370 Mr. J. Prescott on the the two sets of results for the Mark VII. bullet may be regarded) as a justification of the law of resistance used in this paper. Wind Deviation when Velocity of Wind is 10 miles per hour. Winp Deviation. Range in yards. Mark VI. bullet. | Mark VII. bullet. | Mark VII. bullet. 100 1°5 inches. 1°33 inches. 1:34 inches. 200 6:1 5] 58 300 14°4 13°6 13°6 400 26:9 25°5 25°6 500 3°66 feet. 3°49 feet. 351 feet. 600 DAT 5°31 5:34 | 700 tee 763 7°69 800 101 10°35 10°44 / 900 12°8 13°44 13°45 1000 15°7 16°8 16°8 1100 19:0 20°6 20°4 1200 22°5 24°7 24°3 1300 263 29°1 2856 1400 30°3 33:9 | dare 1500 34°7 39:0 le eestsces 1600 39°4 44-7 44-0 1700 44°4 50°7 49°8 1800 49'8 Dine lho ey Gia 1900 556 64:2 1) 163u 2000 618 71:8 1) OSG: | 72. The value of that we should get by integrating: both sides of equation (93) gives the amount of angular deflexion of the line of flight of the shot from the path that an equally resisted spherical shot would follow if no wind were blowing. This deflexion occurs then with every projectile from a rifled gun, and the amount will be the same at the same range for all equal projectiles fired from similar guns. The angle e affects the range only and need not be considered, since it has only the same effect as a slight alteration in the resistance. But the angle 7 indicates a deviation from the vertical plane containing the axis of the gun, and this is something that can be observed as wind deviation can be observed. 73. Now dip de .dy oy at at ae Motion of a Spinning Projectile. 371 Since r contains the factor /, and 7 does not contain 1, it follows that the expression on the right of this last equation does not involve / at all, and is therefore continuous as u passes through the value u. Integrating (103) we get n= =(f-1)9(F) {C+it—(Q4/)(R+f)}r+... .t. (104) If Z, denotes the linear deviation of the shot correspond- ing to this angular deflexion ot the line of flight, then ate =n approximately DAT Therefore, = (; ax. =m al, ee =: al =a = Slate Os c+) ae \. _ (105) The constants C and D have to be determined so as to make Z, and its rate of increase both zero when u=wu,. The constants so determined will, however, only be applicable when wu is greater than 1060 feet per second. ‘The new values of the constants that must be used when u has fallen below 1060 have to be determined so as to make Z, and its rate of increase continuous when w=1060. 74. In the case of the rifle-bullets, which are the worst cases of all spinning shot, we have shown that the value of 7 probably does not become much greater than ¥ at effective ranges. Consequently, the ratio of the magnitude of the term containing 7° in (105) to the term containing 7° pro- bably does not become greater than 1 (2+f)(34f) 3? gn mer” 5 372 Mr. J. Prescott on the which, since fis greater than unity, is itself not greater than - and is probably very much less. There will be very little error therefore in omitting all powers of +t beyond 7°. Then 2 ty=(f- Vg (2) {D+Clog.r + 3rh. Now, by equation (95), lar Kaa a Uy my Ned a fee where a is a constant independent of / and the same for all similar shot of different sizes projected with the same N. Therefore, =4s- Dy (EZ) 4 {0+ Clog. + (2) } Bee Ge! paws vo 4 (2 “ey ; (106) th ( where G is a constant that does not contain J, and has there- fore the same value throughout the motion. The constants C’ and D’, however, change their values at the instant when wy, the change being such that Z, and its rate of increase are continuous then. 75. The constants for the earlier part of the motion, wil u is greater than w,, are determined by the condita that Z, and its rate of increase are both zero when w=Uup. These conditions clearly give D'=-1, C'=—3. Consequently, when u > 1%, — \3 La rt Gl (=) —3 log -1}, G07) which : pall 3x ay Gif tet axa). |), aaa by equation (5). Let the value of Z, when u=, be denoted by Z,’, then, from equation (107), ps ior ey Ada oe } By) =! Gr 4 =) —3log, —1p. . (109) Motion of a Spinning Projectile. 373 Also, when u=uw, dL faa 2 3 Oe nF @ — =a. Hie'sC110) 76. When wu has fallen below u,, the factor /? in (106) must be replaced by 1,7. Then it is convenient to write the equation in the form =e) j 3 at ve Git} D" Ae CO" loge + + G) ie (111) To make Z,=Z,’ when u=u,, we get oe) =—(*). Again, from equation (111), when vu=w, _ Ory i. Star Bt =- Gl {-— - Scare ee 2) and to make the rate of increase of Z, continuous we must make the right-hand sides of .. and (112) equal. Hence v=-a(2) (1) {-(9} Thus, when u< wy, —gradt—! a(m) £ 12(@Y 12 by — ef = 1 @() (4 (“) I 3 —3(WP + Poe —?) log, ub eae i *a() {13° a2 aoe (i240 —1)}. (113) Equations (108) and (113) give the amount of drift at all effective ranges. 77. In the expressions for the drift of a shot we have the very uncertain number /, as well as the quantity c, involved in G, which we can only guess at. But since these quan- tities occur as factors in our expressions, we can at least give the ratio of the drift at any one range to the drift at —3 374 Mr. J. Prescott on the any other ; and if we know the drift at one range our results give us the drift at all other ranges. 78. The numbers in the first and third columns under “‘ Drift” in the accompanying table give the values of jae Zi: “ay f-1G? ic: for the Marks VI. and VII. bullets, and the numbers in the second column are reduced from those in the first column so as to give 7 feet at 1500 yards for Mark VI. All sorts of estimates of drift are given by different people. In the 1909 Edition of the ‘Musketry Regulations’ the drift is stated to be 7 feet at 1500 yards. In Fremantle’s ‘ Book of the Rifle,’ it is stated that the 1898 Edition of the ‘Musketry Regulations’ gives the drift as 11 inches at 1000 yards, and 23 inches at 1200 yards. If we take 7 feet to be correct at 1500 yards for the Mark VI. bullet, the table gives the drift at 1000 yards and 1200 yards as 23 inches and 40 inches respectively. It must be remembered, how- ever, that the shape of the bullet will affect f, and therefore affect the drift. If, therefore, the 1898 and the 1909 Editions are referring to different bullets, no comparisons can be made between the statements in the two EKditions. But we can compare the two values of the drift given in the 1898 Edition for different ranges, for we are told that the muzzle velocity of the bullet to which these results refer is 2037, which is very little different from the muzzle velocity given for the Mark VI. bullet in the 1909 Edition. If 11 inches were the correct drift for the Mark VI. bullet at 1000 yards, then the correct drift at 1200 yards, according to our table, would be 19 inches instead of 23 ag given. The difference is not outrageous when we consider the diffi- culty of the observation. 79. In Fremantle’s book other estimates of the drift are given. Mr. R. L. Tippins’s estimates are 10 inches at 500 yards and 4 feet at 1000 yards. The ratio of these is 4°8, whereas the corresponding ratio for the Mark VI. bullet is 8°2, a rather big difference. Mr. John Rigby gave 10 inches at 1000 yards, which agrees very well with the estimates given in the 1898 Edition of the ‘ Musketry Regulations, but not at all well with the estimate in the 1909 Edition. Possibly these different estimates apply to widely different rifles and different bullets, and may, therefore, not be so discordant as they appear. Besides, it Motion of a Spinning Projectile. 375 is not merely a question of range, for the theory in this paper shows that the muzzle velocity, as well as the shape of the bullet, has its effect on the drift at a given range. Drirt. Range in yards. Mark VI., Mark VI. taking 7 feet Mark VII. at 1500 yards. | So = | F100) | 0-0085 0-064 inch. 0:0053 | 200 0:0384 0-29 0:0249 300 0:0987 0:74 0:0663 400 0:202 esi 0:1910 500). | 0:365 2°75 0:268 600 | O615 Gm 4°62 0°473 700 0:960 (ial 0801 800 1:44 10°8 1-260 900 2:09 15°7 1'946 1000 2°92 21:9 2°905 1100 3°98 29°9 416 1200 5°28 ee a 5°82 1300 6°89 4°31 feet. 795 1400 8°84 5:4 10°63 1500 11:18 7:00 14:00 1600 14:00 87 18°17 1700 17°30 10°83 23°34 1800 | 21:32 13°34 29:67 1900 26°20 16°44 37°46 2000 31°57 19°76 46°90 2100 38°11 23°85 58°48 2200 45°78 28°66 72°58 80. Drift of a Larger Shot.—When X is expressed as a function of the velocity it contains J or J, as a factor, whereas Z,, the drift, contains /? or J,?, and both J and J, are proportional to the dimensions of similar shot. For the same change of velocity, therefore, the ranges of similar shot are proportional to their linear dimensions, while their drifts are proportional to the squares of these dimensions. For a shot similar to the Mark VI. bullet and on n times the scale of the bullet, corresponding ranges will be n times as great, and corresponding drifts n? times as great as for the bullet. That is, X and Z, being the range and correspond- ing drift of the bullet, then nX and n?Z, are the range and corresponding drift of the larger shot for the same change of velocity. To make this rule strictly true, the muzzle velocity uy should be the same in both cases, and, of course, N must be the same in both cases. Thus a 3-inch shot has about ten times the dimensions of the bullet, and it will therefore have a hundred times the drift at ten times 376 Mr. J. Prescott on the the range. For instance, the bullet has a drift 2°75 inches at 500 yards range ; therefore the 3-inch shot will have a drift 275 inches at 5000 yards range. Again, a 12-inch shot will have a drift 2416 inches, that is, 201 feet, at a range of 16,000 yards. These results apply to solid shot, and will need modification for hollow shot. 81. Let us see what information we get about fand ¢ by assuming that the estimate of drift given in the 1909 Hdition of the ‘Musketry Regulations’ is correct for the Mark VI. bullet. This gives an 3 ee GI? (“) = 7 feet, 1 that is, L 6S 59" ( or _)ai=l foot. 2) Ge But a 27 K? Bae Ned? Bd NNR i Sk cNd_ Ha i gba Wan eek Hence, equation (114) gives ihr by QrK? P y ay Oy I Ned ug? ae ay Taking, as before, . Ke 3d, N=3or we get , f-1id 90 4x 33u' - = if CRIB? asks =0°614e i. fii binierigee eet) Here we have one equation connecting the two uncertain quantities cand f. Corresponding values of ¢ and f satis- fying equation (115) are here arranged in tabular form: f. 2 3 4 5 6 oo — [| | | ce (inches) ...| 0:247 0:329 | 0370 0:395 0411 | 0°493 We have here an upper limit for c. With the above estimate of drift our theory leads us to the conclusion that c cannot be greater than half an inch for the Mark VI. bullet. Motion of a Spinning Projectile. 377 82. It should not be difficult to devise experiments that will give fand ¢ with fair accuracy. I have myself found values of f for the Mark VI. and VII. bullets by the follow- ing experiment. The bullet was suspended, with its axis horizontal, by a thin wire. The direction of the axis of the bullet was observed when it was at rest ; then a puff of air was applied at a small angle with the axis, and the direction of motion of the bullet was observed. The bullet, of course, swings as a pendulum, and it is easy to observe the plane of its motion. We have thus all the angles necessary for the calculation of f. In this way I found values of f which had a mean value about 4:2 for the Mark VI. bullet. My apparatus was rather crude, and more refined experiments would probably give a somewhat different value, but I believe my error is less than 15 per cent. 83. If f=4:2 for the Mark VI. bullet, and if equation (115) is correct, then c=0°38 inch approximately. With these values of f and ¢ the product of fe is practicaily the same as if f were 4 and ¢ were 0:4 inch, which are the values used in equation (82) in calculating the upper limit for N in order that the axis should be stable. 84. Angular Deflexion of the Aas from the Line of Flight.— The two components of this angular deflexion are (a—e) and (y—7). We need not consider the effect produced by the wind, for we have already found that its effect is to make the shot point in the direction of the relative velocity. Also, as we are here only considering the mean position of the axis, we shall not take account of the rapid conical motions indicated by the complementary function in equa- tion (72). Therefore, for the rest of the motion, (w—e) +u(y—n)=2— J Side : =F 21 ae by equation (49) =r { = tint @+f)r— (116) i i DE fh ence by equation (93). For the Marks VI.and VII. rifle-bullets duty 7 D2feu We will work out the deflexions when wz has been reduced to $u, which is about 350 feet per second for Mark VI., 378 Mr. J. Prescott on the and 480 feet per second for Mark VII. In this case we must use J, instead of 1. Then gl @ Gre) + iy nas sare 7 C1). t Meee U2 fr Ue * alle! owtelmre } Now, r is such a large number that the first term in the brackets is very much smaller than the third. Consequently, the approximate results are 36g1,d? x—e=(24+f) Fu (117) d36gl,d (118) | aa Tfcug? en With the values of f and ¢ used in the last article, namely, f=4:2, c=0°38 inch, the values of these angles for the Mark VL. bullet, for heed 1, = 6000, are e—e= 24! 119 y—n= 2 4 a The range corresponding to this drop of velocity is, by equation (16), X = 2800 yards, approximately, and the whole angle, «, through which the line of flight turns in the 2800 yards range is 34°, obtained by adding together the angles of elevation and ‘arrival given in the ‘Musketry Regulations’ and allowing for the jump. The angular deflexions of the axis from ‘the line of flight are therefore small compared with « The lateral deflexion, (y--7), 1s to the left for left-handed spin. It is remarkable that this lateral deflexion, which is a consequence of the lag of the line of flight behind the axis, is greater than the lag in the plane of pursuit. 85. When w=z, the values of (e—e) and (y—7) given by equation (116) are discontinuous because / must he used when u>u, and J; when u ‘ 350 ae PP et Modiola phase.? | % Limestone. ———— | Caninia Oolite. | | Lower Caninia Zone C,. Laminosa Dolomite. 550 (yC,) Crinoidal limestones. | | < rinoidal limestones and dolomites: cherts near the 2 Lower Zaphrentis Zone Z. . eat 250 Avonian. | base. —— —————eSa ee = at Se ~ = eas —— —- = Ae oe bs seas cy = S © in at 2) > ® Q® Se % 7 4 9% 9 | | | | | | Mantle. | lia elie | HEE EA . ' es | \ | “it | ; i i i | |" | ) | Cone. t | 5 ~~ Sed eC aay, 1,—Air-coal gas. 2.—Oxygen-—coal gas. 3,—Air—coal gas with addition of oxygen. HemsArkcitt. -4046 Phil. Mag. Ser. 6, Vol. 34, Pl III. Emission of Iron Vapour in Flames. S Lee ® % ‘ij Oo ® 2 @ he : | | : | 1.—Air-hydrogen. 2,—Oxygen-hydrogen. ee a Rex ‘ » ‘a _ " Fi - 1" aah , ; E 4 7 ‘ wv} , ; oy ‘ i « nL, , . |e ‘, = Rte Suvih ais. ee >, 5 leony cbt Aone faa vo [se aim P », eye Ante: o/h spat a A =" hn’ og > eeeitae ree i > @ Z ay P sk Be . i ay ' ; cv a y u td * Z a ¥ * ae ont) . a ‘ q A ? - 7 1 e. a 4 4 t ‘ “! w= “ 1 % y : ‘ , wl ji ” ants 3 i 1 aa ; wr ‘O ‘ + ' P i pies : 4 : Y : | ‘ ‘ euch poe ae i Q J at i } . ‘ z y : j . ie , ! i I 4 My : i : * a Dd r t r 1 -! . i U ¢ > i ' , : { Us G : . on roam ‘ ' , ; ‘ ee iw Rent tae - ae . eG r ‘ : f Eis. : 23 : : 7) , 2 ; * . ‘ A 4 ? = . } f x ® yy - ~ ’ LU , Xs fe 3 ; ; / i , ; : 2 r fr . 5 . J ‘ ut , . . - | a ; Pie i , ‘ ee) : ; , A k ; "4 => se x | Pas 2 she > tte <<" : ’ ' fT ie — ' r * BS NE te ee $F > ve r SS Se eee eee r x 5 t } 5 arto : ~ eo é ' ns S 0 ; ‘ xf i 4 i ‘ < is ae oe 4 in x (oar " a | : r y Pee 1 , a { x " © . = 1 " ‘ Len + or 7 ; 5 F re, i 12 o { 5 bs thas \ #3 ‘ : 5 do més \ 1 i : < A ; Lae } wee ‘ ‘ py f t . ape x . : - es cae ahi 8 : , : < t ‘ * i 4 if 4 a ) d 2 i > y mg } - / a iV ' ew 4 \ ot 4 ‘ ‘ ' 4 x . - = at . - fi . = 4 . ‘ 5 \ J ~* . ~ * a ; ¥ f . a . . 5 \ . ‘ ‘ eI i ) rt * 4 as ipheas : : un) A ‘ . y t ; f ] x r| y, - , ‘ : : ' ' 3 Tie . i Z , in P i ee aay wer iy ie. ‘ " . i *, * i 7 4 ) t . } hha ‘ A ty Sie oy thee ea ae iS aes | eae ‘ ‘ h P , 3 \ vate ay 5 Asie br StL is Nie tdi aN Aa at Oe Prarie? Guta PRG Mao et eC) gt, Oh, ee ein Cee Vite hdlline Hes 1 tie ie aes Wa chae J Vol. 34, Pl. V, 6 ag. Ser Phil. M 3 a a 1e © (ez) 2 A S atl (aa) eee Fre 4 Dee AM HLE MR ry We: ecke “aWat Dpipmpuase oto) {39 in 5 t as ie ? o. Ser. 6, Vol, 34, Pl. VI. a Phil. M Barron & Brownina. Traces of Lower Bob when Upper Bob struck. 12 waveapomAGB s under various initial conditions. Simultaneous traces of both Bi THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES.] NOVEMBER 1917. XXXVII. On the Flux of Energy in the Electrodynamic Field. By G. H. Livens, The University, Sheffield *. ls A GENERAL theory of the energy streaming in an electromagnetic field was first given by Poynting on the basis of Maxwell’s theory, but on apt a | (Ce nf) Jae + + { ood Thus if we now take and since eae T=+ { av{ (CdA), we could assume Ss = dC ; and this appears to be the proper result in the simplest and most general form of Macdonald’s theory. It differs markedly from his result however, for a reason which will subsequently appear. 390 Mr. G. H. Livens on the Flu of Maedonald’s transformation is obtained in a different manner. We start from the relation av tie) ab Tit | Sd =a (BS) - Jao +2 = | BBaas established in the previous paragraph ; Me then, using the relation B= H+47I, where I is the intensity of the magnetization at the typical field point, we write (u®) - 1 dB ee dBy dt 2 dé dt 7 The integral {Ba can then be transformed by the substitution o>, —)Cumlgne so that it becomes { (B Curl A) dv = =| PBA) aa A Curl B) dy wh i { [AB],.df+—= =( (AC!) de iP by Ampére’s relation if CO’ = C+c Curl l is the total current density, including the effective repre- sentation of the magnetic distribution. Thus an we deduce that a ae _ 3 aa and therefore, if we take B fae =. if (AC) dv— { io | (IB), we might assume that S =~ [FH]+ , [AB], ae which is equivalent to Macdonald’s vector. Energy in the Electrodynamic Field. 391 In his theory, however, Macdonald neglects the presence of magnetism, or at least he includes it as effectively repre- sented in the total current of the theory, so that in his case the integral in I does not appear. This procedure is not, however, completely satisfactory as it fails to take into account the fact that a part of the total magnetic energy of the system corresponding to the magnetic polarization of the media of the field, and which represents intrinsic energy of those media temporarily classified as magnetic, is not effectively available mechanically. Further, the magnetic distribution in any arbitrary finite volume of space cannot be completely represented as a continuous current dis- tribution throughout that volume, but must at least be supplemented by a distribution of surface currents over the bounding sur face of the space. For the purposes of a mathematical theory it may, how- ever, be desirable under some circumstances to replace the ‘magnetism by its effectively equivalent current dis- tribution. This can be done by transforming the integral in I. In fact we can write iL (I S) pee { (1 Ciel =) “ls = (ae uaa) (Fe J ) dvs so that we can now use A dhe =] (AC’) 2) (Curl I.dA) and c = — [BH] te S gee eH and in the former expression ¢ Curl I a the current density of the electric flux replacing the magnetism. 4, The transformation adopted by Macdonald 1s not, how- ever, a unique one. Starting again from the relation det 84 = gp ath al Bedv oA ( =) dot iy { [EH],df, we can write eu lBe =7,\ (89 ae = ig | (Cul AZ) de =i)“ G\ et {(ace rl “ae 392 Mr. G. H. Livens on the Flux of and by Ampére’s relation this is again equal to ie \ [Salat 5 Age) so that now we have 1 GH ‘ B T=" ( ay | (AdC") — | aw | (1a), : =~ (em + = [45]. Again, by transforming the integral in I, a variant of this form is got in which | fT =i {a [ (aac) —( (ecu ray | C dA 1 dB S =| ee There is another transformation derivable on the same basis as these last two, but it reduces eventually to the more general form deduced above and need not further detain us. with and 5. So far we have discussed merely the mathematically possible transformations of which the theory is capable, without stopping to examine for each case the physical significance of the results therein obtained. A consi- deration of this other side of the matter will, however, soon show that certain difficulties are involved in Mac- donald’s form of the theory and the alternative one succeeding it in the above discussion. Let us first consider the question of the magnetic energy of the system: the expression ine eC! . B (b= a ao | (Ad) -| do | (raB) ) obtained at the end of the last paragraph cannot possibly represent the energy properly available in the system in q the general case. In fact it represents the total energy | of the system arising on account of the magnetic forces, and this total includes the magnetic energy in the ether } of which the part 1 2 rA is alone available, together with all the intrinsic energy Energy in the Electrodynamic Field. 393 in the media of the field, temporarily classified as magnetic, less that part of it which is associated with the mechanical forces on the polarized media, and which is available only in so far as the presence of the media increases the available energy in the field. ‘This difficulty seems to place the last form of the theory beyond the range of physical possibility; further, the actual form of the theory adopted by Macdonald, which is the mean of this theory and the simpler and more general one first obtained, is open to a similar criticism and must therefore itself be regarded as inadmissible as a physical theory. The point involved in these remarks is further emphasized by the fact that the term in the magnetization which is properly expressed as available energy does not, on Mac- donald’s theory or the one alternative, fit in with the remaining terms in the expression for the kinetic energy, as it does in the simpler theory ; and it may be illustrated by an appeal to the fundamental physical basis of the theory. In this theory the vector Ag C is taken as measuring the electrokinetic momentum in the small element dv of the field: the force assisting the rate of change of this momentum is in consequence 1dA = ar ae On the other hand, the velocity of displacement of the affected medium is assumed to be measured by the total eurrent C, so that the rate of working of the forces on the small element is 1 fx@A é (c A) . dv, which must represent the rate of gain of available kinetic energy in the element, in agreement with the one form of the theory. 6. Of course, if we rely on these physical ideas and agree to retain only the simplest and most general form of the new theory, we must expect to encounter difficulties of a fundamental kind. In fact, we cannot in general assume that {a { (CiaA) = () dv {aq so that, even if there is no magnetism, the intrinsic energy 394 Mr. G. H. Livens on the Flux of of the field is not equal to the available energy. In other words, we shall have to assume something of the nature of a hysteretic quality for the free sther, so that the magnetic energy in any field will be a function of the history of the generation of the field and may not vanish when the field — is again reduced to zero. We shall see presently what this means. . Considering the difficulties which any assumption of this. kind would naturally entail, it might seem desirable to throw over any attempt to reduce the theory to a physical basis on the strength of our preconceived notions on such matters, and to accept Macdonald’s theory as determining all that we can know about such things. In such a theory the irreversible part of the energy distribution is withdrawn from the kinetic energy and attributed to the radiation, so that the energy density would appear in it as determined by the mean of the expressions just used for the intrinsic and available energies in the ether, which contain the irreversible part with opposite signs. But even this procedure is not withont difficulty, as may be exemplified by the discussion of a particular case. Let us examine the circumstances in a part of the field occupied by conducting and dielectric substances, and assume that throughout it the scalar potential is constant in time and space, as would for instance be the case if the field were of the pure radiation type determined in the ordinary way. In this case the electric force is determined simply by its dynamic part, 1dA Uy ie age so that the expression for the current density is e @A oaodA Mite ai) (eras e,o being the dielectric and conductivity constants at the typical field point. Thus on the generalized theory the kinetic energy density is given by 1 (( epanan dA Ae vas Wu, a a ee) a Ne Namen 74? ide he C eee S le ia : — ~ "8c? \ dt Ce (Fi HzO ts = 7) Gar, {cra 1 by im c 7 } E i , Energy in the Electrodynamic Field. 395: which exhibits it as the combination of two parts: the first term represents that part of the energy which balances the potential energy of the field, whilst the second term represents the energy dissipated by resistance. This conforms generally with recognized ideas in these cases. On Macdonald’s theory, however, the density of the kinetic energy is given by If ie GAN ca (, AA me = (A dt? ) te (A dt )|. which hardly admits of any physical explanation. Thus in a case like this, where there is actual dissipation, the generalized theory takes full account of it and gives it its proper expression, whilst Macdonald’s theory ignores it altogether. 7. Up to this point the discussion has centred round a discrimination between the different forms of the theory alternative to Poynting’s theory, which is on an entirely different plane. Ii we take these alternative theories together we see that they possess one great advantage over Poynting’s theory, in so far ag in them the electrons occupy the more prominent positions, so that they should be particularly appropriate in a purely relativist theory. In fact, the particular formula expressing the magnetic energy distribution specifies it explicitly in terms of the work of the electric force acting on the electrons ; and the new formula for the flux of energy shows that it is mainly the flux of static potential energy of configuration of the electrons themselves that is in reality under review. But even in this respect the theories are not so complete in the general case as might at first sight appear. In fact, part of the magnetic energy still remains permanently associated with the ether on account of the ethereal dis- placement current, and this current also contributes a part in the energy flux. There is, however, a very serious disadvantage attaching to any of the new theories. They all involve in their expression the scalar and vector potentials of the field, the definition of which is mathematically incomplete and uncertain. These functions cannot therefore, without further arbitrary restrictions, be said to represent definite physical entities, and a theory interpreted in terms of them necessarily remains indefinite from the physical point of view. Although this difficulty is a very serious one, it may for the present be waived, because it does not appear 396 Mr. G. H. Livens on the Flux of to be @ priort impossible that one of the forms under review does represent the physical facts of the case. There is a further difficulty of a similar kind involved in the general form of theory alternative to Poynting’s and arising from the fact that it is usually possible to determine the static potential of the field only to an additive constant. This means that on such a theory the energy flux vector is determined only to a constant multiple of the total current. But this uncertain term in all cases merely represents a flux vector possessing the usual stream property of hydro- dynamics, and is not therefore relevant to the theory. A similar difficulty besets all theories of this kind, including even Poynting’s, and it is of an entirely different nature to that at present under discussion. Although some of these criticisms may appear in them- selves to be sufficiently decisive, we shall not attempt to draw any definite conclusions from them until we have examined the behaviour of the different forms of theory when applied to definite types of electromagnetic field. — We shall, however, confine our discussion to Macdonald’s own form of the theory and the generalized form of it previously obtained, and in each problem we shall choose those field potentials which have already been found most convenient and appropriate. 8. We first examine the circumstances in asimple radiation field in which the propagation takes place by simple harmonic plane polarized waves. If the direction of propagation is along the axis of < in a rectangular coordinate system and the radiation is everywhere parallel to the axis of y, we may take = peas pias ant —(a+1b)z+20 where 27/n is the period of the oscillation and 6,, the phase constant. Under the present conditions the vector potential is sufficiently defined by the relation dA, — acs ge determining its single component A, in the form iy eum —(a+1b)2z+16y a+ib ; g—- = In this case also it is usual to assume that the scalar Energy in the Electrodynamic Field. 397 potential of the field is constant in both space and time, so that the electric force is determined solely by its dynamic component 2 = i a a Lan: mM C The current density is now where é,o@ are the usual dielectric and conductivity constants of the medium of the field. The condition for the propa- gation is —en*+i.4ane = ¢(a+ib); so that ce (me) E.. Ar ete 1) A is Ane The density of the magnetic energy on the generalized form of the new theory is thus equal to ¢ “ dt Sar and the same result is obtained on Macdonald’s own form of the theory. ‘hus the distribution of kinetic energy in the new theory is identical in both cases with that deduced on Poynting’s theory, although it is of opposite sign. The result that the kinetic energy on the new theory is of negative amount is not confined to the simple type of radiation field here examined, and in the present case it leads to the rather remarkable conclusion that the total energy in the field, consisting of the kinetic and potential energies, is negative in amount and equal to the energy dissipated in the field with its sign changed. Hqually remarkable results are obtained from a discussion of the transfer of energy in these fields. We have seen already that the scalar potential in radiation fields must be taken to be constant, and if this constant were zero there would, on the general form of the present theory, be no transfer of energy at all in the field, for the vector $= 4C 398 Mr. G. H. Livens on the Flux of would vanish at all points of the field. In any other case, when ¢ does not vanish, the transfer would merely take place parallel to the main component of the current, which is generally parallel to the electric force in the field, and would therefore be perpendicular to the direction of propa- gation of the radiation itself. Hven on Macdonald’s theory this conclusion is not appreciably affected. For in that case the radiation vector 1s Lind, Sa dt and in the present case this is id oa at Pa 2 C af == we (CHipia 7) == ( A,H,) ‘ 2 pee 2 : Bit Ny wea Gy pain at bee iOm) ya\as a 2(int —a+ibz +18) | An a+ib 8ir(a+ib) dé =a (2 so that again there is no flux of energy in any direction. Of course this is the only result we could expect in a field where the total energy at each point is zero, or at most equal to the energy developed as heat in the medium, which can under no circumstances be propagated from one point of the field to another except by means outside the electromagnetic scheme. Whether such a view of the processes in operation in radiation fields is physically valid or not, 1¢ is difficult to say: it would certainly lead to new complications in optical theory, where the idea of the transfer of energy along a beam of light is fundamental, but this cannot be regarded as a proper reason for ruling it out altogether. The only method we have of probing such fields is to examine their effect on matter, and in the neighbourhood of all matter the simple circumstances assumed above cease to be valid, and the dynamics of the interaction between the sether and electrons assumes fundamental importance. It will therefore be necessary to inquire further into the problem of the generation and absorption of radiation by electrons before progress can be made in the discussion. 9. We next consider the circumstances in the electro- magnetic field surrounding a simple Hertzian vibrator of strength f(ct) at time t. The complete circumstances of this field are well known, and in the simplest case of a first-order: Emergy in the Electrodynamic Field. 399 oscillation they are determined by the vectors E, H of electric and magnetic force whose components, referred toa spherical polar coordinate system with origin at the vibrator and polar axis along its line of symmetry, are 2 Zone B, = 2288 (op 4 9), Ey es, E,=H.=H,=0. a ae 2g 4 ofl), wherein dashes denote differentiations of f(x) with respect to its argument x, which in all cases is taken to be equal to (ct—r). These expressions are derived in the usual way from a scalar potential and a vector potential with components _ 608 6 6 A, = Ceo), Ap= ind pean Inijthis case the components of the total current at the typical field point are e cos 0 C= SEF Of +f"), cya S80 Gap ap et) the ethereal constituent being the only one existing. Thus on the general form of the new theory, the energy density at the typical field point is given by il dA. C AC a rp") | (cS) = lal Cees) C sin oe ¢ reff + of + fif') |, 400 mh. Mr. G. H. Livens on the Flux of which gives for the amount of energy between the spheres of radii r and r+6r the total T,6r, where t T= A Cede ae ae dt 1 t = (77724 2j'?44re | p"ae). On Macdonald’s form of the theory this expression turns out to be ees 37? Gare a 2F" 7 Qn 7!) : whilst on Poynting’s theory it is simply 1 52 (nf? 4 Arf fata ee The distribution of energy on the various theories is therefore essentially different. At a great distance from the vibrator—that is, in the purely radiation part of the field—both Poynting’s theory and the general form of the new theory give the same energy density, in a form which corresponds to a value of T’, given by | Bei But on Macdonald’s theory the density is such that T. = AT ee which, however, agrees on the average with the former expression in the case of simple harmonic. oscillation ; in other cases it is essentially different, and if the acceleration of the moment of the doublet is constant ie —_ 0, and the density of the kinetic energy in the distant field is then of a smaller order of magnitude than in either of the other cases, although, as we shall see later, the outward flux of energy is more than in Poynting’s case. Up near the radiator the main part of the energy density on either form of the new theory is twice what it is on Poynting’s theory, but on Macdonald’s own form it is not otherwise essentially different. On the more general form of the theory, however, the generally less important Energy in the Electrodynamic Field. 401 part of the energy in the field near the vibrator gradually increases with the time, even to the extent of becoming infinite if there is no damping. On sucha theory, therefore, the process of increasing or reducing an electromagnetic field is an irreversible one, and in order to destroy a field by reducing the forces in it to zero, a certain amount of work is necessary, depending essentially on the process adopted to secure the vanishing field, and the equivalent energy remains stored in the space previously occupied by the field. Such a conclusion involves an idea which is now generally re- garded as inconceivable in any physical theory of these matters. The discrepancy in the result obtained on Macdonald’s form of the theory for the energy in the distant field is further emphasized by comparing it with the distribution of potential energy. On all forms of the theory the density of the potential energy is taken to be 1 2 3, E at any place, and this is easily seen to correspond to a value W,6r for the potential energy stored between the spheres r and r+ 6r, where Wea Le (oifl? t2f +P EIT” + Orff +379), which in the distant field reduces to Wea Thus on Poynting’s theory and the general form of the new theory, the kinetic energy density in the distant radiation field is equal to the potential energy density, but on Macdonald’s own form of his theory these are not equal except in the case of simple harmonic oscillation. 10. The radiation phenomena in the different theories are also essentially different. According to Poynting the energy flux at the typical field point is given simply by the components c sin?@ ty eee x (of? +f'F"), ¢ 2sin 0 cos@ LBs Sa) ec I els Se Ca Aor pe IEae ’ Phil. Mag. 8. 6. Vol. 34. No. 203. Nov. 1917. 2 402 Mr. G. H. Livens on the Flux of which corresponds to a total outward flux over the sphere of radius r amounting to (pastry) Doh 4 3 or simply at a great distance. On the general form of the new theory, the radiation vector has components ¢ cos? Sr = 5 Ges ) (xf +f), csin 6 cos 8 Ss =—___ —_ (r 2 + fll 4 fl), Anrr* which corresponds to a total outward flux over the sphere of radius 7 equal to en th ae aaa, which vanishes at a great distance. Macdonald’s own radiation vector corresponds to a radial flux over the sphere 7 equal to feof fi taf 119 Ye SEU a co eke which in the distant field reduces to ¢ Ft lide This agrees on the average with the result by Poynting’s theory if the field is oscillatory in character, but for the particular case in which 2) = (ax+b)'+ dx f(a) = ( it would give no radiation at all; in other cases, as for instance when the acceleration of ths moment of the: doublet is suddenly decreased, the radiation would be inwards towards the doublet. { i Ki) 7 , Y 1 Sn eae —— ee ee ee ’ 4 i Energy in the Electrodynamic Field. 403 Thus, whereas in Poynting’s theory the transfer of energy in the distant field always exists and is directed outwards if the rate of change of the moment of the doublet is accelerated, there is, on the general form of the new theory, no such thing asa radiation of energy away from the vibrator in the distant field: all that happens is a rearrangement taking place by flux in the spherical surfaces round the vibrator as centre and generally along the lines of electric force at all parts of the field. This conclusion agrees generally with that obtained in the ease of the simple radiation field, and explains the accumu- lation of the kinetic energy in the field near the vibrator. In fact the only difference between the present theory and Poynting’s is that on the latter theory the energy supplied to the field at the vibrator is transferred outwards and radiated away, whereas on the former theory it is stored up in the field surrounding the vibrator and counted there in the kinetic energy. Maecdonald’s own theory forms a sort of mean between the two general theories, and although the energy in the field is definitely determined for each configuration of the field without reference to the past history of the establish- ment of the field and whether there is real dissipation or not, the energy radiation processes involved in it may be very dif- ferent from those mentioned above. The fact that this theory gives a definite transference of energy radially outwards in the distant simple radiation field surrounding the vibrator, when the motion is oscillatory, is not really in contradiction with the result obtained on the same theory in the simple problem first analysed, that there is no such transference in the direction of propagation. In fact, the fields in the two cases, although alike in their general aspects, are of funda- mentally different mathematical origin : the radial component of the zero electric force in the distant field surrounding the vibrator in reality consists in the difference of two finite parts—the one of dynamic origin derived from the vector potential, and the other of static origin derived from the scalar potential. In the simpler case first examined the field was entirely dynamic in character, there being no static potential. Of course, from another point of view, the difference in the two cases must be considered as a disadvantage in the theory. The sealar and vector potentials are merely auxiliary functions introduced to secure analytical simplicity in the relations of the theory, and cannot therefore represent definite physical entities. The real entities of the 2 2 i 404 Flux of Energy in the Electrodynamic Field. field are the electric force and magnetic induction vectors, and any fields which are identical when defined in terms of these vectors must be physically identical in spite of any difference in the mathematical formulation of their relations, and the identity must surely cover the transfer of energy in them. This discussion can be immediately extended to cover the more general cases of motion of individual electrons, but no essentially new results are brought to light. 11. Tosum up, we may say that the generalized form of the new theory of the transfer of energy in the electromagnetic field leads to conclusions which are wholly incompatible with our usual preconceptions of the physical circumstances of radiation problems, although in itself it appears perfectly consistent as a new mathematical and physical theory. On the other hand, Macdonald’s own form of this theory is apparently inconsistent as regards both the physical or dynamical foundations on which it is based and the results which are deduced from it ; and although it avoids some of the fundamental difficulties of the more general theory, it leads to results not entirely free from criticism in certain general cases where these difficulties might be expected to present themselves in tangible form. Under the circumstances there seems to be no other alternative but to reject both forms of the theory. This throws us back on Poynting’s theory as being the simplest and most definite theory which is consistent with the mathematical relations of the field and the physical pro- cesses operative therein. Of course, our physical conception of radiation processes is derived mainly from Poynting’s theory, so that it is hardly fair to bring forward its con- sistency in this respect as an argument in its favour ; nevertheless, I think this theory can stand as by far the most satisfactory yet proposed. Of course no definite proof of the absolute inadequacy of the alternative forms of the theory can be given; and if ever we may decide to revise our fundamental notions of these things in general and radiation in particular, it may be as well to remember that Poynting’s theory is not the only form of theory mathematically consistent with our electro- magnetic scheme. 405 © | XXXVIIL. Some Fundamental Concepts of Electrical Theory. By H. Baruman, IA., Ph.D., Lecturer in Applied Mathe- matics, Johns Hopkins University, Bultimore * Riel. POEN ES. of Force. —The idea of moving lines of electric force or Faraday tubes has been used in brilliant fashion by Sir Joseph Thomson t+ to describe the processes which take place in an electromagnetic field. Scientists are still undecided whether to regard the lines of force as physical realities or merely as useful mathematical tools. The former view of the matter implies, of course, that the ether has a definite structure; and this view is adopted by Sir Joseph Thomson in some of his recent papers f. An electromagnetic theory which is based on the idea of moving lines of electric force may be developed in various ways. We shall begin with a very simple theory which may be regarded as a development of Sir Joseph Thomson’s original idea in a particular direction. Let the equations of the moving lines of electric force be gy. 2 )(= Comme V(x, 7, 2,6) = const. where X and Y are two uniform functions of «, y, ¢, and ¢ which remain constant during the motion of a line of force. let X and Y be regarded as the coordinates of a point in a plane IT; then to each pointin the plane I there corresponds a moving line of electric force and vice versa. Now consider a closed curve C in the 2,Y,2 space, the points of this curve being considered either at one time ¢ or at different times specified by some law t=/(a, y, z), where jf is a uniform continuous function. or each point P * Communicated by the Author. + Phil. Mag. [5] vol. xxxi. p. 149 (1891); ‘ Recent Researches in Electricity and Magnetism, Oxford (1893). See also J. H. Poynting, Phil. Trans. A, vol. “clxxvi. p- 277 (1885); W. Wien, Ann. Phys. Chem. Bd. xlvii. p 327 (1892); H. A. Lorentz, Encyklopidie der Mathematischen Wissenschaften, Bd. v. § 13 (1903) p. "119; Sw Joseph Larmor, Proc. Int. Congr. of Math., Cambridge (1912}, vol. i. . E. Cunningham, ‘ The Principle of Relativ ity’ (1914), Chap. xv.; DSA Mallik, “Phil. Mag. July 1913, p. 144; H. Bateman, Phil. Mag: Oct. 19138 & Jan. 1914; : Messenger of Mathematics,’ May 1915; Amer. Journ. of Math. April 1915; W. Gordon Brown, Phil. Mag. Aug. 1915, p. 282. ile? ‘Blectricity and | Matter,’ London (1904) : Proc. Camb. Phil. Soe. vol. xiv. p. 421 (1908); Presidential Address, British Association, Winnipeg (1909); Phil. Mag, Feb. J910 & Oct. 1913. See also N. R. Campbell, ‘The New Quarterly ’ (1909); ‘ Modern Electrical Theory,’ Cambridge, 2nd edit.: H. Bateman, ‘Blectrical and Optical Wave- Motion,’ Cambridge (1915) ; Bull. Amer. Math. Soc. Feb. 1915. 406 Dr. H. Bateman on some on this curve we shall have one pair of values of K and Y= consequently there will be just one corresponding point Q in the plane II. As P describes the curve C, the point Q will generally describe a curve [. Let us consider the case when both O and [ are small closed curves. The particular type of electromagnetic field which will now be discussed is of such a nature that the flux of force across the closed curve C is equal to Zd(X, Y), where d(X, Y) denotes the area of the curve [and Z is a uniform function of z,y,z. and ¢. If (Z, X, Y) are regarded as the rectangular coordinates of a point M ina space 8, the flux is represented by the volume of a cylinder. Now consider a small closed volume in the a, y, z space, and let a time be associated with each point as before; then, by considering the flux across the boundary of the volume, we see that the electric charge within the volume is repre- sented by an element of volume d(X, Y, Z) in the space S. Expressing the quantities Zd(X, Y) and d(X, Y, Z) in terms of the variables 2, y, z,t, we may write * Zd(X, Y) = E,d(y, 2) +E,d(<, «) + E,d(a, y) —cH,d(a, t) —cH,d(y, t) —cH,d(z, ¢), d(X, Y, Z) = pd(x, y, <) — pv.d(y, 2, t) —pv,d(<, #, t) —pv,d(a, y, t). The transformation is easily effected by writing T da, dyxade , aa, y, 2) =] dx, dy, 82 | 02, OY, OZ dy, dz HOR ics ) Sy, de etc., where dx, 67, Ox, etc. are independent sets of increments of the variables z, y, z,¢t. We easily find that nO 1 OO oe ae | eee C 04, a | | fel OR, . a ae Ca Ou ty Z) oy ede - * The integral forms were introduced into electromagnetic theory by Mr. R. Hargreaves, Camb. Phil. Trans. (1908). See also H. Bateman, Proc. London Math. Soe. ser. 2, vol. viii. (1910). The coefficients E,,, cH,, ete. are generally uniform functions of x, y, 2, and ¢. + This is equivalent to assuming that the element of area isa small parallelogram and the element of volume a small parallelepiped. oa > Fundamental Concepts of Electrical Theory. 407 where the ordinary notation for Jacobians is used. It is easy to verify that these expressions give oe) | / —_ S—-—_—_—_——— — OF. 02 lige =e roldramaye s Ue sete see ow + = _ Ox : Ou. 502 i in accordance with the usual electromagnetic theory *. They also give He Mie, etc. . . | Sule We shall define E,, Hy, EH, as the components of the electric intensity E, H,, H,, H, as the components of the magnetic intensity H; the quantity p then represents the volume density of electricity, and pvz, pvy, pv: the components of the convection current. It is convenient also to regard the vector v as a velocity. If we use the customary vector notation, the relation between E, H, and v may be written in the form ce - (5 ame alias 3 This is the fundamental relation which Sir Joseph Thomson takes as the starting point of his theory. The constant e¢, which has been introduced to bring the equations to the familiar form, will be called the velocity of light. It is important to notice that the three functions X, Y, and Z satisfy the partial differential equations | OX OX ix: fe Ot + Uz oe | Y Oy ane fs ju Ghmeto G OY Oe a pb Pea ay ae = 0, of OL) aanoZ OZ ~~ + Uz A © De Oc = () i * When Z=1 we have p=0, pv=O, and our expressions specify an electromagnetic field in free ether. This case has been discussed in a previous paper, ‘ Messenger of Mathematics,’ Nov. 1915. The con- ditions p=0, pv=O are also satisfied when Z=/(X, Y); but we can generally choose two functions X,, Y, such that d( Xo, ¥) =/(X, Y) dX, Y). Consequently there is no loss of generality in putting Z=1. 408 | Dr. H. Bateman on some These equations are of a type familiar in hydrodynamics : the first equation implies that X has the same value at the two consecutive points (#,¥,2,t), (a+vz,0t,y + vyot, 2 + v,6t,t + dt); consequently, if we regard v as the velocity of a particle of electricity, the first two equations imply thata line of electric force always consists of the same particles of electricity *. When we use vector notation the fundamental equations (2) take the form i DOE BOE roth = —(S° + pv 5) chive — os and the relation (4) signifies that the vectors E and H are connected by the relation (EH)=0: they are consequentiy perpendicular to oneanother. A field in which this condition is satisfied may be called special. § 2. The Lines of Magnetic Force.—The theory of moving lines of magnetic force may be developed along similar lines. Let the equations of a line of magnetic force be V(a#,4y, 28) = const., T(r) y,240)) = eons where V and T are uniform functions. Regarding V and T as the rectangular coordinates of a point in a plane, we assume that the flux of magnetic force through a closed curve OC in the x,y, 2 space is represented by the area d(V,T) of the corresponding closed curve inthe plane of V,T. The factor is in this case taken to be unity because the flux of magnetic force across the boundary of a closed surface is zero if there is no free magnetism. Writing d(V,T) = h,d(y, <)+hy,d(<, x) +h,d(a, y) + ce,d(x, t) +ceyd(y, t) + ce,d(z, t), we have, as before, (oon). . | leeniG —3ya” °~2 3G) These equations give 10h Me rote = ae divh = 0, ).(eh)) = (2 eG) Fundamental Hypothesis—We shall now assume that the * The flux across a closed circuit made up of particles of electricity remains constant during the motion of these particles. Fundamental Concepts of Electrical Theory. 409 electric and magnetic lines of force are connected in such a way that B= ewaad HH ='h, A method of determining a set of functions X, Y, Z, V, T which lead to these relations is given in a paper which has been offered to the ‘ Messenger of Mathematics.’ § 3. Geometrical representation of the various Vectors *.— Consider a sphere of radius ¢ whose centre is at the origin and a force R acting along a line L which cuts the sphere in real points. Let (H,, Hy, H,) be the components of this force and (cH,, cH,, cH.) the moments of the force about the axes of coordinates, then it follows from the relation = |vE] that v,, vy, vz are the coordinates of a point U on the line L. Since the velocity v is generally supposed to be less than c the velocity of light, the point U will be within the spher e, and this is why the line Lswas made to cut the sphere in real points. Ox the line L also passes through the point with coordinates (21, ¥1, %). Similarly it passes through a point derived from the function T. It follows then that we may write ey or Beye NO: Ve ieee aa ammee cc Ot In Sir Joseph Thomson’s relation cH = |vE], v was regarded originally as the velocity of the lines of electric force. If we regard v as defined by this relation it is evidently indeterminate, for (v,, V,, vz) may be taken to be the coordinates of any point on the line L. The velocity of a line of force when defined in this way may be either greater than, equal to, or less than ¢ the velocity of light. If we define the velocity of a line of magnetic force in a slinilar way by the relation cE+[wH]|=0, the components of this velocity w may be regarded as the coordinates of a point on a Jine L’ which is the polar line of L with regard to the sphere. Since L cuts the sphere in real points, L' does not; and so it follows that the velocity w can never be less than the velocity of light, it can be equal to the * Cf. ‘ Messenger of Mathematics,’ Nov. 1915. = kx, Gy = ky, SE = kz, 410 Dr. H. Bateman on some velocity of light if L touches the sphere because then L’ also touches the sphere. . It should be noticed that there are relations of the form for two points on the line L’ can be derived from the functions X and Y, just as two points on L were derived from V and T. § 4. The Electromagnetic Potentials—The fundamental equations oe rot H Te (S: tev): div E = p, eels HL ee ROULE SS ae div.H = 0; are usually solved by writing 1o0A Bl = i = —- 2—— or oe Re ane rotA, E Ae grad ®, (7) where the vector A and the scalar ® satisfy the equation 10® = e e ° e e e e 8 div A+” a 0 (8) In the present case we can obtain the expressions (5) for E and H by assuming aM oT “eee i me Sa Se ee where « and 8 are functions of V and T such that OB _o#_, an OU A second relation must also be satisfied in order that (8) may hold. § 5. Equations of motion of an electrified particle in an electromagnetic field when there is permanent incidence between the particle and a line of electric jorce —Let us now suppose that the volume distribution of electricity is negligibly small in some regions of space, so that the present type of electro- | magnetic field may resemble an electromagnetic field in the sether for these regions. Let a particle of electricity be introduced into one of these regions : the problem is to find out how it will move. Fundamental Concepts of Electrical Theory. 411 If the Newtonian equations of motion were not known, mathematicians might perhaps endeavour to formulate a scheme of equations of motion by using the principle that two particles cannot occupy the same space at the same time. In the present case the principle may be supposed to be violated if the electrified particle cuts through a tube of electric force of the given field, or if it leaves a tube of force to which it has been attached. Let us consider the consequences of assuming that the velocity u of the electrified particle is determined by the equations mi == Ammer = OD. ae OD) Now it follows from the identities een) 13(V,T) eX. Y)_ av, 02) ¢ 0,’ “O@h) °d@,2)’ that both X and Y satisfy the equations ne) 0X OV OV _ 1 ox OV Br or OY OFsmmOrN 02 ' ¢ Ob. Ob OX oF ox of, oX of 1oxo De. 02 Oy Oummecsoe 0? Ot Ot" Hence it follows that fop.€ ox ox Ox ane De ee aS, i 0, and that Y satisfies a similar equation. Consequently X and Y remain constant during the motion of the elec- trified particle ; and this means that the particle remains on the same line of electric force during the whole of its motion, at never cuts through a tute of force *. The law of motion embodied in equations (9) is so simple that it is hardly likely to be correct, nevertheless we are justified in retaining our hypothesis as a possible one until it has been shown conclusively that it is contradicted by * In order to ensure that this may be true it is clearly only necessary to assume that the velocity u is represented by a point on the line L. Our assumption (9) is only one way of satisfying this condition. 412 Dr. H. Bateman on some experience. ‘here are, indeed, some points in favour of the hypothesis in addition to its simplicity. Let us see, for instance, if the above law gives the correct motion when a charged particle moves in a steady uniform magnetic field in which H is parallel to the axis of z. In this case we may write A,= —$Hy+4imcU,, A,=4He+imeU,, As= Emc:, b =4me’, where U is a constant vector. When these forms for A and ® are adopted, the equations of motion take the correct form : a be ax 1 ay 7 ers = Ny, = ae Hy, or DE AEE = rae = dy ih d’y ik saa, mM crs = MUy —— mU, ots 2 Ha, pe = ps di dz dz . ma = mu, = mol). m a2 = (). The quantity m appearing in these equations represents the mass divided by the electric charge with its sign changed ; we assume here that it is a constant. It is clear from the above example that there is some ambiguity in the choice of A and ® when the field is given, for it is easy to write down other potentials such as A,=—Hy, A,=0, A,=0, ®=0, which will specify the same electromagnetic field: consequently the hypothesis is in some respects incomplete. § 6. Incident Frelds.—Two special electromagnetic fields are said to be incident when there is permanent incidence between lines of electric force of the two fields. This means that if «a line of force of one field intersects a particular line of force of the other field at a given instant, it will continue to intersect it. The condition that this may be true has already been found*. If the lines of force of the two fields are given by X=const., Y=const., and Xy=const., Yo=const. respectively, the condition is that the Jacobian should vanish, 2. e. that 0(X, Ms Xa, iG) ta 0 Oe), 206). ee This implies that Yo=const. is a consequence of X =const., Y=const., Xy=const. I¢ should be noticed that when this * ‘Messenger of Mathematics,’ 1916. Fundamental Concepts of Electrical Theory. 413 condition is satisfied two lines of force which intersect twice can separate if the points of intersection approach one another, coincide fand then become imaginary. The con- dition may sometimes imply that two lines of force touch one another, 7. e. have a common tangent. If (KE, H), (Ey, H,) are the electric and magnetic intensities in the two fields the condition implies that (EH,) + (EH) = 0. This equation may also be written in the form OUN, Ate Vo, To) a! J O(2, y, 2)? consequently it also implies that there is incidence between the lines of magnetic force. Our condition may also be obtained geometrically by considering the lines L and Ly, which are used in the representation of the vectors belonging to the two fields at a point (2, y, z,t). The condition implies that the two lines intersect and that there is consequently a velocity v which can be regarded as the velocity of the two lines of force through the point. The condition also implies that the polar lines L’ and L,’ intersect. It should be noticed that if d is the shortest distance between the lines Land I and @ the angle between them, we have (EH,) +(E)H) = 6, dsin 8, where G and G denote the magnitudes of the electric intensities. Now d can be regarded as the relative velocity of the two lines of force which pass through the point 2, y, z at time t. Hence if (EHy) + (E,H) is not zero one of these lines of force cuts through the other. If, on the other hand, (EH) + (E,H) vanishes and changes sign as ¢ varies, the relative velocity changes sign and so the two lines of force meet and re- bound without cutting through one another. Finally, if (EH,))+(E,H) vanishes but does not change sign, the two lines of force generally cut through one another. It is evident that if L cuts through Ly as ¢t varies, then L’ will also cut through Ly’. Hence, if there is cutting of lines of electric force there is also cutting of lines of magnetic force; if lines of electric force collide without cutting through one another, some lines of magnetic force do the same. If there is permanent incidence between two lines of electric force along a curve OC, there is generally A414 Dr. H. Bateman on some permanent incidence between two lines of magnetic force in which the point of intersection moves along the curve C. In practice it is frequently easier to ascertain whether there is a relation between V,T; Vo, I) than whether there is a relation between X, Y ; Xo, Let us consider, for instance, the fields of two moving point charges in free ether. The exact expressions for X, Y; Xo, Yo have not been found in the general case, although differential equations for finding them are known. Appropriate expressions for V and T have, however, been given by Mr. R. Hargreaves *, and with their aid the present problem can be solved. Let the motion of the first point charge be specified by the equations oa EG); Ya n(T), Bima Ga); then T=r, where 7 is given by the conditions [e—&(r) + [y—n(r) P+ [e--& 7) P=PC—7), rst. (10) Let us write 20 = P47? +C— 07? ; then an appropriate value of V is T _ af! + yn! +26'— een!" —0" = afl Lyn! +20 — ete yl) where the primes denote differentiations with respect to r. We shall use the suffix 0 to distinguish the corresponding quantities in the second field. We then have the three equations ee Ve alg | 26 Ve —t[ r'’—Vr'] th (io Ve] ae 0, | wl Eo’ — Vk | +y[1o —Vono | +2[ 0 — Voto | \ (11) — Ct) t''— Voto | Sg = ae | = 0, | | w[E—£))+yl[n—m] +2L$—Soi —et[7—70] —[9-@] = 07) the last one being obtained from the two equations of type (10) by subtraction. es When 7, 7), V, and Vo are given, it is generally possible to solve the last three equations and (10) for x,y, z,t. In some cases, however, the equations are satisfied by an infinite number * Proc. Camb. Phil. Soc. 1915. + The charge has been assumed for simplicity to be 47, Fundamental Concepts of Eiectrical Theory. 415 of sets of values of 2, y, z, t when 7, 7), V, and Vo are con- nected by one or more relations, and they may perhaps be inconsistent when the relations are not satisfied. The three equations (11) give 0+ sets of values of a, y, ¢, and ¢ which generally represent a point moving along a straight line with constant velocity, but if all the deter- minants of the array ee Vals. Cae ss" —Vr', ot Me, Eo —Vobo. no’ — Vono's Co’ — Vobo', T—Vot> A) —V00', | eases 7 — Nos g= Co, T—T09, 0—8O,, vanish, the three equations are not independent and give oo” sets or even «? sets of values of x, y, zc, and t. In the first case equation (10) is satisfied only if the velocity is the velocity of light and &, 9, €, 7 is one position of the moving point. For a similar reason &, mo, Go, 7) must also be a position of the moving point. This case may be rejected as trivial. In the second case the determinants of the array may all vanish on account of either three, two, or only one relation between V, Vo, 7, and 7). If three relations are required for the vanishing of the determinants, it means that it is possible to pick out a set of «+ lines of magnetic force of one field such that each of these lines of force is permanently incident with a finite number of lines of force of the second field. The condition (EH,) +(E,H)=0 is then satisfied for co” sets of values of x, y, z,t; 2.e., at points of a moving curve. Itfollows from § 6'that there is also either permanent incidence or contacts of a tangential nature between lines of electric force of the two fields at points of this moving curve. Two relations are sufficient for the vanishing of all the determinants of the array when two equations of type lE+mnt+n€+ pt+g@ +r= 0, lE, + my tnOo>tpm+q+r= 0, are satisfied, J, m,n, p,q, 7 being constants. If we regard _(E, 0, ©, ict) (&, 0, So, tT) as the rectangular coordinates of two points in a space of four dimensions S.4, the equations signify that the paths of the two point charges are repre- sented by two curves which lie on the same hypersphere or hyperplane. In the latter case the two point charges lie at any instant in a plane which moves with uniform velocity in 416 Dr. H. Bateman on some a direction perpendicular to itself. In particular, the con- ditions are satisfied if the two point charges move in one plane. If only two relations are required for the vanishing of the determinant, it means that each line of magnetic force of one field generally has permanent intersections with a finite number of lines of magnetic force of the second field. The intersections lie on a moving surface, and lines of electric force of the two fields have permanent intersections on this moving surface*. When the two point charges move in one plane, it is evident that the plane must be the surface to which we have just referred ; and if we generalize this case by means of the transformations occurring in the theory of relativity, it appears that the moving surface is represented in S, by the hypersphere or hyperplane on which the repre- sentative curves of the two paths lie. One relation is sufficient for the vanishing of all the determinants of the array when four equations of type le mae no + on GU eae LE + mn t+ m64+ pitt MO+7= 0, 1E, + myo + no + pto> + GA +r =, Eg + myo + 1400+ Pitot QiTo+ 71 = O. In this case the paths of the two point charges are repre- sented in S, by two curves which lie on the same sphere or plane. In the latter case the two charges lie at any instant on a straight line which moves in a plane with uniform velocity, retaining the same direction all the time. In particular the conditions are satisfied if the two charges move along the same straight line. When only one condition is required for the vanishing of all the determinants, each line of force of one field is permanently incident with o' lines of force of the other field. There is thus complete incidence of the two fields and no cutting of lines of force. It should be noticed that when two incident special fields are superposed the total field is also special +, for we have (E+E’, H+H’) = (EH) + (EH’)+ (EH) + (E'H’) = 0. §7. A system of mutually incident special fields—The last theorem may be generalized as follows :—Zf a system * It may happen that there is a set of co! lines of force such that each line of force meets ©? lines of force of the second field. + It is often easy to find the lines of force in the total field when the lines of force of the constituents are known. The example considered in § 8 will illustrate this. Two point charges moving along the same straight line provide another good example. - Fundamental Concepts of Electrical Theory. 417 of special fields is such that there is incidence between each pair, then when all the fields are superposed the total field is special. The question naturally arises whether the fields of a number of moving point charges can be mutually incident. If we represent the path of a moving point charge by a curve in a space of four dimensions as in the preceding paragraph, any two of the curves must lie on either a plane or asphere. A system of mutually incident fields is obtained when the paths of the charges are represented by straight lines or circles related to one another in a suitable way. The following cases are of some interest, the paths being repre- sented by (1) A system of straight lines through a point in the space of four dimensions ; (2) A system of circles through two points ; (3) A system of straight lines in a plane, or curves in a plane ; (4) A system of curves on a sphere. Another case of some interest occurs when all the fields of one system are incident with all the fields of another system but are not incident with oneanother. Paths represented by the two systems of generators of a quadric surface give rise to such a system. The systems of moving particles given by cases (1) and (2) have already been studied in connexion with the theory of time * § 8. The lines of electric force in a certain type of electro- magnetic field—If{ we adopt the customary idea of the field of a moving point charge and imagine it to be specified by Liénard’s potentials, it is clear from the foregoing analysis that the fields of a number of moving point charges can only be incident in certain rather special cases. Moreover, the idea of incidence loses its importance to some extent when the tubes of force issuing from a single point charge are supposed to fill the whole of space. Sir Joseph Thomson has already suggested that this may not be the case, and has used this idea in some of his physical theories. The mathematical analysis which will now be given may perhaps elucidate matters a little. Using the same notation as in § 6, let us write a—E+u(y—n) roy Rie = Cheese a=xX+iY, B= X-1Y. * ‘Messenger of Mathematics,’ Nov. 1915. Phil. Mag. 8. 6. Vol. 34. No. 203. Nov. 1917. 2G 418 Dr, H. Bateman on some The function e satisfies the partial differential equation d7 O2,d7 de, dr 0% _ 1arda az d2' Syn oc Be Oo and @ satisfies a similar equation. The functions X and Y are to some extent indeterminate, since certain functions of them will serve the purpose of § 1 just as well as X and Y. We shall suppose them to be chosen so that « depends only on s and 7, both of which satisfy (12). Choosing & so that the condition OV dV 9s, dV d«_ 19V dz OF OF) Oy oy 02:02) Cam is also satisfied, we find that there is a relation of the form F(s, 1)da = 2(e—£?—4?—£")ds menLit(e’ tin \le— 0) tea, sesyeg +e ea — Pf (E" in! (e+ 0) —O'(E'—i')3], where F'(s, 7) is a function whose value need not be specified here. The condition a=const. or da=0 leadsto the Riccatian equation which is used in finding the lines of electric force™, and which has been solved by Dr. Murnaghan f in a large number of cases. For our present purpose we need only the fact that a depends only on s and 7. An important consequence of this is that « satisfies the three partial differential equations of type Om) 2 Ola. a) a Be) Oy because both s and 7 are solutions of these equations. The function ££ satisfies a similar set of equations with —7 in place of 2. The following expressions may now be adopted for the * H. Bateman, Amer. Journ. of Mathematics, April 1915. + Johns Hopkins Circular, 1915: Dissertation, Johns Hopkins (1916). Fundamental Concepts of Electrical Theory. A19 components of the electric and magnetic intensities in an electromagnetic field in the ether * :— (Oy pees (a, 8) O(a, T) oe T) BY = i228) —ifla, ? a4 2) +48; 7) = ie) mS Olcgay ie sin : Tee a) ae —: * fle a j O(a, 8)” o_ + 0(4, 8) a foe O(a, ee gow ay ~¢ O(a, t) aes a ”) SG@ri ‘ * (B,7 Wea eles) (7) us 2) = 5,2) the ) T) 5 je) OS T) where o has been written for 2V. To find the lines of electric and magnetic force in this type of field we must obtain identities of the form (a, B) —f(a, T) d( a, T) + iS, T) ONC oa) = dd, vr), d(o, tT) +f(4, T)d(4, 7) +f(8, 7) d(B, 7) = d(y, T 7 ila, 7)= 2 gla 7) the moving lines of magnetic force are given by the equations X=c+ g(a, 7) +9(8, 7)=const.. t=const. The lines of electric force are given by the equations (a, 8, T)=const., ab(a, Bsr) =consts, where these equations give a pair of solutions of the differential equations Ebi) dB ae =j(S, ae ie = f(a, 7) Let us now write r=T as before and consider in the plane of X, Y a two-dimensional irrotational motion of an incom- pressible fluid in which the velocity potential ® and current function VY are given by the equation D+ 1V=9(X47Y, T)=9(a, 7). * The function f(z, y) is supposed to be real when x and y are real, so that f(a,r) and /(8,7) are conjugate complex quantities. 2G 2 420 Dr. H. Bateman on some lf uw and v are the component velocities, we have uw fe PY) Pera and the paths of the particles of fluid are obtained by writing _ dx Baie wot oa Hence the paths of the particles are given by the equations dp da) ae =](4, T), As =/(8, T), which are the same as those obtained above. Since the equations «=const., @=const., T=const. repre- sent a point which moves along a straight line with the velocity of light, it follows that a line of electric force in the present type of electromagnetic field can be regarded as made up of a system of particles which are projected from the different positions of the moving point charge in direc- tions which vary with 7 according to a certain law. These particles are supposed to travel along straight lines with the velocity of light, and their positions at any instant give a line of electric force at this instant. This geometrical description of the motion of the lines of force has already been given in the simple case when /=0, 2. e. when the moving point (€, , €, 7) is the only real singularity *. The only difference in the present case is that the law for the variation of the direction of projection is different. In the simple case a point with coordinates (X.Y) in the plane II corresponds to one line of force, and if we associate a time T with the point and consider the line of force at one instant ¢, there will be just one point P on this line of force which was projected from the moving point charge at time Tea In the more complicated case we have the path of a particle of fluid xa T); ye as the image of a line of force, and we may obtain a line of force from the lines of force in the simple case by connecting up the points on these lines which correspond to the different positions of our particles of fluid. In this more complicated case there is still a constant electric charge 4m associated with the point charge * Bulletin of the American Mathematical Society, ae ‘American Journal of Mathematics,’ April 1915. Fundamental Concepts of Electrical Theory. A21 (€, m, € 7), and the lines of force at any instant start rom the position of the point charge at this instant. The point charge is, however, not generally the only singu- Jarity in the field, there are other singularities which move along straight lines with the velocity of light. A good idea of the nature of these singularities is obtained by con- sidering the singularities in the associated two-dimensional fluid motion, assuming that sources and sinks correspond respectively to positive and negative charges travelling with the velocity of light. Now in the theory of two-dimensional fluid motion singu- larities are often avoided by the introduction of free surfaces or boundaries made up of lines of flow, and a similar idea may be adopted with advantage in the present case. Since a line of flow corresponds to a line of electric force, the free surfaces or boundaries in the electromagnetic field will be lines of electric force; this is just what has been suggested by Sir Joseph Thomson. It should be noticed also that if the boundary of the two-dimensional motion is stationary or fixed, the corresponding boundary of the electromagnetic field is made up of lines of force of the simple field, 2. e. the ordinary electromagnetic field associated with a moving point charge. The idea that the tubes of electric force do not fill the whole of space is a very natural one if we wish to retain the principle that two tubes cannot occupy the same space at the same time, for then some room must be left for the _ tubes of force issuing from other charges. This principle cannot be regarded as an axiom at present because it is not generally accepted. Sir Joseph Thomson has, for instance, considered the work done when one charged particle penetrates into a tube of force issuing from another*. It is clear, then, that there are several directions in which the theory of moving tubes of force may be de- veloped. The following possibilities may be cited as worthy of discussion :— (1) Permanent and mutual incidence of the lines of force of the different point charges. This need not apply to all the lines of force. (2) Temporary incidence of lines of force of different point charges but no cutting of lines of force. The analogy with a system of perfectly elastic moving spheres may be useful in the study of this case. (3) Motion in which the lines of force can cut right through one another. * Phil. Mag. Oct. 1913. ) orm 422 Some Fundamental Concepts of Electrical Theory. To discuss the first case we must find the condition that the generalized fields of two moving point charges may be incident. In other words, we must find the condition that there may be a relation between y, T, Yo, and 7, where x and 7) refer to the second point charge. Writing 2 tere) OT oie e—C+¢(t—T)’ ~ e—C+c¢(t—T)’ and remembering that @ depends only on a and 7, we may write o(t—7)[ E" (a+b) —in'(a—b) + C'(1—ab)] + (1 4+ab)(@— £7 —97—6") e(t—7)| Eat b)—in (a—b) + 6(1—ab)—c +46) | + F(a, 7) + F*(Q, 7). The expression for yp is similar to this; on the other hand, we have 7 ; ath an am nM a—b_, 19 —b iysint ot) 7 =item) 74 ieee (13) z= €+¢(t—7) a = %+¢(t—T) 2 Let us write E—f)+ia—g)=p, €- &—i(n—m)=9, C—O te(7—%) =”, ia ee tr) =, PYtrs=k then we find from equations (13) that 2c(t—7)|aqgt+bp—abr+s|+x«(1+ab) =0, 2c(t—T)) [ag + bp—abr+s|+(gats) (pb+s) + (rb—q)} (ra —p) =0, b+s ats Bee by aoe! “ e rb—g ra—p Now put E+in =A, E-—mM=p, FC +¢e7 =v, 6 —OT =p, Ey ting=A, E—MmM=Ho SoteTo=Vo, So— CT )= fo; Colours diffusely reflected from some Collodion Films. 423 then _ e(ap!' +br" + p'!—abv'') + 2(aqt bp+s—abr)(r'p' +0'p') x «(ap + br! + p'—abv') + F(a, 7) + F*(, 7), Xo at K(Ayby a Dana ., oe po’ = AgoVo'') = 2(aog + bop + s— Apdo”) (Ao’ fey. + Vo Po ) K (Gg + boAo + Po — 20000 ) + Fo (a0, 7) + Fy (4, To): Let us regard a and 6 as thecircular coordinates of a point in a plane then when y, yo, 7,7) are given constants, the above equations represent two circular curves which generally inter- sect in a finite number of points and with each of these points we can associate just one set of values of 2, y, c,and¢t. This means that the line of force (x, r) intersects the line of force (Yo, To) a finite number of times. These lines of force will thus generally cut through one another, but if the two circular curves touch at all their common points the lines of force will meet and rebound. It may happen that the two circular curves are the same or have a common part. In this case there is permanent incidence between the two lines of force. The number of relations which must be satisfied by yx, vp», 7, and 7) in order that the two curves may have a common part, may be either one, two, three, or four. The first three cases may be dis- cussed geometrically in the same way as before; in the fourth case there is permanent incidence between only a finite number of lines of force of the two fields. If such cases exist in nature, it is possible that the moving points of incidence are occupied by electric charges and that when the fields of a large number of moving charges are super posed each charge is a point of incidence of a number of lines of electric force. XXXIX. On the Colours diffusely reflected from some Col- lodion Films spread on Metal Surfaces. By Lord RaYLeicnH, O.M., F.RS.F T is known that “ when a thin transparent film is backed by a perfect reflector, no colours should be visible, all the light being ultimately reflected, whatever the wave- length may be. The experiment may be tried with a thin layer of gelatine on a polished silver plate” t. An apparent exception has been described by R. W. Wood§: “A thin + Communicated by the Author. J Wave Theory, Enc. Brit. 1888 ; Scientific Papers, vol. ili. p. 67. § ‘ Physical Optics,’ Macmillan, 1914, p. 172. 424 Lord Rayleigh on the Colours diffusely film of collodion deposited on a bright surface of silver shows brilliant colours in reflected light. It, moreover, scatters light of a colour complementary to the colour of the directly reflected light. This is apparently due to the fact that the collodion film ‘ frills,’ the mesh, however, being so small that it ean be detected only with the highest powers of the microscope. Commercial ether and collodion should be used. If chemically pure ether obtained by distillation is used, the film does not frill, and no trace of colour is exhibited. Still more remarkable is the fact that if sun- light be thrown down upon the plate at normal incidence, brilliant colours are seen at grazing emergence, if a Nicol prism is held before the eye. These colours change to the complementary tints if the Nicol is rotated through 90°, 1.€. in the scattered light, one half of the spectrum is polarized in one plane, and the remainder in a plane perpen- dicular to it.’ I have lately come across an entirely forgotten letter from Rowland in which he describes a similar observation. Writing to me in March 1893, he says :—“ While one of my students was working with light reflected from a metal, it occurred to me to try a thin collodion film on the metal. This not only had a remarkable effect on the polarization and the phase but I was astonished to find that it gave remarkably bright colours, both by direct reflexion and by diffused light, the two being complementary to each other. I have not gone into the theory but it looks like the pheno- menon of thick plates as described by Newton in a different form. The curious point is that I cannot get the effect by making the film on glass and then pressing it down hard upon speculum metal or mercury although I think the contact is very good in the case of the speculum metal. Possibly, however, it is not. Gelatine films on metal give good colours by direct reflexion but not by diffused light: only faint ones. It would seem that the collodion film must be of variable density or full of fine particles. However, I leave it to you. I send by express two of the plates used.” Probably it was preoccupation with other work (weighing of gases) that prevented my giving attention to the matter at the time. Wishing to repeat the observation of the diffusely scattered colours, I made some trials, but at first without success. On application to Prot Wood, I was kindly supplied with further advice and with a specimen of a suitably coated plate of speculum metal. Acting on this advice, I have since obtained good results, using very dilute collodion reflected from some Collodion Films. 425 poured upon a slightly warmed silvered plate (plated copper) warmed again as soon as the collodion was set. That the film is no longer a thin homogeneous plate seems certain. Wood speaks of “ frilling,” a word which rather suggests a wrinkling in parallel lines, but the suggestion seems negatived by the subsequent use of “mesh.” I should suppose the disintegration to be like that sometimes seen on varnished paint, where under exposure to sunshine the varnish gathers itself into small detached heaps. At any rate there is no apparent change when the plate is turned round in its own plane, showing that the structure is effec- tively symmetrical with respect to the normal of the plate. As regards Rowland’s suggestion as to the origin of the colours, it does not seem that they can be assimilated to those of “thick plates.” The latter require a highly localized source of light and are situated near the light or its image, whereas the colours now under consideration are seen when the plate is held near a large window backed by an overcast sky, and are localized on the plate itself, the passage from one colour to another depending presumably upon an altered scale in the structure of the film. The formation of well-developed colour at the various parts of the plate requires that the structure be, in a certain sense, uniform locally. The case is similar to that of coronas, as in experiments with lycopodium, only that here the grains must be very much smaller. When examined by polarized light the behaviour of different plates is found to vary a good deal. We may take the case where sunlight is incident normally and the diffuse reflexion observed is nearly grazing. In the case of the specimen (on speculum metal) sent me by Prof. Wood, the light is practically extinguished in one position («) of the nicol, that namely required to darken the reflexion from glass. In the perpendicular position (8) of the nicol good colours are seen, and also of course when the nicol is removed from the eye. At angles of scattering less nearly grazing there is some light in both positions of the nicol, ek fainter light in («) showing much the same colour as in (8). It will be noticed that this behaviour differs from that observed by Wood (on ancther plate) and already quoted. On the other hand, one of the (silvered) plates prepared by me shows a better agreement, more light than before being scattered at a grazing angle when the nicol is in the (a) position, while the colours in the («) and () positions of the nicol are roughly complementary. 426 Lord Rayleigh on the Colours diffusely No more than Rowland have I succeeded in getting dif- fusely reflected colours from collodion films on glass or, | may add, quartz, either with or without the treatment with the breath suggested by Wood. The latter observer describes an experiment (p. 174) in which a film, deposited on the face of a prism, frilled under the action of the breath and then afforded a nearly three-fold reflexion. But,as I under- stand it, this augmented reflexion was specular. The only thing that I have seen at all resembling this was when I treated a coated glass with dilute hydrofluoric acid with the intention of loosening the film. Even when dry, the film remained out of optical contact with the glass, except I suppose at detached points, and gave an augmented specular reflexion, as was to be expected, inasmuch as three surfaces were operative. Two views are possible with regard to the different behaviour of films on metal and on glass. One is to sup- pose that the actual structure is different in the two cases : the other, apparently favoured by Wood, refers the differ- ence to the copious reflexion of light from metallic surfaces. The first view would seem the more probable a prior and is to a certain extent supported by Rowland’s experiment. I have not succeeded in carrying out any decisive test. On either view we may expect the result to be modified by the metallic reflexion. As to the explanation of the colours, anything more than a rough outline can hardly be expected. We do not know with any precision the constitution of the film as modified by frilling. And, even if we did, a rigorous calculation of the consequences would probably be impracticable. But some idea may be gained from considering the action of an obstacle, e. g. a sphere, of material slightly differing optically from its environment and situated in the neighbourhood of a perfectly reflecting plane surface upon which the light is incident perpendicularly. Under this condition the reflected light may still be supposed to consist of plane waves un- disturbed by the previous passage through and past the obstacle. The calculation, applying in the absence of a reflector but without limitation to the spherical form of obstacle, was given in an early paper*. In Maxwell’s notation, f, g, are the electric displacements. The magnetic susceptibility is supposed to be uniform throughout; the specific inductive capacity to be K, altered within the obstacle to K +AK. * “On the Electro-magnetic Theory of Light,” Phil. a vol. xil. p- 81 (1881) ; Scientific Papers, vol. i. p. 518. le * i i; ; i , 4 a ft i 3 reflected from some Collodion Films. A427 The suffixes 0 and 1 refer respectively to the primary and scattered waves. The direction of propagation being supposed parallel to 2 and that of vibration parallel to z, we have fo=9o.=0, and Near. 2) ot he pea aie es EL, e’™ being the time factor for simple progressive waves. For the scattered vibration at the point (a, 9, y) distant r from the element of volume (dz dy dz) of the obstacle, we have ; k?P ies . Sy Ip n= ga By, —(¢ +e) h, Oa Gb P= — a ff | age-ae Ap AZ es. ea eos and the integration is over the volume of the obstacle. If the obstacle is very small in comparison with the wave- length (A) of the vibrations, hjye~“” may be removed from under the integral sign and PAK hye K ; T denoting the volume of the obstacle. In the direction of primary vibration «= 8=0, so that in this direction there is no scattered vibration. It will be understood that our sup- positions correspond to primary light already polarized. Tf, as usually in experiment, the primary light is unpolarized, the light scattered perpendicularly to the incident rays is plane polarized and can be extinguished with a nicol. The formation of colour depends upon other factors. When the obstacle is very small, P is constant, and the secondary vibration varies as k*, so that the intensity is as the inverse fourth power of the wave-length, as in the theory of the blue of the sky. In this case it is immaterial whether the obstacles are of the same size or not, but for larger sizes when the colour depends mainly upon the variation of P, strongly marked effects require an approximate uniformity. If the distribution be at random, the colours due to a large number may then be inferred from the calculation relating to a single obstacle; but if the distribution were in regular patterns, complications would ensue from the necessity for taking phases into account, as in the theory of gratings. For the present purpose it suffices to consider a random distribution, although we may suppose that the centres, or more generally corresponding points, of the obstacles lie in a plane perpendicular to the direction of the primary light. where Pee (4; 423 Colours diffusely reflected from some Collodion Films. When the obstacle is a sphere, the integral in (3) can be evaluated *. The centre of the sphere, of radius R, is taken as the origin of coordinates. It is evident that, so far as the secondary ray is concerned, P depends only on the angle (y) which this ray makes with the primary ray. We suppose that ~=0 in the direction backwards along the primary ray, and that y=7 along the primary ray continued. Then with introduction of the value of hy from (1), we find = (5) m—2nk cos5y. Eye m? m? Ds NIRA reln? et) (= m COS ”) Me ’ where The secondary disturbance vanishes with P, viz. when tanm=m, and on these lines the formation of colour may be understood. Some further particulars are given in the paper just referred to. The solution here expressed may be applied to illustrate the scattering of light by a series of equal spheres distri- buted at random over a plane perpendicular to the direction of primary propagation. The effect of a reflector will be represented by taking, instead of (1), i ert ( ema 4 Crt Caio) et. (7) 2 expressing the distance between the plane of the re- flector and that containing the centres of the spheres. The only difference is that m7* sin m—m7?cosm is now replaced by sinm cosm /sin m!' oe) . Tea ar +) paisa + ey = 3 m? m3 2 (8) where m is as before, and m’=2nRsin4y. In the special case where, while the incidence is perpendicular, the scattered light is nearly grazing, y=4$7, sin}y= cos 3y=1/,/2, and m=m'=,/2.kR; so that (8) becomes ne mm e sinm cosm oe Vpn m mf" (9) This vanishes if cos kag=—1; otherwise the refiector merely introduces a constant factcr, not affecting the character of the scattering. At other angles the reflector causes more complication on account of the different values of m and m’. October 2, 1917. * Proc. Roy. Soc. A. vol. xe. p. 219 (1914). [ 429 | XL. The Distribution of the Active Deposit of Radiwm in an Electric Field. By 8. Ratner (Petrograd), Research Student, University of Manchester * AS CONSIDERABLE amount of work dealing with the question of the distribution in an electric field of the active deposit from the emanations of radium, thorium, and actinium has led to the general conclusion that the. carriers of the active matter are positively charged. Great difficulties, however, have arisen from the fact that even with large electric forces a certain fraction of the active deposit appears on the anode. This anode activity varies with the conditions, but when small quantities of emanation are employed it usually does not exceed 10 per cent. of the total activity. In case of large quantities of emanation, however, the anode activity becomes very large and may reach the value of the cathode activityt. In order to account for the origin of the anode activity, it has been suggested either that a certain fraction of the recoil atoms carry a negative charge, or that some of the posi- tively charged atoms lose their charge by recombination with negative ions and reach the anode by the process of diffusion. Considerable disagreement in the results of different observers suggests, however, that the origin of the anode activity is of a more complicated nature. Several years ago, using a vessel in which large quantities of radium ‘emanation had been constantly kept previously, the author noticed that the activity concentrated on the electrode introduced into this vessel did not depend upon the direction of the field applied, even when minute quantities of emanation were used. A closer study of this phenomenon led to the supposition that the distribution of the active deposit may be in some way influenced by the electric wind, which is always produced in an ionized gas when in an electric field{. Experiments carried out in this direction have established the predominant réle of the electric wind in the process of the distribution of the active deposit of radium in an electric field. In the present paper these experiments are described and the results given. * Communicated by Sir E. Rutherford, F.R.S + Wellisch, Phil. Mag. [6] xxviii. p. 418 (1915). t Zeleny, Proc. Camb. Phil. Soe. x. p. 14 (1898); Ratner, C. R. elviii. p. 565 (1914); Phil. Mag. [6] xxxii. p. 455 (1916). 430 Mr. S. Ratner on the Distribution of the Active Distribution of uncharged Radioactive Atoms in an Electric Field. 2. The progress of the present work has been greatly facilitated by preliminary experiments carried out by means of a method which permits the investigation of the distri- bution of radioactive recoil atoms in a vessel not containing emanation. This may generally be realized by an arrangement shown in principle in fig. 1. Four insulated brass plates A, B, C, and D form two condensers AB and CD. The plates B Fig. 1. pte oe B) C t B R corr emcee and C are provided in the centre with circular apertures, and are connected by a tube t. The central part of the plate D may be taken out and replaced by a similar plate. A small disk R coated with radium A is placed in the centre of the plate A and the plate B is negatively charged, the experiments being carried out at atmospheric pressure. Under these conditions a stream of positively charged recoil atoms of radiuin B passes from the disk R towards the plate B with the velocity of ordinary gaseous ions. A certain part of them lose their charge by recombination with negative ions before they reach the plate B, and then they are carried by the electric wind which arises in the con- denser AB owing to the intense ionization produced by radium A and to the electric force between the plates. It has been shown in a previous note* that under these con- ditions the electric wind is directed towards the plate B, passing by inertia the tube ¢t, so that the neutral atums of radium B are earried towards the plate D. Experiments show that practically the whole of the activity is deposited on the plate D, the activity remaining on the inner surface of the tube ¢ being very small, even when it is so long as 150 cm. Various experiments have been undertaken in order to * C. R, elviii. p. 565 (1914). | Deposit of Radium in an Electric Field. 431 show that the radioactive atoms entering the tube ¢ are really uncharged. Thus, a strong stream of air maintained between the plates C and D carries away completely the active matter even when large potential differences are applied to the plates. Further, the activity received by the plate D does not depend upon the sign or the strength of the electric field in the condenser CD. Also, if the plate D be replaced by another provided with a central aperture, the recoil atoms are carried by the wind through this aperture, and the plate remains inactive in spite of the electric field established in the condenser CD. 3. In order to investigate the distribution in an electric field of uncharged radioactive recoil atoms by applying the principle described in the previous section, the following apparatus has been constructed. ‘Two square metal plates A Fig. 2. and B (fig. 2), provided in the centre with square apertures aa and 66, are insulated from each other by four rectangular pieces of glass g, forming thus a closed vessel, which may be 432 Mr.S8. Ratner on the Distribution of the Active easily opened by taking out one of the glass windows. In the aperture aa of the plate A fine copper wires are stretched 6 mm. apart in order to support a small disk R, and so as not tv interfere with a current of air. In the apertures aa and bb are fixed two tubes ¢, and ¢, of square section (for ease of construction), made from flat pieces of tin sheeting in the form shown in the figure. By means of pieces of rubber-sheet S they are insulated from two other tubes ¢; and ¢, of the same shape, which are fixed in two square apertures cc and dd, cut in the two opposite faces of a tin cube K. The face CD of this cube may be easily removed, and is provided in the centre with an ebonite plug N through which passes a brass rod L carrying a plate P. The whole apparatus forms a closed vessel allowing free circulation of air through the condenser AB, the tubes ¢, and the cube K. The current of air is produced by a potential difference established between A and Band a source of ionization. The small disk R, after a long exposure in an electric field to a large quantity of emanation, is quickly removed from the emanation vessel and placed in the centre of the plate A. The plate B is charged to a negative potential, while A and the cube K are earthed. A certain part of the recoil atoms of radium B, as shown in the previous section, are carried by the electric wind in the direction indicated by the arrows and are constantly circulating through the apparatus, so that the cube K forms a vessel filled with uncharged radioactive recoil atoms in slow and constant motion. The preliminary experiments consisted in studying the distribution of these atoms in an electric field. ; The dimensions of the apparatus are as follows :— The plates Agamd ID... 0. ees 12 x 12 xemir Aperturesja@amd ce ...4 2250). 6) > oe Ov 1 fo / =p 1 COU oy Ye = poy ne tee Pres) Ow LO | sees —kV/2w = —gq—-— = Lkoo), i V*w=—9 ; Bi kkp6). } with the equation of continuity 1 Op wee 2h es Convection Currents in the Atmosphere. 451 Now in the undisturbed state vw=v=w=0, 6=0. Hence 1 Op \ oe OP o Wx Po Ox” | ; 1 OPo Ce ee ——_—, 5 nee Po OY ©) il OPo | 0 =e San Po O# J Hence Pu=kpoo? (2? +y°)—gpoz + const. . . . (6) Then putting p=po+p’, p=Potp, and substituting for . ! / 3 a Po, we have, remembering that p’, p, and o are small quantities, 1 OF — wv = kV = is, 2 — 3kp,9), | 1k = + 2eou—kV7?v = tp Sp! ~$hp.8), (7) Ow we mee er ( 234 | ee ee HP) In the case we are at present considering, p varies only on account of temperature variations ; so that if V is the temperature and a the coefficient of volume expansion, we have p= —pozV and 6 = 29 Vi0# re Dee (8) From equatious (7) we find, by differentiating with regard to «, y, and z respectively, adding and substituting for 6, 2 77 1_o4ip 6) pct a a OV ONGwO Vi 20700 Ou =9 5 tI SS +2 (E-S). - @ UB eae NG Ot of a Vor, OF (9) Transform to cylindrical co-ordinates 7, , z, and let the corresponding velocities be (R, ®, w). Then u=Rcosd—Psind, v=RKRsingd+Pcosgd. . (10) 452 Dr. H. J effreys on Periodic On account of the condition of symmetry, R and ® must be independent of @. Then ov ou ko aTaerieee U ae Ow, = 5 B+ 5, agen and CRG aes i(WeR- 3) = — : p73) a ot Onegin Oe ig oF + 20k -k(V%b—5) = 0,0) ele ~, a 1 ‘ ou yon = gaV—2 (2 318), (15) eer iV? (E=3i8)— Jeter =e a I9z Ot Of” ~ 5 being regarded as known in consequence of V being known, these equations have to determine p’, R, ®, and w. In the previous paper we had V =e (ber? + ce) Jo Arey) 20a aan alee where m is the root of m?=)?-+¢y/k with a positive real part. In this case V has the saine form, and m is unaltered, since in the differential equation for the temperature * the rotation does not appear. Define the quantities I, W, A, and B, which will be found to be functions of z only, by the equations p Soe) == IANO, & Po (ib) = WI (Arey, Ge C ‘ é - (18) R = A J (Arey, ® = B Ji(Arjem. J Then F CA (hr? + oy) A — 20B —k 3G ee ALL 8. tee VB (kM? + ty) B+ 20A—k ae O)y°s0)) ace Oak eset) 4 iW (kX? oy) W a ae = ya(be-" + c0-") —O, se 1) * Hquation (2) of the previous paper. Convection Currents in the Atmosphere. 453 (Ge az” —w)-= why Wy — g(bve- v2 + ce-™) + y? (be + ce—me) + k{b(v?— )e-” + c(m? — r?)e-™ 1, , (22) r+ = aury(bem* + 007m), SUS OPE These five equations are evidently not independent, as they have all been derived from the three equations of motion and the equation of continuity. Any four of them may therefore be used to find their solution. Consider first the complementary functions, where II, W, A, and B are all proportional to e**, where yw is a constant. We find that the last four lead to the determinantal equation 20 —kp?+kriwy 0 0) =e 0 0 pe ak +k? oy 0 —2onr w= ) eae 0 0 be which reduces to (—hyt+ kd? + oy)?(w?—d7) +4y?o? = 0... . (24) _ In general kd? is small compared with a. The roots are found to be approximately y+ - My : HC +0 (Se tocaaw ~~ 29) We shall make A, the height of the free surface in the undisturbed state, large compared with (k/)?, but small compared with 1/X. ‘hus the only admissible values of pu are the small roots containing » as a factor and the large roots that have a negative real part. They are then four in number, but which of the large roots are retained will depend on the sign of y—2. Let these be pw, and po, and let the smaller ones be #3 and pa, which are equal and opposite. Then we can write for the complementary functions IL = py? + poet? + pers? + pset?, . . . (26) and for the particular integral we find gubve—”* he gacme—™ v?—yr'|(y?— 40”) mm? — yd? /(y?—40") y As m is not equal to any of the values of pw when rotation is taken into account, it will be unnecessary to treat the i= — (27) A454 Dr. H. Jeffreys on Periodic e-™ term separately, as had to be done in the previous problem. Then W= a ae opse M328 puzz_ PAP 4 suse v a Biyo gun? ¢ =e? ee Fis 2, > ene pe : Fp daj— pee? Beane! frp oraz EP ane 20Aps Be 2 ella 2anr . 20Xr y’ —4o y?—40° 2@rgaby jay P(y?—4ew”) —9?r? Amc The boundary conditions are that there is no motion at the ground, so that A=>B=W=0 when z=0, and that P=JPo\ wat when z=h. Now 6 is insignificant at the free surface, and hence the last relation is equivalent to yl = gW. 02). 2S ee ees The first three lead to wa Ho's @higeby w*( ps + Pa) SS Ee Oe Fee 2 22 | Onegai (32) y+ 2@ y—20 v*(y?—407) -¥ y’—4o FPO 1 )( gabv es aye ( 2p 2 po v2(y?7—40") — 972A?’ — 4” Ms } ) = aaa gun = ( +E (PoP) (1- eee =: )=0. (33) Now gp;/y? is large even for bodies of continental dimensions; gv/y? also is large. Hence the equation (31) is found to lead to 2 ab 22 o—vh M3( p3— ps) = G a yeh) (pst+ps) — Feta Tat = ne" (34) Substituting in (33) we have, remembering that A/v is small, Ya y+ a2 —2w r? gab sg a = Ona cz ) yy? — 4a? ; } Aho (p3+ ps) i : gi ae (1-2) — = (pst ps) =0. (35) v’ - — 40”) al a Convection Currents in the Atmosphere. 455 The ratio of the term y(p3+p,)/¢ to psh(p3t+ps)/y 18 of order (4w?—y’)/d’gh, or about 10-2-?. Thus, if the radius of the region considered is large compared with 10,000 km., the ratio is large; similarly, the ratio of the two terms in gab/y is of order (4@?—y?*)v/Ng, which is of the same order but somewhat larger. Hence for large areas we can neglect the last two terms of (35), and we have approximately Da ipy — gave “pi ps== 05 We (36) Similar results will hold for the term in e~™, Hence the pressure variation is practically given by egal) 73) meme GEC 74. ee Te el en Thus the pressure variation in the upper air has the same sign as the temperature variation; if vy is real the phase of the first term is the same, but the term in e€-”* would give rise by itself to a lag of 45°. There is no pressure variation at the surface. The vertical velocity can be neglected in the eouation of vertical motion, as the terms depending on it are of the order of 10-* of the others. . If the radius is less than 1000 km., and it is justifiable to neglect the frictional terms in (35), we should have Pst ps = gab(1—e-”)/v?h+ a similar term, while p, and p, are of order (kA?/w)(p3+jyu). Then, as these are small, we have approximately bj1—e~™ ae (/1—e7™ eet eae) y ( vh ‘ ¥ m mh ; Oe) The pressure variation in the upper air then has the same sign as the temperature variation, and that at the surface the opposite. As before, the velocity can be neglected in | the equation of vertical motion. In the extreme case when vh is very small, the pressure variation at the free surface arising from the first bracket is given by Il=4gabh and that on the ground by —4gabh, vanishing at a height z=th. The neglect of second order terms needs justification. In place of the accurate operator d Oren Gm? o>)... 0 di gel Od ob” OF we have always used oO and it is necessary to show that ot 456 Dr. H. Jeffreys on Periodic the neglected terms are small compared with that retained. The largest of them is RO/Or, and the ratio of this to 0/d¢ is of the same order as that of yA7II/4o? to y; so that the neglect is justified so long as II is small compared with (w/d)*, which is usually true. Thus, for the annual variation in Asia, II is of order 10‘(cm./sec.)?, and (@/d)? about 10” (cm./sec.)*, and in most cases the neglect is similarly justifiable. Interpretation of the Results. The first effect of heating is that the air near the ground expands and forces up that above it. On this account alone, the vertical displacement in the upper air would contain terms like wbe'y/v, and thus the free surface would change in height and the pressure variation in the upper air would contain a term gpoxbe''/v; there would be no pressure varia- tion on the ground. This corresponds exactly to the case when (4@?—97)/Mgh is large; if y>2o this motion is mostly radial, but if y<2m the transverse wind increases in magnitude as the pressure in the upper air rises, and remains always near to the geostrophic value. This trans- verse wind thus helps to maintain the pressure differences. This ideal ease only arises, however, when A is of the order 1079/1 cm., corresponding to areas as large or larger than Asia. For smaller areas two further factors need to be considered. The uplifting of the upper air by the expansion of the lower is of the nature of a tidal wave, which would in the absence of rotation tend to spread out with velocity /(gh), comparable with the velocity of sound. If such a wave can spread from the centre to the circum- ference within a single period, the upper air in the heated area has time to flow out over the lower, and the pressure in the upper air may thus be very much reduced, the surface pressure at the same time being caused to vary in the opposite sense to the temperature. This corresponds to the case when gh is large compared with y?/A*?. If the rotation is rapid compared with the temperature variation, the velocity of such a wave is {ghy?/(4@?—y’)}2, and the outward flow beeomes important if gh is comparable with (4eo? — 97) 22. If friction is taken into account, the outward flow is much hindered, and if the friction were large enough there would be no outward flow and no variation of pressure on the ground. The ratio of the frictional terms in (35) to the last term is, however, of order (k/2@)*h71, which has already been regarded as small, being of order 1/50, Convection Currents in the Atmosphere. A457 Further, the case here considered has been the extreme one of no slipping, whereas in reality the velocity quite near the surface is not on an average Jess than about half of that at a considerable height ; if, then, friction can produce only this reduction close to the ground, it must produce much less at a great height (especially as turbulence diminishes with height), and hence its effect in reducing the redistribution of mass must be small. It thus appears that the effect of a rise in temperature in a region whose linear dimensions are large in comparison with the height of the atmosphere and small compared with 10,000 km. is first to cause an expansion near the yround, lifting up the air above into a tidal wave. ‘This causes a system of horizontal winds in the upper air, and the wave spreads out; the pressure variation is affected more and more by this the smaller the area. Friction presumably produces little effect on this movement, so that it will be justifiable to omit it in calculating the motion of the upper air and to calculate the surface winds from the upper winds subsequently. A second approximation may then be resorted to if necessary. The velocity can always be neglected in the equation of vertical motion, and the geostrophic condition holds in the upper air if the variation is slow compared with the earth’s rotation. Comparison with Observation. The actual variation of temperature in the atmosphere does not follow the simple exponential law here considered, but the motion arising from any given symmetrical variation of temperature can be found from the results here obtained, provided this variation can be expressed in the form \\\ FO, vw y)To(ar) ee dr. dvdy, which is true of most functions met with in practice. Qualitatively such a difference does not seem likely to make much change in the results obtained. The effect of the actual steady falling-off of density with height, instead of the discontinuous change here assumed, is more dificult to predict. From what has been said, however, we should expect that the geostrophic condition would hold except close to the surface in almost all latitudes for the annual and other slow variations ; but for diurnal variations (y=) and semi-diurnal variations (y=2Q.) y can never be small compared with 20 cos, and hence the geostrophic condition cannot hold. This is, on the whole, in accordance Phil. Mag. 8. 6. Vol. 34. No. 203. Nov. 1917. 2K 458 Periodic Convection Currents in the Atmosphere. with observation, as the wind at a considerable distance above the ground is known to have xpproximately the gradient magnitude and direction appropriate to the pressure distribution at the same height. From the results of this paper we should expect the divergences from this rule to be important in temperate latitudes when the period is not long compared with 15 hours. The type of disturbance here considered cannot be expected to bear any resemblance to the cyclones and anticyclones of Western Hurope; for it deals only with a standing oscillation over a fixed area, whereas a cyclone 1s not usually generated in the region where it is observed. It, may, however, find a counterpart in the annual variation of pressure in the continents of North America and Ae ae the well-known tendency for wide-spread depressions and elevations to form over these in summer and in winter respec- tively, agrees with the theory, as does the average direction of the observed winds round them. ‘The further result, pre- dicted by the theory, that the pressure difference between two points at the same great height should have the opposite sign to that between their projections on the earth’s surface, awaits confirmation, as the upper-air observations usually record only the relation between temperature and pressure, the height being found only by calculation from the pressure and not directly. Wind velocity at a great. height should also be opposite to that. near the surface * The Southern Hemisphere can yield little direct infor- mation, as everytbing is there dominated by the planetary circulation. The physical interpretation that nas been given of these results suggests that making allowance for compressibility and the ditierence between the troposphere and stratosphere will not affect their character, save by numerical ache that will not change the order Ob magnitude. The wave velocity is of order (gH)?, where H is the height of the homogeneous atmosphere, for all modes of variation of temperature with height; so that possible temperature variations will not lead to any change in the order of MGh/(4w* — y*), which determines the mode of approximation to the result. * The existence of the counter-trade winds above 3000 metres, observed by A. L. Rotch and L. Teisserenc de Bort, Hann-Band der Meteor sogesohen Zeitschrift, 1906, p. 274, on the whole confirms this, 1 woes 6 Wi Soh 2 mK gabe : eiaor | XLII. On the Theory of Osmotic Equilibrium. To the Editors of the Philosophical Magazine. Foxcombe, nr. Oxford. GENTLEMEN, — Sept. 18th, 1917, M® SHORTER’S letter as published in the July number differs from that which I saw. In his original letter he failed to distinguish between the two cases mentioned on p. 268 of my paper; his present communication is, however, more precise and possibly it may be thought to need an answer. Mr. Shorter says, “ Suppose a mixture of two liquids A and B to be in equilibrium with the mixed vapours through a membrane permeable to A only, under conditions of pressure and concentration such that they would also be in equilibrium, if placed in communication through a membrane permeable to B. Let p and be the pressures of the liquid and vapour respectively. Let us increase p to p+dp and yp to w+ oy, adjusting the increments so that there is no disturbance of equilibrium.” He then goes on to say, ‘“‘Suppose now that this liquid and vapour are put into communication through a membrane permeable to B. There will not in general be equilibrium.” It is here that we disagree. His conclusion is easily shown to be wrong ; for, assume that, on altering the pressures as above, the original mixture (of concentration a) is not in equilibrium when the “B” membrane is opened, then there must be some other mixture (of concentration «' say) which will bein equilibrium. That is to say, the vapour mixture @’ under pressure + Oy is in osmotic equilibrium with the liquid under pressure p+ 6p when both membranes are open. Now by proposition (a) on p- 267 of my paper we may close the membrane permeable to B without disturbing the equilibrium, Thus we have two different mixtures « and a’ in equilibrium with the same liquid through membranes permeable only to A; hence by the equivalence theorem (and it is to be observed that this theorem is here only concerned with membranes permeable to one component) these two mixtures, each of which is under the pressure w+ dy, would be in equilibrium with one another through an “A” membrane—clearly this is impossible. The rest of Mr. Shorter’s letter, doubtless, would not have been written had he realized that I have imposed no restriction on the magnitude of the individual partial pressures of the two components except that their sum is to be equal to w. 460 Geological Society. Thus equation (6) p. 270 of my paper, which in the full notation should be written Sap/ spp =F apl7 By» is established by another method. This equation, as already stated, is of theoretical interest. and I hope, when suitable experiments have been made, it will throw light on the action of dissimilar molecules on one another both in the liquid and in the gaseous states. I am, Yours faithfully, BERKELEY. XLII. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from p. 384.] May 2nd, 1917.—Dr. Alfred Harker, F.R.S8., President, in the Chair. A pete following communications were read :— 1. ‘Supplementary Notes on Aclisina De Koninck, and Aeclisoides Donald, with Descriptions of New Species.’ By Jane Longstaff (née Donald), F.L.S. 3 2. ‘The Microscopic Material of the Bunter Pebble-Beds of Nottinghamshire and its Probable Source of Origin.’ By Thomas Harris Burton, F.G.S. As shown by the distribution of the heavy minerals, com- bined with (a) the direction of the dip in the cross-bedding, (6) the evidence adduced by boreholes and shaft-sinkings, a main current from the west is indicated. In the neighbourhood of Gorse- thorpe this current bifurcated, one division flowing eastwards, the other running south-eastwards. A large quantity of the material is derived from metamorphic areas, as shown by the presence of staurolite, shimmer-agegregates, microcline, sillimanite, and kyanite. The source of the bulk of the material is probably Scotland, and the westward adjoining vanished land, from rocks similar in the main to those of the metamorphic and Torridonian areas known in that country. Minor supplies came from the neighbouring Pennine ridge, and from other surrounding tracts of high land. The material was transmitted by means of a north-western river and its tributaries, flowing into the Northern Bunter basin. During certain flood-periods this river overflowed across Derby- shire, carrying its load of sediment, much of which was deposited, as it is now found, in the Pebble-Beds of Nottinghamshire. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SHRIES.] *, *. a DECEMBER‘1917. ‘S XLIV. The Application of Thermionic Currents to the Study of Ionization by Collision. By Frank Horton, Sc.D., Professor of Physics in the University of London * Mae theory of the ionization of gases by collision has been experimentally verified for several gases by Prof. J. S. Townsend and his pupilst. The method used consisted in measuring the currents between two parallel plate electrodes when the ions are set free initially at the suriace of the negative electrode by the action of ultra- violet light. Townsend has shown theoretically that under these conditions if 7) ions are set free at the surface of the negative plate, and if, on the average, each negative ion creates « new pairs of ions, and each positive ion creates 8 new pairs of ions, by collisions in moving through 1 cm. of gas, the number of ions reaching the positive plate is given by BA = B)e(e-#)4 a—B t—Bd : In this formula d is the distance between the parallel plate * Communicated by the Author. + J.S. Townsend, Phil. Mag. [6] iii. p. 557 (1902); v. p. 389 (1908) ; i. p. 598 (1903). "J. S. Townsend and H. E. Hurst, Phil. Mag. [6] vill. p. 738 (1904). H.E. Hurst, Phil. Mag. [6] xi. p. 535 (1906). E. W. B. Gill and F. B. Pidduck, Phil. Mag. [6] xvi. p. 280 (1908) ; and xxiii. p. 887 (1912). Phil. Mag. 8.6. Vol. 34. No. 204. Dec. 1917. 21 462 Prof. I’. Horton on the Application of Thermionic electrodes, and must be large compared with the mean free paths of the ions. It was found by experiment that when the gas pressure p and the electric intensity X between the plates are kept constant, the currents obtained by the action of a constant source of ultra-violet light on the negative electrode are in agreement with the above formula. The values of a and 8 depend upon those of X and p. They were determined by measuring the currents with different dis- tances between the electrodes, keeping X and p constant. It follows from the theory of ionization by collision that pond : are functions of = and this result was verified experimentally by obtaining values of « and 8 for a wide range of values of X and p. The values of = were then plotted as abscissee against the corresponding values of AG e as ordinates, and it was found that the resulting points all lie on a continuous curve, showing that the functional relation is satisfied. The experiments described in the present paper were made two years ago, and were the commencement of an investigation of the ionizing properties of the ions emitted by glowing solids. The completion of these experiments has been suspended while investigations in another direction are being made, but the fact that the new problem is closely related to the present subject has led to the collection of the earlier results. The method of studying ionization by col- lision is, in principle, that of Townsend, but a glowing solid is used as the initial source of negative ions, instead of a zinc plate illuminated by ultra-violet light. The apparatus is represented in fig. 1. It is contained in a glass vessel made in two pieces which fit together by a ground joint rendered air-tight by a mercury seal. The two parallel plate elec- trodes, A and B, are circular in shape and each 1:9 em. in diameter. The upper plate A is of aluminium 3 mm. thick. It has an aluminium rod 2 mm. in diameter and 4 em. long screwed into the centre of its upper face, and by means of this rod the plate is suspended. The upper end of the rod is turned down to a small diameter and has a hole drilled transversely through it. A small platinum hook goes through this hole and the hook is attached to a fine platinum wire ‘025 mm. in diameter. The wire passes through a small platinum tube E, making contact with the right-hand side of it, as seen in the figure. A few centimetres above EH the 463 to a silk thread, the upper end of groove cut in the plug of the stop- a windlass for raising or lowering t Currents to the Study of Lonization by Collision. he Fig. 1. which is wrapped round a platinum wire is tied on cock F which serves as plate A. The platinum tube E is connected to a wire sealed through the wall of the platinum glass tube at O. The 2L2 464 Prof. F, Horton on the Application of Thermionic end of the wire at O thus serves as a terminal for making electrical contact with the plate A. The glass tube in which A is suspended is 3°5 cm. in diameter and about 20 em. long. It has a connexion at the top through a drying tube to a mercury pump and McLeod gauge. ‘The phosphorus pent- oxide used in the drying tube was purified from traces of phosphorus by heating for several hours at 250° C. in a stream of ozonized oxygen, a method recommended by Mr. J. J. Manley. The lower plate, B, is of aluminium *5 mm. thick and fits. into a depression turned in the end of the thin brass tube C, the outside diameter of which is slightly larger than that of B. Fixed to the lower end of C are three brass spring clips which hold it in position on the upper end of the inner glass tube of the ground joint. The spring clips are not shown in the diagram. The plate B with the cylinder C can thus be taken out of the upper glass tube. When placed in position in that tube, C presses gently against a spring § made of a platinum wire, the other end of which is sealed through the glass tube at P, and serves as a terminal for making electric connexion with the plate B. In the centre of the plate B there is a small square hole, each side of which is 1:°3 mm. long. About 1:5 mm. below the plate B a platinum strip, which can be heated electrically, is held in a horizontal position by two leads of stout platinum wire, H and K, to which it is welded and which for the greater part of their length are covered with glass to insulate them and to make them more rigid. A short distance below the platinum strip the two leads are connected together by a glass cross-piece so that there can be no relative motion of their ends. The platinum strip is about 1 cm. long; it is 1 mm. wide and ‘02 mm. thick. Its central part is bent upwards through the square hole in the middle of the plate B, and arranged so that the upper surface of the strip inside the hole is in the same plane as the upper surface of the plate. The part of the platinum strip which is exposed fills the small square hole except for a space about ‘15 mm. wide all round it. The placing of a new strip in position was a matter requiring considerable patience. The central part of the strip was first bent into the shape required by means of a small press, so that the central raised part was made exactly the right size—1-0 mm. square. The ends of the strip were then welded on to the platinum leads, and these had usually to be bent slightly to get the square. in the centre of the end of the brass tube C and in the same plane as that end. The aluminium plate B was then. Currents to the Study of Ionization by Collision. 465 carefully placed in position in the circular groove cut to fit it. The temperature of the heated platinum strip was measured by means of a thermocouple of wires of platinum and platinum with 10 per cent. of rhodium. These wires are ‘025 mm. thick and the junction is very small. It was welded on to the under side of the platinum strip—as nearly as possible at the middle of the raised central portion. The welding was accomplished by raising the strip to a bright red heat and pressing the thermojunction against it. If the strip was hot enough the junction stuck to it, but if this did not occur the strip was raised in temperature very cautiously until welding took place. That the thermojunction wires do not tap off any difference of potential from the heating circuit was shown by leaving the platinum strip heating until a steady temperature was obtained and then suddenly reversing the direction of the heating current, when the same reading was obtained on the d’Arsonval galvanometer used to measure the thermoelectric current. The temperature of the central portion of the strip was obtained from the deflexions of this galvanometer which were standardised by observations of the melting-point of pure potassium sulphate, making use of the curve given by Callendar™ for this purpose. The thermocouple wires were insulated by fine glass tubes and passed out of the apparatus at L and M. The platinum strip was heated by a current from a battery of three large accumulator cells, the current being regulated by wire resistances. When the platinum strip was tested as a source of ionization, it was found that it had to be raised to a very high temperature in order to obtain currents measurable on a galvanometer. This caused the strip to sag, so that the raised central part was no longer in the same plane as the upper surface of the plate B. For several reasons it is desirable to work with the strip at as low a temperature as possible, and it was therefore decided to make use of the much greater thermionic emission which is obtained by covering the platinum with lime. A small piece of sealing- wax and about an equal quantity of lime were finely powdered and intimately mixed in a mortar. A little water was added and a small drop of the mixture was placed in the middle of the central raised square of the platinum strip. This was gradually heated to a bright red heat and the combustible material of the sealing-wax burnt away, leaving a very thin deposit of lime, which adhered firmly to the platinum surface. * H. L. Callendar, Phil. Mag. [5] xlvii. p. 519 (1899). 466 Prof. F. Horton on the Application of Thermionic Suitable thermionic currents were obtained with this electrode at temperatures of 900° C. to 1000° C. The thermionic currents were measured by means of a sensitive d’Arsonval galvanometer, which was arranged in the circuit as shown in fig. 2. Bite a Pe udyunedeuaunuud A. Upper plate. B. Lower plate. |G. Galvanometer. The galvanometer was insulated on paraffin blocks and was connected to the movable plate A and to the positive terminal of a high potential battery, the negative terminal of which was connected to earth. ‘The lower plate B was also con- nected to earth. It is desirable that the part of the glowing filament which is exposed in the plane of the plate B should be at the same potential as that plate. his equality cannot be obtained along the whole length of the 1 mm. of strip exposed, because the heating current has to be maintained, and there was a drop of °05 volt along it due to this cause, the total difference of potential between the two ends of the strip being about ‘5 volt. The potential of the middle point of the exposed portion of the hot strip was kept at zero by shunting the strip with a high resistance and earthing this resistance at its middle point. That the desired equality of potential of the plate B and the spot of lime on the hot Currents to the Study of Ionization by Collision. 467 platinum strip had been established by this arrangement, was tested in the following way. The plate A was raised to a positive potential of 4 volts and the thermionic current was measured when it had become practically constant. The heating current was then reversed and the thermionic current again measured when it had become steady. ‘This was repeated several times and the mean of the galvanometer readings for each direction of the heating current was found to be the same. In this test it was necessary to take the mean of several readings, as the thermionic current from the lime was slowly decreasing with time. This gradual decrease in the electron emission from lime is most marked in the early stages of heating. After several hours’ heating it becomes small, and at the temperatures used in the present research the current was practically constant during the course of ten minutes or so. The experiments consisted in measuring the currents between the two plates A and B at different distances apart. The plates were accurately parallel to each other. This was tested by lowering the plate A until it nearly touched B and then examining the width of the space between them with a microscope, looking in two directions at right angles. The cylinder © could be moved slightly on the spring clips which held it in position, and this movement allowed B to be adjusted parallel to A when examination showed that such adjustment was necessary. The distance apart of the plates was measured to ‘001 cm. by means of a travelling microscope. A piece of white paper was placed behind the apparatus and this was illuminated by a lamp placed on the further side of it. The image of the edge of the plate seen through the microscope was very sharply defined, and it could be adjusted on to the horizontal cross- wire with considerable accuracy. In the experiments, the distance between the plates was always adjusted so as to be.an exact number of millimetres as measured by the microscope. The apparatus (seen in fig. 1) was firmly supported in a strong iron retort-stand resting on a slate slab which also supported the measuring microscope, so that no error was introduced through relative motion of the two supports. The measurements were taken in the following order :— The distance apart of the plates was first adjusted to 1 mm., and a difference of potential V was established between them. When the thermionic current was steady, the reading of the galvanometer G was observed. The travelling micro- scope was then raised 1 mm. by means of its micrometer 468 Prof. F. Horton on the Application of Thermionic screw, and the plate A was raised until the image of its lower surface was again on the cross-wire. The difference of potential between the parallel plates was then increased to 2V and the thermionic current was again read. The distance between the plates was increased in this way 1 mm. at a time up to 5 or, in a few cases, 6 mm., and the electric intensity between the plates was kept constant at the value V volts per mm.ateach distance. ‘The distances were then gradually decreased down to 1 mm., corresponding reductions being made in the potential of the upper plate, and observations of the thermionic current were made at each distance. In this way any alteration in the emission from the hot lime could be detected. As a rule, the currents for increasing and decreasing distances were only slightly different and the mean was taken for each distance. When the differences were not small the set of readings was rejected. The constancy of the temperature of the platinum strip was ascertained from the reading of the galvanometer in the thermocouple circuit, and, when necessary, adjustments were made in the resistance of the heating circuit to secure a constant temperature throughout the series. The time required to take a complete set of readings was usually about ten minutes. The observations recorded in this paper were all taken with dry air in the apparatus. The pressure varied from 0-8 mm. to 10 mm. in different experiments. The electric intensity between the plates was usually 410 volts per em., the lowest potential difference between the plates being 41 volts when they were 1 mm. apart, but in some experi- ments larger electric intensities were used. It has been found experimentally by Townsend that when X/p (the electric intensity in volts per cm. divided by the gas pres- sure in mm. of mercury) is small, 8 (the number of new pairs of ions formed by one positive ion in travelling 1 cm.) is very small, and the production of new ions by collisions is carried on practically entirely by the negative ions. If 6 is zero, the general equation lor the current between the two parallel plates reduces to ees ad R= ner Under these conditions the currents 7, 79, ng, ...... measured with distances w, 2, 3a, ...... between the plates, increase in a constant ratio, for No/ Ny = N3/No=...... =€ Currents to the Study of Ionization by Collision. 469 In the experiments, the results of which are summarized in the following five tables, the ratio between successive values of the current was found to remain practically con- stant over the range of distances used. The mean of these ratios was taken, and from this a2 was found. The currents at different distances were then calculated, using this value of a, and the calculated values of the current are tabulated for comparison with those observed. The currents are ex- pressed in arbitrary units, they may be reduced to amperes by multiplying by 6°9x 107°. Pressure of air, p= 9°86 mm. Hlectric Intensity, X=410 volts per cm. Distance between plates, in cm. — Pressure of air, p=8'12 mm. a'97. | | Observed | Calculated arene current. | current. é ee | observed currents. | 23-0 23:0 1:09 Daina k 25°3 | 1:12 28°0 27°9 1-11 31:0 30°8 1:10 34:0 309 Electric Intensity, X=410 volts per cm. a=1°63. ES ay) 12°5 14:7 173 20°4 24:0 470 Prof. F. Horton on the Application of 7 hermionie Pressure of air, p=5°32 mm. Hlectric Intensity, X=410 volts per cm. a= 208, Distance i | Ratio of between plates, ec Uae ee ‘ - successive in cm. : " } observed currents. a 21:5 23°7 1:44 2 31:0 31:0 1381 3 40°5 40°5 1-31 4 53:0 53°'0 1°31 5 69°5 69°3 Pressure of air, p=4:00 mm. Hlectric Intensity, X=410 volts per cm. a a=4-09: 18°4 18-4 28:2 277 43°0 41-7 64:2 62°8 94°5 94°6 1°53 153 1:49 ur Pressure of air, p=1°81 mm. Hlectric Intensity, X =410 volts per cm. a6 22, aS oC B® 16:2 16:2 30:0 301 55:0 56:1 103°5 104:6 193 194 364 363 1°85 1°83 1-88 1:87 1:89 Currents to the Study of Ionization by Collision. ATL In comparing the values ef « given in these tables with the results obtained by Townsend, it must be remembered that the temperature of the air in these experiments is much higher than the temperature in experiments where the ionization is produced by ultra-violet light, and at a given pressure under these conditions the number of mole- cules per unit volume of the gas is less than it is at the same pressure in Townsend’s experiments. The number of collisions made by an ion in travelling a distance of 1 cm. under the electric force is thus reduced by the increased temperature of the gas, and unless this reduction is neutralized in some way by an increased production of ions on account of the higher temperature, the value of « will be less than it is at lower temperatures under the same conditions of electric intensity and gas pressure. The experiments of H. A. Wilson on the thermionic current between a glowing platinum wire and a surrounding platinum cylinder * have led to the conclusion that the number of collisions, resulting in ionization, made by the negative ion, varies inversely as the absolute tem- perature of the gas when its pressure is constant—that is to say, is proportional] to the number of molecules present in unit volume of the gas. It follows, therefore, that an increase of temperature of the gas, apart from the alteration of density which it causes, has no direct influence on ionization by collisions, which, in a given electric field, appears to depend solely on the nature of the molecules and their distance apart. This result also follows, on the modern theory of sparking, from the earlier experiments of Snow Harrist and of Cardani{ by which they showed that sparking potentials are independent of the temperature of the gas so long as its density is constant. For the purpose of comparing the results of the present experi- ment with those of Townsend, it is thus only necessary to reduce the pressures to the values they would have if the density of the gas were kept constant and its tem- perature were reduced to that of the laboratory. If the temperature of the laboratory in Townsend’s experiments is taken as 17° C., and the temperature of the gas in the present experiments is ¢°C., the observed pressure must be multiplied by arate to obtain the “ corrected pressure.” H. A. Wilson, Phil. Trans. A, vol. 202, p. 243 (1903). . W. Snow Harris, Phil. Trars. vol. 124. p. 213 (1884). P. Cardani, Rend. della R. Acc. dei Lincet, yi. p. 44 (1888). ++ 472 Prof. F. Horton on the Application of Thermionic In order to make this correction, it is necessary to know the temperature of the gas subjected to ionization by collision. I had at first hoped to be able to ascertain this by means of the thermojunction attached to the platinum strip, by cutting off the heating current and observing the dying away of the thermoelectric current. The thin platinum strip cools very rapidly to the temperature of its sur- roundings, and the thermoelectric current falls rapidly at first and then more slowly ; but the method (which gave results in rough agreement with those finally obtained) was discarded for the more accurate method of measuring the temperature with a mercury thermometer. For this purpose the wide glass tube seen in fig. 1 was replaced by another of the same diameter, and having a horizontal side tube at such a level that a thermometer sliding in it could have its bulb placed just over the central hole in the plate B and about :5 mm. above it. The thermometer used had a small cylindrical bulb 7 mm. long and 2°5 mm. wide. Its stem passed through a hole (which it just fitted) along the axis of a small brass cylinder fixed in the side tube. The further end of the thermometer was fixed by sealing-wax into a small cylinder of soft iron which could be moved along the side tube by means of an electromagnet. In this way the bulb of the thermometer could be placed in position for temperature observations. An upper aluminium plate similar to A (fig. 1) was suspended above the lower plate, the distance apart of the plates being about 4°5 mm. The thermometer could be moved between them without touching either plate. Observations of the temperature were taken when the middle of the bulb of the thermo- meter was vertically above the centre of the exposed square of glowing platinum, and also when the bulb was drawn to one side so that no part of it was over the hole. The temperatures registered in these two positions usually differed by about 8° C. With this apparatus, observations of the temperature of the air were taken at various gas-pressures, the temperature of the platinum strip being adjusted in each case to 1000° C., the value it had when the results recorded in the tables were obtained. The gas-pressures at which these investigations were made were adjusted so as to be near to the pressures used in the ionization experiments. A curve was drawn connecting these pressures with the observed temperatures of the air, the temperatures plotted being those registered by the thermometer when its bulb was immediately over the exposed square of glowing platinum. From this curve Currents to the Study of Tonization by Collision. 473 (reproduced in fig. 3) the temperature of the air at the pressures actually used in the ionization experiments was obtained, and was used in the way already indicated to deduce the pressure of air at 17° C. which would have the Fig. 3. {20 7 aie. a ee a = eae | eee {ase ee wee pes | he & | Bea | | 2m -aaaaeo ae Aer # s (ee | pee || aane ca ee oO i 2 6 7 r=) =) 10 Pressure in nim. same density as that experimented on. In the following table the values of a, p, and X are collected. The tem- perature of the gas experimented on, as deduced from the subsidiary experiments, and the “ corrected pressures ” are also given. Electric Intensity, X=410 volts per cm. Corresponding | pressure if gas Temperature nee were cooled with of gas. P , constant density | a. oC, ewe to 17°C | mm. of mercury. P mm. of mercury. 117 9°86 7:33 ‘97 1138 8:12 6°10 1:63 107 5°32 4-06 2°68 103 4:00 3°09 4:09 95 1°81 1-43 6-22 Fig. 4. pore HCH cee ae = ic sic Doe's 474 Prof. F. Horton on the Application of Thermionic The values of «/P are plotted as ordinates against the corresponding values of X/P as abscisse in fig. 4. The curve shown in that figure is drawn from Townsend’s values *. It will be seen that all the points (marked Hee ee MMe. | -CEECEE CEE CHECEE ae CEE me Pe Leal & TO | a = le EERE Rb ed fsspeecusvensnesenst= eee 100 200 250 300 /p by circles) representing the results of the present expe- riments fall very near to the curve, showing that the results of the two methods of experimenting are in good agreement. “In the experiments, the results of which have so far been given, the ratio of the successive currents, when the distance between the plates was increased by equal increments, remained constant over the range of distances used , showing that the ionizing action of the positive ions was negligible. Two sets of observations were made with larger values of X/p than those already recorded, and under ‘conditions in which the effect of the positive ions could no longer be neglected. In these cases the values of « and @ were * J.S. Townsend, Phil. Mag. (6) iii. p. 557 (1902). Currents to the Study of Ionization by Collision. 475 obtained from measurements of the currents at three dis- tances by means of the equation (2—f) ele-P 0 a— Beebe * The results of these experiments are given in the following tables. From the method of calculation, the values of « and 8 when substituted in the above equation give values of n which are equal to those of the observed currents at three distances. It will be seen that the formula expresses the values of the current over the whole range with fair aceuracy in the first table. In the experiments where X= 820 volts per cm., observations at three distances only were taken, as the potential differences required for the larger distances were not available. n=n Pressure of air, p=°820 mm. Hlectric Intensity, X=410 volts per cm. n= 20 B= 049. es Observed | Calculated | Ratio of | between plates, cuaererit’ RABY i successive in cm. observed currents. | 1:62 2 20°3 | 19:0 | 1-68 3 34°] | 309 72 “4 58°5 | 58°5 | 1-77 "D 103°5 | 103°5 | 1°85 6 191°5 191°6 Pressure of air, p='810 mm. Hlectric Intensity, X=820 volts per cm. a=6'9O0: B= -195. “I | 22°2 22:2 1:98 “2 | 44-0 44:0 2°10 3 | 92°2 92:2 SS SSS eS SS SS == SSS SSS SS SSS SS SEE ae a 476 Prof. F. Horton on the Application of Thermionic The temperature of the gas at these pressures was found to be 86°C., and the pressures were corrected to the values they would have if the gas were cooled at constant density to 17°C. The values of X/P, 2/P, and B/P were thus found to be :—- x | a p im | Pi Py 619 7°85 0077 1253 9:94 "193 The values of 8/P are much smaller than those obtained by Townsend. For X/P=619, Townsend’s value of 8/P is ‘104. The greatest value of X/P in Townsend’s experi- ments was 787, at which his value of 6/P is ‘18, and, from the general shape of the curve *, it would appear to be more than *3 when X/P=1253. It therefore looks as though the positive ions obtained by collisions in these experiments were less active ionizers than those produced in the ultra-violet light experiments ; but it seems desirable to verify these small values of 6/P by further experiments before attempting to explain them. In order to compare the values of «/P obtained in this research with those Fig. 5. 100 200 300 400 500 600 700 800 900 .. 1000 Woo 8 1200 %Y/p obtained by Townsend, for values of X/P greater than 300 (the higher limit of the curve in fig. 4), the values of X/P and a/P have been plotted in fig. 5. From the curve * J.S. Townsend, Phil. Mag. (6) vi. p. 598 (1903). - ey ee Currents to the Study of Ionization by Collision. 477 drawn the values of /P corresponding to the values of X/P given in Townsend’s paper have been read off, and are given, together with Townsend’s results, in the following table :— oi aw x = o Pp (J. S. Townsend, (none). Ph. Mag, 1903, yall vi) 350 5°28 5°25 395 5°86 a8 437 6°32 6:3 480 6°77 68 530 T21 C28 662 8:18 8°42 787 | 8:82 8-9 It will be seen that the values of a/P corresponding to these higher values of X/P are also in good agreement with the values obtained by Townsend. The ionization in these experiments takes place at temperatures up to 100° C. higher than the temperature in Townsend’s experiments, and this general agreement of the results is direct evidence that, in the case of negative ions, the power of ionizing by col- lisions in a gas of given density is independent of the temperature within these limits. The author wishes to acknowledge his indebtedness to the Government Grant Committee of the Royal Society for the means of purchasing some of the apparatus used in these experiments. Royal Holloway College, Englefield Green. Phil. Mag. 8. 6. Vol. 34. No. 204. Dec. 1917. 2M L478 4 XLV. Atomic Frequency and Atomic Number.—Frequency Formule with Empirical Constants. By HW. StTaniEy ALLEN, M.A., D.Se., University of London, King’s College. § 1. Introduction. A RELATION between the characteristic frequency of an element in the solid state and the atomic number is to be expected on theoretical grounds. In a paper communicated to the Royal Society the author has sought to establish such a relation, and it will be convenient to summarize briefly the conclusions already reached. 1. If N denote Moseley’s atomic number for an element and v the characteristic frequency deduced from determinations of the specific heat in the solid state, simple relations are found to hold between the values of the product Nv for different elements. 2. For 26 metals it is found that the product can be expressed in the form ND ae where n is a whole number and p, a constant having the value 21°3 x 10” sec.~! (approximately ). 3. The same rule is obeyed in the case of certain non- metallic elements. 7 4. Similar results are found when the characteristic frequency is calculated from the elastic constants of the solid by Debye’s formula. The value of the ‘frequency number,’ n, thus obtained is not in all cases the same as that deduced from the specific heat. 5. Application of the theory of probability shows that there is but a small chance of the product approaching so nearly to integral multiples of a constant frequency by a mere accident. 6. It is found that the atomic numbers of Moseley give better agreement with the proposed relation than do the atomic ordinals of Rydberg. 7. The empirical results have been discussed from the standpoint of the Quantum Theory, and it may be suggested that the integer n is related to the number of electrons concerned in determining the crystalline space-lattice of the element in the solid state. In the present communication an attempt is made to extend these results by considering other methods of deter- mining the characteristic frequency. * Communicated by the Author. Atomic Frequency and Atomic Number. 479 § 2. Frequency Formule with Empirical Constants. Various formule have been suggested for the purpose of expressing the atomic frequency in terms of other physical properties. Several of these have been discussed and compared in a paper by Blom*. They are based on grounds partly theoretical, partly empirical, but are all variants of the expression for the frequency in simple harmonic motion ee) ay Qa A’ where A is the atomic mass, and D is the restoring force for unit displacement from the equilibrium position. It has been overlooked by later writers that the first formula of this kind was given by Sutherland +, who claimed that he had found simple relations between the periods of vibration of elements at the melting point for several chemical families. His reasoning may be summarized as follows :—When a molecule, mass M, and specific heat (mean) c, is heated from rest at absolute zero to its melting- point T,, it receives heat McT,, which is taken to be propor- tional to its mean kinetic energy $Mv’. Thus v is propor- tional to ,/(McT,/M). The length or amplitude of the vibration is assumed to be equal to (or proportional to) #T',(M/p)*, where a is the mean coefficient of linear expansion and p is the density. Hence the periodic time is propor- tional to a T'.(M/p)*/,/(McT./M). Now it isa characteristic feature of Sutherland’s theory of the process of fusion, that melting occurs when the space between the molecules attains a certain value relatively to the size of the molecules. Thus a1’, is constant, a relation that has been verified by Griineisen { for monatomic elements. Again, according to the law of Dulong and Petit, Me is constant. Hence the periodic time is proportional to (M/p)*/,/ (T./M). Putting M/p=V, the molecular (or atomic) volume, the frequency is proportional to * Blom, Ann. d. Physik, vol. xlii. p. 1402 (1918). + Sutherland, Phil. Mag. vol. xxx. p. 318 (1890) ; vol. xxxii. p. 524 {1891). t Griineisen, Ann. d. Physik, vol. xxxix, p. 257 (1912). 2M 2 = 480 Dr. H. 8. Allen on Atomic This is exactly the relation obtained later by Lindemann * for the atomic frequency at the melting-point. For an element of atomic weight A the ae may be written vaka/ (<3)= es: The value of the multiplying constant 4, which is found empirically, was given by Lindemann as 2°06 x 10”. The same formula was afterwards employed by Nernst for the calculation of the characteristic frequency, using for & the value 3:08 x 10”, which is almost exactly half as large again as the factor employed by Lindemann for the trequency at the melting-point. §3. Results obtained by using Lindemann’s Formula. With Nernst’s value for the constant and with experimental data from the Smithsonian Physical Tables (1914), the atomic frequency, v, and the product, Nv, have been calcu- lated for a large number of elements. For our present purpose Lindemann’s formula has an important advantage in the fact that the melting-point in many cases is known accurately, whilst the value of V*, the only quantity in the formula varying with temperature, does not differ greatly for most of the elements. For a test of the formula we select seven of the metals for which results derived from the specific heats have been given in the former paper. Jor these metals the data required in Lindemann’s formula are known with consider- able accuracy. The results. are co ollecton for comparison in the following Table. Tapia tT: Nv x 10-?2, Element. N. Specific Heat. Lindemann. ys Dae Ai ee 13 1075= 5x21°5 1082) 552s SW Aa aANoAece | 26 209:0=10 x 20°9 236'8= 11215 ORM Au Aas Passes 29 1Ot =" Ssc2is 214-5=10 x 21-4 LDPE MERA BEA 30 1442— 7x206 143-6= 7x205 oe a eudalerae 47 211:0=10 «21:1 225°2—=11x20°5 CO /o Wipe ei) 48 168°8= 8x21-1 1445= 7x206 IBS alae See | 82 1540= 7x220 163°2= 8x 20-4 Mean value of De from Lindemann’s formula =?0°9X 10 see. =?. * Lindemann, Phys. Zeisschr. vol. xi. p. 609 (1910). Frequency and Atomic Number. 481 An examination of the figures in the last column of the table shows a maximum divergence of about 3 per cent. from the mean value of y,. This is too large to be accounted for by uncertainty as to the experimental data (T,, A, V) used in the calculation of the frequency, so that we are forced to conclude either that Lindemann’s formula is not exact *, or that the proposed relation Nv=ny, is only an approximation. A comparison of the two columns of integers in bold type reveals the unexpected result that the integer required for the frequency determined from the specific heat is not in all cases the same as the integer required in connexion with _ Lindemann’s formula. For Al and Zn the same integer appears in both columns of the table; in the other cases there is a difference of one unit between the figures in the two columns. There does not appear to be any obvious regularity in the relative position of the larger unit. This surprising result at first produced some degree of scepticism as to the trustworthiness of the relation in question, but similar results have been found in such a large number of different cases that it is almost impossible to doubt the general accuracy of the relation. The atomic frequency has been ealculated by means of Lindemann’s formula for all elements for which data are available. The results are given in the following Tables. Where the data are uncertain the frequency is in brackets. In such cases the value of Nvis to be regarded merely as giving the most probable value of the integer n, which may be called the frequency number. An examination of the figures in the column headed Nvx10-¥ makes it difficult to escape the conclusion that the product Nv may be expressed as an integral multiple of a constant frequency of about 21x10” sec.-! It is true that little weight can be attached to those cases where the product has a large value, for then it is always possible to choose the integer to give a suitable value for y,; but even if we exclude all elements for which the frequency number is greater than 10, sufficient instances remain to make our conclusion secure that the coincidences cannot be accidental. At the same time we must admit that for 14 elements not * A conclusion reached from theoretical considerations by Griineisen, Ann. d. Physik, vol. xxix. p. 298 (1912). The same result would follow from Sutherland’s argument if «T, were constant only for similar elements. 482 Dr. H. 8. Allen on Atomie in Table II. the results given by Lindemann’s formula are not in agreement with the proposed relation. Various suggestions might be made to account for these discrepancies. Tasxe IT. Atomic Frequency by Lindemann’s Formula. Element.) N. | vx10-?. Nvx 10-12.) Hlement.| N. | yx1071*. INvx10-13, Be samba 5 | (281) ©x 2071 Mice 45 701 15x21°1 (OR ea smeeice 6.|: C64) 7 BO% 21-8 |) Pdi 2. 46 6°16 14 x 20°2 Qos ote 8 ama el C20' 3) Ti Aon ete: 221) 4:792 | 1120-48 ALG se 13 8°33 ODO2L "E> Idan. 48 301 7 X 206 Il ease 14 105 Be 21-0 Sia ee: 50 2°50 6 x 20°9 Carre: ac: 20 4°28 224 1 Sbyes. ol 3°22 8 x 20°5 Se 91| (684) | 7x205 || Te ...... 52| (2:69) | 7x20-0 ENT Sense 22 Del 20°2" i Cag cea 59 112 3x 20°6 Vi risect 23 9°26 TOD21°3: Bane 56 2:66 7X21°3 Grid. 24} (9:98) aI20°2 Miia oo 57 | 304 | 8x27 Mins 20 25 8°35 OS 20'9:. |i eres 58 2°86 8x 20°7 Re peceuee 26 9:11 p< 21D) Paes 59 324° 9x 21:2 Gorn s o7| 887 | 11218 || Nd...... 60; 311 | 9x208 INA eee 28 S80, MIs<20°7 | Satees.s 62 3°76 1x22 Cares ye. 29 ESOT MOD? 1°45)|\ Mal ees | 73 O72 20 x 20°9 Ga ee 30 |) 479) Vag 205 | We 74| (6:06) | 2220-4 Gate aaee 31 2°82 2 SC 7allete ya GS) (oso 76 5°96 22 x 20°6 Geese 32 (5°23) S020 9 | as | v7 547 20x 21°1 DO Gases: oF (2°94) >< 20°O': (|| tet aeeees 78 475 | 18x206 Sae 38| (344) | 6x21'8 || Au ......| 79 | 369 | 14x208 Wi) 80 | 407s 2-7 | Oe 81} 200 | 8x203 Tein, 140 | (463) | 9x206 || Pb ...... 82) 199 | 8x20-4 Chu e 8 Al (6:73) Pals 212. |B ee | 83 1-80 Tx2e3 INO sac 42 TOT alo 21 +2 ||| Wha 90 (3:06) | 138x212 IW gee 44 6:99 15xX20'5 || U | 92 (467) | 20x214 In the first place it has been assumed in the application of Lindemann’s formula that the solid is monatomic. If the inolecule contains several atoms the result must necessarily be modified, and it would obviously be possible by properly choosing the number of atoms to be assigned to the molecule to obtain agreement with the proposed relation. In the present state of our knowledge of the constitution of the solids in question little would be gained from such a procedure, and to the writer it appears better to adopt, tentatively at least, a suggestion made in the earlier paper, and admit that in certain cases simple submultiples of the fundamental frequency, v,, may occur. This has been done in the following Table (III.) in which fractions 4, $, or ¢ have been introduced into the frequency number to secure agreement between the values of v,. It is significant that the table includes four alkali metals, which are peculiar in their large atomic volume, and three or four elements which are known to exist in allotropic modifications. Frequency and Atomic Number. 483 TABLE III, Elements with Fractional Values for n. Element. N. pel) *. Ny X<105%7: MME ices son 50 cance 3 10°65 12x 21"< . BE GR ee 4 23°63 43x 21-0 2d) Reeicee See Recon 11 4°31 22 21:0 Ie eos erence 12 7°88 45x 21:0 12 [erat ) eee 15 672 5. X 20°2 (yellow) ............ “an 3°83 22 X 20°9 S (chombic) ......... 16 4°30 32 21:1 (monoclinic) ...... wet 4-24 34 X20°9 Le eee 19 2°53 24 xX 21°4 2 1 eee 33 4°20 6421-3 Lh) ae ae 37 1:54 22 x 20°7 Lt ae 49 | Por Bee 2a eee ata ee 53 1°82 435X215 ISIN Se 80 1°38 54x 21:0 It should be noted that the product Nv is the same for beryllium as it is for magnesium, and practically the same for sodium as it is for potassium. The data for hydrogen and nitrogen are not known very accurately, but it may be mentioned that v for hydrogen (4°88 x 10”) is approximately 4y,, whilst for nitrogen (N=7, v=2°5x 10”) the product Ny is approximately ?v,. §4. The Formula of Einstein. Hinstein* has put forward an equation for the atomic frequency depending upon the elastic properties of the solid. He takes the restoring force proportional to the distance between two molecules and to the bulk modulus of elasticity. Thus D=const.V*K~-!, where K is the compressibility, and therefore Vs v=ta/(Ex): The constant & was evaluated by Hinstein on certain assumptions as to the arrangement and interaction of the molecules, and found to be 2°8 x 10’, but the empirical value 3°3.x 10" is found to give more satisfactory agreement with the frequency determined from the specific heat. The values of the frequency calculated by Blom with the use of the latter constant have been employed for the determination of Nvx10-¥. The results are given in Table IV., and an inspection of the column headed “ Hinstein” shows that the product can be expressed in the form nyv,, where n is an integer in 23 cases, and differs from an integer by 4 in 4 cases. * Kinstein, Ann. d. Physik, vol. xxxiv. p. 170 (1911). eoecee eeecce eoecse eeccece LAGX6 0-1GXG1 6.08X< eocece Dr. H. S. Allen on Atomic ‘okgoq, “G 484 puer’ T-06XTI "USSIOUNI) “f eee ee L:23 X ¥1 ‘unyWoITV “e ee Soe SANINANNANAANAAN ee eS SOoGoenne mk MV a Ye) :XXKXKXXXKXXX GI Hid Edd BD Oo * kiica “UUVUEPUTT °Z ‘y21-O1 X@N FO SUATVA "AI Wavy, esnece 9.06X OI eoeeee we eeee eersee eeecoe 9:06X L SI6X 6 v-06X IT 8:06X OT 8-12X OL 9-I1GX 8 PIGX TI eeesee ecee 01G% 11 eeceece “ule sur ‘qeoqyT oytoedg a sireeeeeeeeseeeees gig teeteereeseneeeee icy Heteeteereerees ug ee poy La *JUOWO TH —- Number. to L Frequency and Atom eeenee a eeeee ee : eeeee . ee eeeee 6-16 X%6 6:06 X8 9-06 X OT ‘padisop oq 0} YoNUI ouv Sorngeloduio, MOT 4B WINIYIT] JO yVey OYloeds oy Jo suoTyvuTUtA | 6166X%8 seneee 8-06 X 8I 6-06 * OT 9-06 X 1 1-06 X GT ‘toded g wo[g WIM puodse1109 0} potoqminu erv sULUNTOS OY], » ‘soanqeacodmioy MOT AIGA OJ BI ONTVA JOSAR] ONT, “9 T-13X 1 GIGX 8 616 8 0:66 8 0:06 GT ee ewes seeree eeeree ¥-06% II 906% 8 9:06 L 89-06% OT G1GX C1 08% 1 He toe P-0GX 8 8.0GX 8 0-16 X ¥4 8.06% FI 9.06% SI LIZX 06 9.06 6 P-0GX GZ 6:06 0 repeal I 8.06% 6 Z1SX 6 1-06 X 8 LIZX 8 Gia 9.06% § G16 XP 0:06% 4 G.0GX & 6:06 9 L136 Xo 9-06 L 8F:06% TL Z0GX FL Teles oo Q.03X GT GISX GI SIZX Sl 9.06X 6 LIEX 8 8-13 9 1-06 X 8 eeoree 0ZX ¢ 8-06 X No) S oS x SOrDAan Nr GIsX L G.06% IT 0-06 FT eereee G13 FI eee eee eee eee Bare GIZX BI L1BX 6 a 21 TIX 8 G1ZX OL 6.06% IT eeeeee O-1GX GI eeererr eyeqy “D 68 . Cree roene . . eee ee ete ence ee ee ee rs . sweet enere eeeeee re teeeee ee er eee eee eeer ee eeteees ee seer ewer eseeeree Pe Ig qa IM set 98 486 Dr. H. 8. Allen on Atomic §5. The Formula of Alterthun. A formula for the atomic frequency involving the coefficient of linear expansion, a, has been proposed by Alterthun*. For a number of elements the product eT, is found to be approximately constant. If, then, for T, in Lindemann’s formula we substitute the reciprocal of the coefficient of expansion, we find v=h,/(1/AaV*), The constant is taken as 4°2 x 104. The frequencies calculated by Blom by means of this formula have been used in the preparation of the corre- sponding column in Table IV. As might have been antici- pated from the fact that the coefficient of expansion at ordinary temperatures has been employed in the calculation, the agreement here is only moderately good. §6. The Formula of Griineisen. On the basis of his theory of the solid state Griineisen + has obtained a relation between the coefficient of expansion, the compressibility, and the specific heat of a substance. The relation, however, involves a coefficient which, according to Griineisen, is approximately but not strictly constant for different elements. By using this relation in connexion with Hinstein’s formula an equation for the atomic frequency is found, which may be written y=2°9 x 10"4/(C,/3aV), where C, is the specific heat expressed in gram calories per degree. Taking the values of the frequency given in Table V. of Griineisen’s paper, the values of Nvx10-” have been calculated and are recorded in Table IV. It should be mentioned that the value given for the frequency of silicon is regarded as open to question. The majority of these results show good agreement with the proposed relation. For the sake of completeness the results obtained from the specific heat and by the formula of Debye have been included in the Table. It must be remembered that the fractional values in the Table are in all cases provisional. § 7. Comparison of Frequency Formule. A comparison of the frequency numbers given in Table LV. for a specified element shows that in many cases the numbers differ according to the formula employed. When allowance * Alterthun, Deutsch. Phys. Gesell. Verh. vol. xv. pp. 25, 65 (1918). t Griineisen, Ann. 1. Physik, vol. xxxix. p. 257 (1912). Frequency and Atomic Number. 487 has been made for want of accuracy in the formula or in the experimental data, it still appears probable that the same element may have different frequency numbers according to. the physical conditions and the particular modification of the solid state that is under examination. It has usually been assumed that the various frequency formule are equivalent to one another. Thus, if the formule of Hinstein and of Lindemann give the same frequency, kyy/ (VBA) = hay/(TIAVY), and therefore V/KT, should be constant for different elements. The experimental values, however, are in agreement with this conclusion only in particular cases *. A reason for this may now beassigned. Instead of identifying the frequencies given by the two formule, we must put ING — 10, jeanne, 70 ane where the subscript = or L refers to the author of the formula employed. Hence Vp ne vy, My,’ : i neV from which it follows that must be constant for = el different elements. It is only in those cases in which my, =n, that the simpler expression can legitimately be employed. § 8. Conclusion. The frequency formule here considered, unlike the formula of Debye discussed in a former paper, have an undetermined constant, the value of which must be found experimentally. The formule themselves are to some extent empirical, and it cannot be said with certainty that they are applicable to all elements without modification. It is, therefore, the more remarkable that for the majority of elements the frequency calculated by these formule conforms to the relation Nv=ny,. It appears from the results that, for a particular element, the frequency number, n, is conditioned by the physical state of the solid. Broadly speaking, the number n varies in a periodic way with the atomic number, but the discussion of the dependence of the value of n on the place in the Periodic Table is deferred till it can be dealt with more completely. * Einstein, Ann. d. Physik, vol. xxxv. p. 679 (1911); Griineisen, Ann. d. Physik, vol. xxxix, p. 300 (1912); Richards, Journ. Am. Chem. Soc. vol. xxxvii. p. 1643 (1915). f 488: 4 XLVI. Electronic Frequency and Atomic Number. By H. STaNLEY ALLEN, M.A., D.Sc., University of London, King’s College *. § 1. Introduction. | a paper on the rélation between atomic frequency and atomic numbert the writer has shown that the product of Moseley’s atomic number, N, and the characteristic frequency, v, for an element in the solid state can be expressed in the form Nv=ny,, where nis an integer and py, a fundamental frequency, which is constant and equal to about 21:3 x 10’ sec.~* for metallic elements. The object of the present papert is to show thata similar relation holds for certain electronic frequencies, but when the vibration of an electron is in question, it is necessary to replace the constant v, by the fundamental electronic frequency, v,=3°289 x 10% sec.~1, which is Rydberg’s con- stant in spectral series usually expressed as the wave-number 109679°22 (Curtis). The relation then takes the form Ny ere or Ny (ies In these cases, as in the case of the characteristic atomic frequency, v refers to some limiting frequency or to a frequency associated with a maximum value of some variable quantity. § 2. Lhe Maximum of the Photoelectric Effect. When a photoelectric current is produced by light in which the electric vector is parallel to the plane of inci- dence, the curve showing the relation between the photo- electric activity and the wave-length has a maximum for a particular value of the wave-length. According to Pohl and Pringsheim § this “selective” effect has been established with certainty only in the case of the alkali metals. The value of the wave-length corresponding to the maximum has been determined by these investigators with an accuracy of about 2 or 3 per cent. The results have been employed in the construction of Table I., which gives the value of v and of Ny for the four alkali metals examined. * Communicated by the Author. + Supra, p. 478. { A preliminary account of the results has been given in a paper read before the Royal Society, Nov. Ist, 1917.. § Die lichtelektrischen Erscheinungen (Vieweg, 1914). Electrome Frequency and Atomic Number. 489 succes aaa | 3 Element. N. din pp. | vxlo714. | Nyx107™. Tics one Raa 3 eon) LOT 1x 3-21 Natt. ay 11 340 8-82 3x 3°23, Tau aos | 19 435 | 6:90 4X3:27, My oN, | 37 480 | 6-25 7 x3:30 | | | The figures in the last column of the table shew that Nv may be expressed in the form mnv,, where n is a simple integer and v, is very nearly constant. The mean value of v, for these four elements is 3°255 x 10% sec.-!, which is so near the Rydberg value 3°289 x 10 sec.~! that there can be no reasonable doubt as to the identity of the two numbers. Pohl and Pringsheim also record a maximum photo- electric activity in the case of the four metals magnesium, aluminium, calcium, and barium; but they do not regard these elements as showing a true “ selective” effect in the sense in which they use that expression. Sometimes the maximum is not observed till some time after the first pre- paration of the metal surface. Provided, however, the maximum is in some sense characteristic of the metal in question, we might expect to find that the corresponding frequency is related to the atomic number in a way similar to that already found for the alkali metals. That this expecta- tion is realized, at least approximately, is shown in Table II., TasieE II. | | Element. N. Nin pp. | vx10—}4, ra elOm e. | | ais | teh cea 12 250 12:0 43x 3°20 LE | Yee 13 254 11:8 5x3:07 i | 220 13°6 5X 3°54 2. Cay (2.5 20 350 8°57 5X 3°48 ik Cae recs: 55 250 12:0 20 x 3°30 2. Bat. ce. 56 280 10:7 18 Xx 8°33 i | | 1. Pohl and Pringsheim. 2. Richardson and Compton. which also includes results obtained by Richardson and Compton *. The values of y, in this Table do not show the * Richardson and Compton, Phil. Mag. vol. xxiv. p. 575 (1912), vol. xxvi. p. 649 (1913). A490 Dr. H.S. Allen on Electronic same constancy as those in Table I., and in themselves could not be regarded as sufficient to justify the proposed relation. The mean value of vy, obtained from these results is 3°31 x 10" sec.~1, which is not far from the Rydberg value. In the case of sodium Compton and Richardson found two maxima in the sensitiveness frequency curve. The first maximum at 360um presumably corresponds to the maximum of the “‘selective”’ effect observed by Pohl and Pringsheim at 340 uu. The second maximum at 227up is supposed to correspond to the “ normal” effect. The value of Nv deduced from the second maximum is 44x 3°26x 10", so that the frequency number (43) is half as large again as the frequency number (3) for the ‘‘ selective” effect. Mention must also be made of a- paper by Souder *, who worked in Prof. Millikan’s laboratory. He states that the maximum suggesting the “selective” effect is found under conditions which are supposed to yield only the “ normal ” effect. No figures are given for the wave-length corre- sponding to the maximum, but from the curves published in the paper the wave-length in the case of sodium appears to be about 350 yy, which agrees with the results already recorded. For lithium, the maximum is somewhere in the neighbourhood of the wave-length, A= 280 pu», given by Pohl and Pringsheim. With a freshly-cut surface of potassium, however, the maxi- mum appears at about 380yy, which differs considerably from Pohl and Pringsheim’s value, 4354p. From Souder’s maximum the value of Ny is found to be 44x3°33 x 10", indicating a change in the frequency number, n, from 4 to 43. § 3. The Limiting Frequency of the Photoelectric [ffect. Tt has been established by the results of a number of in- vestigators that the photoelectric effect ean be observed only when the wave-length of the exciting light is less than a certain critical value—the long-wave-length limit. Thus for the emission of electrons to take place the frequency must exceed the limiting frequency, vp. Hinstein tf suggested that the energy of the electron liberated by light of frequency v could be expressed in the form 4 mv?= Ve=hy— hy. Here V denotes the potential necessary to prevent the emission of an electron (charge e, mass m), and h is Planck’s * Souder, Phys. Rev. vol. viii. p. 310 (1916). + Einstein, Ann. d. Physik, vol. xvii. p. 182 (1905). Frequency and Atomic Number. 491 constant. and the fundamental 2 4 electronic frequency is determined by vr= sent . Hence W, which is eV,,, is equal to hyp. Thus Vy, the ionization potential for atomic hydrogen, is equal to hv,/e or Vz. The value of the ionization potential determined experimentally for hydrogen gas, that is for the hydrogen molecule, is of the same order of magnitude, being 11 volts instead of 13°55 volts. The following Table (V.) gives the value of the ionization potential, Vo, measured by Franck and Hertz *, and also the value of the product NVo. The table includes all gases for which direct experimental determinations are available. The results in the last column show that in the case of the monatomic gases there is remarkable agreement with the relation NY, — ee TABLE V. Element. | N. V, (volts). | NV,. | | | Hydrogen ...... 1 11:0 ESGEI-O Perch 2.2.3: 2 20°5 3X13°7 | Nitrogen ...... i 795 | 4x 13-1 OC a7 Cee eee 8 9°0 | 5x144 1 2) re 10 16:0 | 12x13°3 || Argon eee 18 12:0 | 16135 Puereury <2. 80 4-9 |} 380x131 In the case of the diatomic gases, hydrogen and oxygen, the agreement is not so good; nitrogen, however, falls in line with the monatomic elements. The experimental values of Franck and Hertz for the ionization potentials have been confirmed for certain gases * Franck and Hertz, Verh. Deutsch. Phys. Gesell. vol. xv. p. 84 (1918) ; vol. xvi. pp. 457, 512 (1914). Phil. Mag. 8. 6. Vol. 34. No. 204. Dec. 1917. 2N 494 Dr. H. 8. Allen on Electronic by other investigators+. Special interest attaches to an accurate determination of the ionization potential of helium recently carried out by Bazzoni. He concludes that the value 20°5 volts is slightly too large and that the true value is nearer 20 volts. If we assume that NV)>=nVsz is an exact relation, and that N=2, n=3 for helium, the calculated value of Vo 1s 20°33 volts. In the case of mercury, ionization of a second type occurs for a potential of 10 volts{, which is almost exactly double the value recorded in the table, so that the corresponding frequency number, n, would be 60. So far as the verification of the proposed relation is concerned, no great importance can be attached to cases involving such large integers. §5. Thermionic Potentials. Intimately connected with the potentials discussed in the present paper are the potentials observed in dealing with the emission of electrons from glowing solids and the contact potentials between different metals. In these cases the results obtained depend to such an extent on surface con- ditions and the presence of gaseous films, that as yet it is hardly possible to assign to the various elements reliable values that shall be characteristic of the elements themselves. The work that an electron would have to do to escape frem the substance may be measured by the equivalent potential dif- ference, ¢. ‘he values quoted in Table VI. for ¢, the “ electron TasLe VI. Element. N. @ (volts). | Authority. No. Carbon ......... 6 4°14 Langmuir. 2x 12°4 4°51 Deininger. 2135 Wallen .e: 20 3°04 Horton. dxX122 Titanium ...... 22 24% Langmuir. 4x 18:2 (ren) 6 coe eiinn 26 3°2* Langmuir. 6 x 13:9 Nickel 2.02.52.6. 28 2°9 Schlichter. 6xX13°5 Molybdenum. . 42 4°31 Langmuir, 14x 13:0 Tantalum ...... 73 4°31 Langmuir, 23 X13°7 Tungsten ...... 74 4°52 Langmuir. 25 x 13°4 Platinum ...... 78 5:02 Deininger. 30x13:1 BY IL Horton. 30 x 13°3 AD) cvoyphayae yi gaaee 90 3°36 Langmuir, 22 X13°7 * Preliminary measurements by Dr. Dushman. + Pawlow, Proc. Roy. Soc. A, vol. xc. p. 398 (1914); McLennan and Henderson, Proc. Roy, Soc. A, vol. xci. p. 485 (1915); Goucher, Phys. Rey. vol. vill. p. 561 (1916) ; Bazzoni, Phil. Mag. vol. xxxii. p. 566 Sar t Tate, Phys. Rev. vol. vii. p. 686 (1916). Frequency and Atomic Number. 495 affinity ’’ of the elements, are derived from thermionic mea- surement *, and for the reason stated must be received with some reserve. It is, however, interesting to find that the values of N@ approximate fairly closely to multiples of 13°55 volts. § 6. The Physical Significance of the Empirical Relation. The empirical relations here discussed may be summarized in the formula NV. where vy is some characteristic frequency, and n is an integer (or in some cases an integer +4). On multiplying each side of the equation by h, this gives Nhv=nhv,, or, by using the quantum relation hyv=eV, NeV =neVz, But Ne is equal to the charge, EH, on the atomic nucleus. Hence EV —neVz=0. This suggests that in the limiting conditions which arise in all the physical phenomena under discussion, we have to deal with a minimum value of the energy of a system comprising the nucleus and a certain number of electrons f. The fundamental electronic frequency, y,, is found to be about 154 times the fundamental atomic frequency, »,. It is perhaps worthy of notice that the hypothetical positive electron of Prof. Nicholson, derived from the study of funda- mental spectra in Astrophysics, has a mass 151 times that of the negative electron. * O. W. Richardson, ‘The Emission of Electricity from Hot Bodies,’ pp. 69-79, 164-178 (1916); Langmuir, American Electrochemical Society, pp. 8341-396 (1916). + In this connexion it is interesting to note that the frequency number 3 is found from the ionization potential of helium. As the helium atom contains only two electrons, this may possibly indicate that a third electron is temporarily associated with the atomic system when it is subjected to bombardment by electrons. In this way an unstable arrangement may be produced and ionization result. 2N 2 496 Mr. T. Chaundy: Method of Line-Coordinates for Nate Summary. In various physical phenomena connected with the sepa- ration of an electron from an atom a characteristic frequency, vy, or a corresponding potential V (determined by the quantum relation Ve=hv) is met with. There appears to be a simple relation between this quantity and N, the atomic number of Moseley. In a number of cases the product Nv can be expressed in the form nv, or (n+4)vz, where n is an integer and vz is a fundamental electronic frequency. This tunda- mental frequency is identical with Rydberg’s constant in spectral series, 3°289 x 10 sec.~?. This relation is found to hold in connexion with (1) the maximum of the photoelectric effect, (2) the threshold. frequency or long-wave-length limit of the photoelectric effect, (3) the lonization potential of a gas, (4) the “‘ electron affinity” deduced from thermionic measurements. The relation is discussed from the standpoint of the Quantum Theory. XLVI. A Method of Line-Coordinates for Investigating the Aberrations of a Symmetrical Optical System. By THHo- poRE CHaunDy, M.A., Christ Church, Oxford (attached to Munitions Inventions Depariment)*. HE coordinates of a straight line in space of three dimensions, though, of course, well-known to mathe- maticians, seem scarcely to have received that general use to which from their fundamental character they are entitled. The present application of line-coordinates to the theory of Geometrical Optics takes the standpoint of considering a typical ray of a beam of light passing through the optical system, regarding that ray as completely determined by knowledge of its six coordinates. The effect, then, of any series of refractions undergone by such a ray can be stated. in terms of the contemporaneous transformations undergone by the coordinates of the ray. Since, in knowing the behaviour of every ray in its passage through an optical system, we are in full possession of the geometrical facts of the system, itis clear that formuls expressing the coordinates of a typical ray, at any stage of its passage, as functions of its coordinates before incidence, constitute a complete conspectus of the geometrical properties of the system. * Communicated by the Author. Investigating Aberrations of Symmetrical Optical System. 497 The construction of such formule in the general case, though doubtless the ideal aim of the method, leads of course to results sufficiently unmanageable as to be of little value. The present paper therefore considers only the case of chief practical importance, namely, that of a symmetrical optical instrument composed of media bounded by spherical inter- faces with collinear centres in which only those rays are regarded whose inclination 0 to the optic axis is small enough to require the retention of low powers only of 0. (1) Line-Coordinates. In space of three dimensions with the ordinary rectangular Cartesian frame of reference, the equation to the straight line whose direction cosines are 1, m, n and which passes through the point (a, 8, y), is ! (e—a)/l=(y—B)/m=(z—y)/n. . « . (1) Now these quantities a, 8, y, 1, m, n here employed to define the straight line are, unfortunately, not unique for a given line, since in place of (a, 8, y) we may set the coordi- nates of any other point on the line. The three quantities l, m, n, however, are organically connected with the line, since they, and they only, can define its direction, but we require further coordinates to distinguish our chosen line from among the double infinitude of parallel lines having the same direction (1, m, n). Reference to equation (1) shows that ny—mz=nB—my, le—ne=ly—na, me—ly=ma—IP. That is to say, the three quantities ny—mz, lz—nx, me—ly are the same whatever point (2, y, z) is chosen: in other words they are invariants for the line. We write a=ny—mez, b=lze — nz, Rite Baber fies (27) c=me—ly, and choose [/, m,n; a, 6, «] as the coordinates of the straight line. 498 Mr. T. Chaundy: Method of Line-Coordinates for These six quantities clearly satisfy the two identities 24m? + n2=1, 3) al+bm+cn=0. (3) In this way the six degrees of freedom represented by the six coordinates are reduced to four, corresponding to the four degrees of freedom known to be possessed by a straight line in space of three dimensions. Considerations of symmetry, however, render it in general more convenient to employ these six coordinates obedient to two identities than to employ four independent coordinates. The quantities a, b, c may be given the following physical meaning, if it is desired to visualise them. If unit force be supposed to act along the line, then a, 0, c, respectively, are the turning moments of the force about the three axes of coordinates, at the same time that, of course, J, m, n, are the components of the force along the axes. It is thus some- times convenient to speak of a, 6, c as the “moments” of the line about the axes. (2) Refraction at a single surface. Suppose a ray whose direction cosines are (1, m, n) to be incident at a point of a refracting surface at which the normal (drawn into the second medium) has direction cosines (L, M,N): then we shall require formule giving the direction cosines (l’, m’, n’) of the refracted ray. The first law of refraction, namely, that the refracted ray is in the plane of the incident ray and the normal, is expressed analytically by the fact that I’, mm’, n’ can be written in the form ’=Al +BL, l m'=Am+ BM, n'=An+BN, j for some values of the multipliers A, B. To express analytically the second law of refraction, namely, that sin(angle of incidence) =psin (angle of re- fraction), suppose that the incident and refracted rays make angles 0, 6’ with the normal and that L', M’, N’ are the direction cosines of a line in the plane of incidence normal to (L, M, N), e.g., the section by the plane of incidence of the tangent plane of the retracting surface. The various directions are then related as marginally indicated. (4) Investigating Aberrations of Symmetrical Optical System. 499 Thus LL'+MM'+NN’'=0, since these directions are at right angles. Also LL'+mM’+nN'=sin @, since these directions make an angle 90°—6@. Finally W’L' + m'M'+n'N’=sin 6, since these directions make an angle 90°—@’. Fig. 1. MIN A LLM.N If then we multiply the equations (4) in order by L’, M’, N’ and add, we obtain, in virtue of the results just established, that sin 0’=A sin 0. But sin @ =gisin 0". Hence A= Aw. Again multiply the equations in order by L, M, N and add. We obtain Ll' + Mm'+Nn’=A(LI+ Mm+Nn)+ B(L?+ M?+4 N?) ; te cos 0’ =A cos6+B; De: B= cos 6’— cos O/p. Thus the fundamental equations for refraction at a single surface are cos 4 ii } be | cos 0 = jae Mees (5) : | i ~ + (cos gS FN. be po S i - (cos §' — m m= —+ (cos o/ — ————— 900 Mr. T. Chaundy: Method of Line-Coordinates for (3) Lhe Symmetrical Optical System. The optic axis we take as axis of a, the direction of ongoing light being the direction of x | increasing. The frame of reference will Fig. 2. always be a right-handed set of axes Ay arranged as marginally indicated. a Deviations from the optic axis will be ~ regarded as small quantities ; quantities measured along the axis as of appreciable = > =~ me magnitude. | “a Hence, m,n, y, 2, are small quantities AZ of the first order; 2, / are not small quantities, / differing little from unity. In fact, 1—l=(m?+n’)/(14+/) is a small quantity of the second order. Again, b=lz—nx and c=ma-—ly are evidently small quantities of the first order, while a=ny—mez is a small quantity of the second order. Consider refraction at a spherical surface of radius 7 whose vertex is at the origin and whose centre is accordingly at the point (r, 0, 0), if we adopt the usual convention that surfaces convex to the oncoming light are of positive curvature. The normal at (2, y, z) is the line proceeding from that point to the centre of the sphere. Its direction cosines are accordingly ay (a—r)/7, ee te Substitution of these values for L, M, N in equations (5) gives for refraction at a spherical surface mage — (cos 6’— 25) e—Tah be a7) be f | m cos 0) y n= — — cos g'— SE) 2, . m= a - > (6) i It follows that ny—mz n'y —m'g= Vz—n'a= ge + z( cos iets “) : m Investigating Aberrations of Symmetrical Optical System. 501 But since we have defined line-coordinates a=ny—maz, etc., we can write these equations as a’ =alp ; 6 =b/u+ 2\e0sd —(cos 0)/pp sei). . (I) c' =c/u—yicos 6! — (cos 8)/p}. f Hquations (6), (7) are equations which give the coordinates of the refracted ray in terms of the coordinates of the incident ray and also of the coordinates of the point of incidence and of the angles @, 6! of incidence and refraction. Our next task is to express x,y, z, 0, 6’ in terms of l,m, n, ff,.0,.¢. Before so doing we may conveniently call attention to the first equation of (7), namely a’=a/u. Writing it as a'u=a, we see that the moment about the optic axis of the ray in any medium, multiplied by the refractive index of the medium, is invariant for a single refraction and hence for any series of refractions at coaxial spherical surfaces. In other words, the moment of a ray about the optic axis of an optical instrument varies inversely as the refractive index of the medium in which the moment is measured. It should be observed that this is true exactly and not merely as an approximation. This fact leads to certain useful identities subsequently stated. We may also observe that the two fundamental identities P4+m+n27=1 and al+bm+en=0 are sufficient to fix 7 and a, when we know the four quantities m, n, b, c. It is sufficient therefore to work in terms of m, n, 6, c only: from a knowledge of these the position of the line is completely determined. Jt is to be remarked that it is only the possession by the optical instru- ment of a rotational symmetry about its axis that enables us to discard / and a and work in terms of the remaining four coordinates alone without serious violation of symmeiry. Reference to the second and third equations of (7) shows that this symmetry is better served, if we work with —e, and 6 rather than with 0, c as corresponding to m,n. We therefore write U=—c; v=b. Our defining equations are now a=ny—mze: u=ly—me: v=lze—ne: and the identity al=un—vm. . ... =. . (8) 502. Mr. T. Chaundy: Method of Line-Coordinates for Hquations (7) are rewritten a=aly; u =u/p+yicos 7 —(cos@)/ut; >. eee) v =v/m+z2{cos 6’ —(cos 8)/p}. Now it is known that if @ is the angle between the directions (1), m,, ”,) and (l., me, ne), cos P=11l, + mymg + NyNo, and that sin? b= (myNg— Men, )? + (Myly— gly)? + (Lym — Lym,)?. Hence, since @ is the angle between (J, m,n) and V7); Yt erie sin? = {(mz—ny)/r}? + {(na—lz—nr)/r}? + {(ly—me+mr)/r? = {a?+ (v+nr)?+ (utmryt/r? = {a?+ (u? +07) + 2r(mutnv) +72(m? + n?)}/7?, while sin 0'=(sin @)/m gives 0’. . ».)..) 2a (4) The First Approximation. Collecting our four fundamental equations of refraction m' =m/jp— {cos 6'— (cos @)/uhy/r, ) man 4’ — (cos @)/phz/r, : alias u' =u/ + {cos 6 — (cos @)/why, | v' =v/ w+ {cos 0'—(cos @)/utz, J with the defining equations u=ly—me; v=lz—ne; we may now proceed to our first approximation. In equation (11) m,n’, uw, v,; m, n, u, v, are all small of the first order, as also are 0, 0’. We may therefore replace cos 0, cos @ by their first approximation, 7. e. we may write cos — 1 — coro - } This leads to m'=m/p—(1—1/p)y/r ; n'=niu—(1—1/p)e/r ; wsufp+d—lfe)jy; v=e/u+(U—I/p)z. Now, since (a, y, z) is a point on the refracting surface which has its vertex at the origin, # is a small quantity (in point of fact of the second order). Certainly then to the first onder) 7—0, 117 Investigating Aberrations of Symmetrical Optical System. 503 Hence the identity u=ly—mza becomes u=y ; so v=z. Substituting for y, z in the equations above, we have m'=m/p—(1—1/p)u/r3 n'=n/w—(1—1]p) 0/7 ; u'=ufpt+(l—l/p)u; v=e/p+U—1/p)r; meow —wand,o —v. Itis clear from this result and from general considerations. of symmetry that it is sufficient to restrict attention to two only of these equations, say those for m’ and wu’. The other two are of precisely similar form and can be at once written down, replacing m, m’, u, uw’ by n, n’, v, v! respectively. From this on we shall then consider only the transformation effected by refraction in the two coordinates m and u. The transformation in n and v is then known, mutatis mutandis. The jirst approximation is thus represented by m’=m/w—A—1/p)u/r. . . .. .. (12) eA Amal os Cami yn aia Ream ea) (5) The Second Approximation. For the second approximation we can no longer write cos@=1; instead we must set cos @=1—4 sin’. So Ah sin G. 2b" cos 6’ =1—4sin? #’=1— Hquation (10) gives sin? = {a?+ (uw? +7) + 2r(mu+nv) +77(m?+n’)} 5 here a? is of the fourth order and is to be rejected. Thus! cos 6! —(cos 6)/w=1—1/44 4(1/pw—1/p?) sin? 6 =(w—1) [wt 3(u— Dwi + o%)/r° : +2(mu+nv)/r+ (m?+n*)}. A gain the equation to the sphere is (e—rP + year, i.e. 2ra=a?+y? +27. Hence to lowest order v=(y? +27) /2r. But also to lowest order y=u, z=v. Thus to the second order v= (u?+0*)/2r. 504 Mr. T. Chaundy: Method of Line-Coordinates for Again, since w=ly—maz, it follows that y=(ut me), = {ut+m(u?+v*)/2r}{1— (mi? + n2)1-3, =ut mu? + v*)/2r + ulm? + n?)/2, to the third order on expansion. Accordingly icos 0’ — (cos A) /phy =[(¢—1)/et+ @—1) ip? fm? +? + 2(mu-+ nv) [7 + (uw? + v7) /77)] x [wt m(u? + v*) [27+ ulm? + n?)/2 | =u(w—1)/wt (w—1)/2u{u(m? +n?) (1+ 1/p) + 2(mu-+nv)/ur + (m/r+u/pr?)(w? + v?)} to the third order. Hence by substitution in the first two equations of (11). =m/p— (1—1/p)u/r— (u4—1)/2u{ (1+ 1/p)u/r(m? + 2?) + 2u(mu + nv)/pr? + (m/7? + u/pr?) (wu? +07)f¢ . (14) ul =u + (w— 1)/2pef C1 + Ufpe)ee(en? + 02) + 2(u/wr)(mu + nv) + (m/r + u/pr?)(u? +07)}. (15) These are the equations of the second appre correct to the third order. (6) The General Optical Instrument. The first order formule given for a single refraction by equations (12), (13) contain, of course, no more information than is accorded by the Gaussian first order geometry. For a general optical instrument they will clearly take the form m =Pm+ Qu, u' =Rm-+S8u, where P, Q, R, S are optical constants of the system. In fact, applying these equations to a ray passing in the plane yOx and recognizing that, for such a ray, to the first order, m is the inclination of the ray to the optic axis and u the ordinate of its point of crossing the plane of reference, we see that we have merely reproduced the fundamental equations of Cotes’s method, as demonstrated, for example, in Herman: ‘Optics’ (Chapter 6), with m, u written for «, y. I shall therefore suppose the theory of the calculation of the optical constants P, Q, R, 8 sufficiently known and pass on Investigating Aberrations of Symmetrical Optical System. 505 to consider the second approximation, involving terms of the third order. I shall show that for this approximation the formule for m’, uw’ are always of the form m' = Pm-+ Qu+4{ (Am + Bu) (m? +n’) > —2(Cm-+ Du) (mu + nv) + im + Fu) (uv? + 0?) , | He ul =Rm + Sut+4{ (Gm + Hu)(m? +n?) r - (16) —2(Im+ Ju) (mu+nv) + (Km-+ Lu)(uw?+v?)}, J weer Oo, C.D, HE, F, GH, 1, J, K, li are optical constants of the system. Now these forms are invariant for any number of coaxial spherical refractions so long as we are content to retain small quantities up to the third order only. At any rate it is evident that the equations (14), (15) are of this form, though certain of the constants, namely A,C,G, I, are absent, i. e. are zero in this case. For a series of refractions suppose m’, n', u’, v' expressed in terms of intermediate coordinates M, N, U, V by equations of the form (16), at the same time that M, N, U, V are themselves expressed in terms of the final coordinates m, n, u, v by a second set of equations of similar type. Now, in the third order terms it is sufficient 10 substitute the first order expressions in M, N, U, V, 2. e. expressions of the form pm+qu, pnt+qv, rm+su, rn+sv. Thus the third order terms in m’, n', u’, v' have the same form in m, n, u, v as they had in M, N, U, V. Again, in the first order terms we replace M, N, U, V by expressions in m, n, u, v of the form of those on the right of (16). Hence, after both these substitutions we have expressions for m’, n’, u’, v' in terms of m, n, u, v of the form stated in equations (16). Hence the form persists through a pair of refractive systems, if it persists through each separately. Since we have seen that it holds for a single refraction, it follows that it holds for our general optical instrument. The third order functions of m, n, u, v, namely 44(Am-+ Bu) (m? +n?) —2(Cm-+ Du) (mu + nv) + (Em + Fu)(u? + v?)} and 4{(Gm-+ Hu)(m? + n?) —2(Im+ Ju) (mu + nv) + (Km + Lu)(u? +02) $, which must be added to the first order expressions Pm + Qu 506 Mr. T. Chaundy: Method of Line-Coordinates for and Rm+Su to give the second approximation formule for m',u', may conveniently be termed the “aberration functions of the third order.” Their coefficients A, B,....K, L may likewise be termed the “aberration coefficients of the third order.” They are optical constants of the system and in terms of them all the third order aberrations can be stated. The aberration coefficients of the third order are connected by the seven identities : . SA—RB—QG+PH= (1—P’)u/p’ or 1—P? if first and last media are the same| 17) SC—RD— QI +PJ=PQp/p’ or PQ ” Pr) SH—RF—QK + PL=—Q?u/p’ or = 2 29 SA +RC—QG— PI=0, 1 SB+RD—QH—PJ=—21/y/, the Petzval Sum, | SC+ RE — QI—-PK=—3]1/p/, the Petzval Sum, f he SD+ RF—QJ-—-PL=0. J I omit the proof of these equalities from reasons of brevity. The first three follow from the fact noted in § 3, that the moment about the optic axis of any ray varies inversely as ~ the refractive index of the medium. (7) The Correction Conditions for the Second Approzi- mation. Since the formule (16) contain complete information regarding the aberrations of the second approximation for an optical instrument in which P, Q, R, 8, A, B,...K, L have been calculated, it is clear that we must be able to state in terms of these quantities last-named, the conditions that the optical instrument be perfect to the second approxi- mation. The method ‘is sufficiently illustrated, if we limit ourselves to the simplest case, that of an incident parallel beam. I shall suppose also the first and last media to be of refractive index unity. Without loss of generality we may take the infinitely- distant source generating the parallel beam to lie in the plane zOz. Every ray of the beam will thus have the same direction cosines /, 0, 2; the other coordinates u, v will, however, vary from ray to ray: in fact, they represent the two degrees of freedom enjoyed by rays of the beam. To visualise them we may remember that to a first approxi- mation they are the Cartesian coordinates of the point of Investigating Aberrations of Symmetrical Optical System. 507 incidence of the ray on the plane of the first surface of the system. With an infinitely-distant object in the plane zOz, the image, if one exist, is also to be found in this plane. We shall therefore study the position of the point in which the emergent ray strikes this plane. Denoting by [U', m', n’; a’, u’, v'] the coordinates of the emergent ray, we suppose it to cut the plane zOw in the point (£’, 0, &’). Then from equations (8) w=—mE sa =—m'e, Putting m=0, u’=—m/é', our fundamental formule become ml fu=Q-+${ Bn®—2Dnv + Fe? +0°) —m'&'lu=S8+3{Hn?—2Inv+ L(v?+r)}. _ (i.) Incident beam parallel to the axis. Here n=0, so that m’/u=Q+4F (w?+v’), —m'&u=S84+$L(u?+v’). Thus @&=—{S+3Ll(w?+o)t/{Q+sF'+r")}, = —§/Q4+ (FS— QL) (u? + v?)/2Q’. if &’ is to be the same for every ray, 27. e. is to be inde- pendent of u, v, we must have FS—QL=0. This is the condition for no spherical aberration. If fh is the semi-aperture, then h?=u?+v?, and the amount of spherical aberration when the above condition is not satisfied 1s seen to be h7(FS —QL)/2Q?. It is worth while to remember that —Q is the power of the system. (ii.) Point-to-point correction. Returning to the case in which n40, we have ee S4+4{Hn?—2J3nv+ L(u? + v7) } ~ Q4+3{ Br?—2Dnv+ Fu? + v’)} = —§/Q+ {(BS —- QH)n?—2(DS —JQ)nv + (FS — LQ) (u? + v?)}/2Q? to the second order. 508 Mr. T. Chaundy: Method of Line-Coordinates for Again, ¢’=—a'/m'=—a/m’, since a is an invariant. But al=nu—vum=nu, since m=0. Thus C'=—nu/lm' n 1 Fag Bae Q+4{Bn?—2Dnv + F(v? + v7) } = —nflQ +n{Bn?—2Dnv + F(u’? + 0”) }/2Q?. In order that all the emergent rays may converge to a common focus we require that &', €’ be each independent of U, Uv. This necessitates DS JOOS —LO=—0) Donna ee. D0) = Jt, In virtue of the identity DS + FR—JQ—LP=0 already stated (18), it is clear that these conditions, apparently four, really represent three independent conditions alone, 7. e. two in addition to the condition of no spherical aberration. When these conditions are not satisfied, we can state the aberrations under the guise usually recognized, namely, error against the sine-condition and astigmatic error, in terms of our aberration coefficients. With light parallel to the axis, the sine-condition requires that the ordinate of the incident beam bear a constant ratio to the sine of the inclination to the axis of the emergent beam, 7. ¢., in our notation that v/n' be the same for every ray. We may measure error against the sine-condition by variation in v/n’ between marginal and axial rays. Now for an incident paraxial ray, 2. e. for m=0=n, we have n'=Qv+43F(v?+0")v, so that v/n' =1/Q— E(u? + v?)/2Q?. Thus error against the sine-condition is measured by —h?F'/2Q’. Satisfaction of the sine-condition requires F=0. ; The third component of departure from point-to-point accuracy usually considered is astigmatism. This is an odd sort of quantity, having little organic relation with the system, since it is necessary to specily the position of a stop ; its discussion is correspondingly awkward, and I accordingly Investigating Aberrations of Symmetrical Optical System. 509 content myself with stating that for a central stop an axial distance & in front of the first surface, the astigmatic separation, 7. e. the distance between primary and secondary foci, is 4(DR—JP) +4(DS —JQ—FR + LP) —/2(FS—LQ)}/i222Q?. (iii.) Distortion and curvature of field. If now the previous aberrations have been corrected and the conditions D=0=F=J=L thus satisfied, the optical system is point-to-point perfect for parallel light, 7. e. an infinite point source will produce a definite point image. The coordinates of this image are now given by E'= —§/Q+n?°(BS—QH)/2Q’, O'= —n/1Q+7n°B/2Q’, where n denotes the direction of the infinite object. For a flat field the z-coordinate of the image must be the same for every source at infinity, z. e. ’ must be independent of n. The condition for this is Bs—QH=0. On the other hand, for absence of distortion we require that the distance from the axis of the image-point be pro- portional to the tangent of the inclination to the axis of the infinitely-distant source, 7. e. that €’ be proportional to n/l for every infinite source. This requires that B=0. The system is corrected for both curvature of field and distortion if we have B=0=H. In the uncorrected system Newton’s form for the curvature of the field gives 2(E' — &y')/E?={ (BS —QH )n?/Q’} /{n7/PQ?} = BS— QH. The distortion can be measured by ({' — &') /(n/1) = Bn?/2Q? to lowest order. For an optical system completely corrected for parallel light the conditions are B=O0=D=F=H=J=L. These six conditions in virtue of the last identity in (18) contain only five independent conditions, tallying of course with Setdel’s five conditions. The foregoing analysis makes attempt at no more than outlining the method; I hope to have subsequent occasion, to show its application, on the one hand to such theoretical questions as discussions of the form of an uncorrected image, and on the other to practical questions of optical instrument design. Phil. Mag. 8. 6. Vol. 34. No. 204. Dec. 1917. 20 Lo co kO Fadl XLVIII. Note on the Action of Coupled Circuits and Mechanical Analogies. By H. C. PLumMErR*. 1. 7HXHE paper by Prof. Barton and Miss Browning (p. 246) suggests a doubt whether a simple elec- trical problem really is made easier for the average student by a complicated mechanical analogy. It seems as if the avowed purpose is to remedy the difficulty of one problem not only by setting another but by requiring the relation between the two to be understood. Yet surely the real difficulty begins, and almost ends, in the electrical problem, in establishing the fundamental equations. The mechanical analogy does not assist in this but only in the subsequent interpretation. Curiously enough, the authors do not seem to have pre- sented their analogies in the clearest light. Thus, for example, their equations (27) and (28) may be re-written in the form (P+.Q+ AP) SY + (P+ Q)gl-y=— BOS, t 2 2 (P+Q48Q) SY + (P+ Qglte=— PD. The analogy is now exact, and requires i Peet 1 P+Q+8Q LR 7 EO? NS 7 Pe i _8Q fe cle =e = —MS=-.5.—,, iia g P+Q’ : Gp leas which give for the coupling SPQ 1= (PEQ+AP)(P+Q46Q) more distinctly than the authors seem to have shown. At the same time LR--NS=M(R-—S), a limitation of the con- ditions apparently overlooked. The second analogy does not suffer from the same defect, but in general terms is even more complicated. The coupling may be made positive by reversing the sign of y or z. * Communicated by the Author. On the Action of Coupled Circuits. Gu 2. The equations of the problem may be written aa dr LIR—, -MR=: =—2, d?x dy Hven from the point of view of the student, the method of normal coordinates seems the simplest in this case. For all values of A, da d2. (LR+2.MS)75 —(MR+2.NS) +3 = —wthy. Let MR+A.NS=A(LR+2. MS), and then a (LR+2. MS) Pe (a—ry) + (@—Ay)=0, so that the solution is x—hy =k cos : +e) pe=ULR+A.MS. But the equation for 2X, .MS—A(NS—LR)—MR=0 ... (1) is a quadratic with two roots ry, A». The complete solution is therefore given by &— yy = ky cos i, +4), L—DAgy = kz CO8 ( +6), (2) pr=LR+X,. MS, p2=LR+2,. MS, where 27p,, 27p, are the periods. Let 27P, 24P, be the natural (uncoupled) periods, so that P)?=LR, P.?=NS. Then prtp?=2UR+4+(A,4+A.)MS=LR+NS, pipe = LR{ LR + (Ay + Ay) MS} +A,A, . M*S? =LR.NS—WM’RS, or, the coupling being given by M?=y’. LN, prt+pP=PP+P?, prpr=PYP21—y’). ~. (3) The method seems so obvious that it is probably well known even in elementary teaching. Ina case like this it can be understood when the simple general theory of linear equations with constant coefficients is not familiar. 202 512 Prof. H. C. Plummer on the Action of 3. If initially the currents are zero, ¢,=e,=0 ; if at the same time the charges are v=a, y=0, (2) give immediately 2—yy =a cos py 't, 2—Agy=a COS Po It. The equations (3) hardly exhibit the variations of p,, po with y very clearly. The quadratic in p’, pi—(P/? + Ps?)p? + P?P22(1—?) =0, is no better in this respect, and its solution conveys the rela- tions even more obscurely. There seems, therefore, to he some excuse for a graphical representation. Higa: ¥ o = = = wz & . 7 ms . ~ ~ ~ oe On OA, the diameter of a circle (fig. 1), let OD represent P,2, and DA represent P,?.. The perpendicular BD re- presents P,;P,. Take G on BD so that GD/BD=y. Let ECGE be the diameter through G. Then HG+GEF=0D+ DA=P,?+P.,?; HG ° GF =BG e GB'=P _P.(1—y) . P,P,(1+y) = P/’P.2(1—y’). Hence EG represents p,? and GF represents p,”. Thus as the coupling increases from D to B, the squares of the periods OD, DA change in a way easily apprehended until one disappears and the other is represented by the diameter of the circle. ve aan : peat i bs Coupled Circuits and Mechanical Analogies. 513 Or again, since BD represents P,P,, and HG represents Pipo, HG being perpendicular to EF, OB'=P(P+PZ), BA=PA(PY+P,"), EW? =py(pr+p2'), HE’ =po*(pi'+ ps’). Hence on the same scale by which P,, P, are indicated by OB, BA, EH represents p,; and HF represents p .. Thus if OB, BA represent the natural periods, the periods under coupling are shown by EH, HF and their changes can be traced in a much more instructive way than the equations alone, simple as they are, naturally suggest. 4, There is another elementary point. If a, y are con- sidered as coordinates the fundamental vibrations are per- pendicular to the lines w—A,y=0, x—A,y=0, which are not at right angles. Let MASS =A/R so that (1) becomes M,/(RS) —m(NS— LR) —n?M “(RS) =0,7 or 2m 2y,/(LNRS) yy. PiPs l—-m? NS—-LR ~ 3(P.2—P,’)’ In fig. 1 this fraction is represented by —GD/CD, and thus m=tan$GCD=tan FOC. The lines Y=M4L, Y= MQ are therefore represented in direction by OF, OE. But the required lines are Ry=m,S'z, R?y=m,S*z, and the latter are easily derived from the former when the ratio R/S is given. They are represented by OX, OY in the figure. The simple oscillations are perpendicular to OX, OY, and their periods are 2ap;, 27p,. The resultant compound oscillations are to be found from their components perpendicular to DA, DB’. The fact that OX, OY are not rectangular axes is an inconvenience which can be avoided by taking other vari- ables &, 7 such that go> We ee One, ON Ty) 2 Wal ae CB) As this change of variables isindependent of the coupling, it simplifies the problem without involving any complica- tion, and can therefore be introduced with advantage at tlie beginning. The problem is then brought to its simplest terms as the projection of two rectangular harmonic motions ae! Prof. H. ©. Plummer on the Action of on two other rectangular axes. The fundamental periods are shown graphically in fig. 1, and the inclination of the two sets of axes, which is now simply AOF. 5. The problem may also be considered in a slightly less elementary way. The fundamental equations may be written in the form (LN —M?*)RS2 =—NSx«—MRy, (LN —M’)RSy = —MSa— LRy. Now these equations will also follow in a dynamical problem defined by the kinetic energy T and the potential energy U, where 2T= (LN — M?) (84? + Ry’), BS Re iy 7o,\ Rm? +2May+ gv: The common conjugate diameters of the ellipses 2U=1, Sz? + Ry?=1, are MS2?— (NS— LR)azy —MR7’?=0, or (2 —Ayy)(a@—Agy) =0, by (1). Normal coordinates are obtained by taking these as axes. But it is simpler to take &, 7 as the variables accord- ing to (4). Then 2T =(LN—M2)RS(£2 +7”), 2U=NS# + 2MR2S°£y + LR7?. Since P’=LR, P,."=NS, M’=y’.LN, these imay be written 2T=(1— 9?) PPE" + 9) 2U =P 4 dy . PyPoén + Pea} The axes of U are | Whe PP (2 —7’)= Gee a Pere and if these are taken as the axes of X, Y, 21" Do (X? + Y?), 2U =p," X* + pi Y?, where p.?+p=P2+Py’, py7p.2=PP.2(1—y7), as before, by simply comparing the last two forms of U. The rest of the solution follows immediately. 6. The obvious analogy suggested by (5) may just be worth noticing. On the surface 2(1—y7) Py? Porqf= P2E? + 2y . Pi Po&n + Pym)’, the axis of € being vertical, a particle is oscillating under (9) Coupled Circuits and Mechanical Analogies. d15 gravity in the immediate neighbourhood of the origin. It is evident that its kinetic and potential energies are propor- tional to T and U as defined by (5). Hence its motion presents an analogy to the problem considered. The surface is fully defined by the section on the definite plane =497}, and this section is (L—y’)PPo? = Po2k? + 2y. Py PoéEn + Pin? =(Poé+y. Pin)? +d —y) Pi? | = (19) PSPS? + (y Pod + Pin)’, or clearly an ellipse inscribed in the rectangle €=+P,, n=+P,. The ellipse, which is also represented by E=P,cos 0, n=—P, sin (0+e2), sin a=y, indicates by its axes both the periods and the directions of the fundamental oscillations. The area of the ellipse, com- pared with its maximum, is cos « or /(1—y’). In the limit the ellipse degenerates into one diagonal of the containing rectangle. The effect of the coupling is very clearly shown by the change in the shape and area of the ellipse, and the slewing round of the axes seems to show very well how the fundamental oscillations must enter into both systems dependently. The two degrees of freedom of the electrical system are represented by the displacements of a single material particle parallel to two fixed rectangular axes, and what may he lost in distinctness seems to be compensated in the very clear connexion between the two displacements. It is the natural type of all small oscillations with two degrees of freedom, and if it is not well adapted to detailed experiments it has the advantage of being most readily apprehended without any experiment at all. Fig 2. eee | be: K Fig. 2 shows the corresponding sections of the surfaces when P,/P,=1°5 and the coupling y is 0, 0°5, and 1. wasIaS 516 _ On the Action of Coupled Circuits. 7. The differential equations of the problem are also formally reproduced when 2T= Le? —2Mry4+Ny?, 2U=27/R+y7/8. These suggest the mechanism represented in fig. 3. Here a Fie. 3. heavy ring is pivoted on an axis AB, about which it can turn with a spring connexion. It carries the axis CD on which is pivoted a heavy concentric sphere, also with a spring attachment. If the axes CD, AB are inclined at an angle @, while the ring rotates about AB through an angle y and the sphere rotates about CD through an angle ¢ relative to the ring, the mechanism presents a framework of Eulerian angles. The rotation of the sphere about AB can be resolved into components yr cos @ about CD and w sin @ about the perpendicular axis. Hence if C is the moment of inertia of the sphere about any axis and B is the moment of inertia of the ring about AB, the kinetic energy of the system is given by 2T=C(d+v cos 02+ Cw? sin? 6+ By? = O¢?+ 20d cos 0+ (C+B)w’. Astronomy and Electrical Theory of Matter. 517 On the other hand, if the spring factors about CD, AB are R-1, S~1, the potential energy of the system is given by 2U = 6? +87". Now let 0=90°—B, G=2, p=—y; and C=L, B=N—L(L 3 dv F= F(a) = m1" aT tmov%(1—v’) eons where the second term expresses v din /dt. For transverse acceleration, m is constant because speed is constant, the second term is zero, and the transverse inertia is simply as above. For longitudinal acceleration, however, we must use the full expression, which writes itself algebraically vw \dv piss oil aia” | af aa F= am ( uy 1 —v") di ido dt’ and the coefficient of dv/dt represents the longitudinal inertia. But there is no sueh complication about momentum unresolved into factors: that always has the value m,u(1 —y?/¢?)-?. The following way of putting the matter has perhaps some merits :— =m,(1—v*)~ Let v=c sin 2 then m=m, sec D and mu=m,¢ tan B d dp = Pkt ae Se (ES z, APT ae lis while a ae (mu me SEC” BT dv aes Da Nge eas =m, sec fp. dt which, read as mass-acceleration, gives the longitudinal inertia. L. On the Kinetic Theory of the Ideal Dilute Solution. To the Editors of the Philosophical Magazine. GENTLEMEN,— N a recent number of the Philosophical Magazine*, Dr. F. Tinker puts forward a theory of the Binary Liquid Mixture, deducing expressions for the partial vapour pressures of the components, and obtaining in the case of a dilute solution the relation expressing Raoult’s law. Nowa direct kinetic explanation of Raoult’s law would constitute a very great advance in the theory of the Ideal Dilute Solution. At present the only kinetic foundation for the theory is the law of the proportionality between the partial vapour pressure of the solute and its concentration (Henry’s law). Raoult’s law and the “ Gas law ” of osmotic pressure * May 1917. 522 Dr. 8. A. Shorter on the Kinetic can only be deduced from Henry’s law by kinetically obscure thermodynamical methods. It is evident, therefore, that any theory which claims to give a direct kinetic interpretation of Raoult’s law must be subject to very careful consideration. I wish to point out, with respect to Dr. Tinker’s theory, that when itis applied correctly to the special case of a dilute solution, it yields a law for the lowering of the partial vapour pressure of the solvent quite different from Raoult’s law; and that Dr. Tinker’s deduction of Raoult’s law from his general theory is due to a mathematical error. Dr. Tinker first obtains expressions for what he terms the internal partial liquid pressure of each of the components of the binary mixture. This is the pressure which would be exerted by the molecules of the component if they behaved as molecules of an ideal gas and were confined in a space equal to the total “free space” in the solution. Dr. Tinker obtains for the ratio of the internal pressure 7 of the pure solvent, to the partial internal pressure 7,’ of the solvent in the solution, the following expression, m= Set (1- Sa) flea where N and n are the number of molecules of solvent and ~ solute respectively in the solution, V; and V,—20, are the molecular volume and “free space’? respectively of the pure solvent, V, and V,—0, corresponding magnitudes relating to the pure solute, and ne the expansion on forming the solution from its components. Dr. Tinker next considers the relation between these partial internal pressures and the partial vapour pressures of the components, arriving at the following expression for the ratio of the vapour pressure p, of the pure solvent to the partial vapour pressure pi of the solvent in the solution, Pi 2, oe Q ere Noe N eee ) bem, [10] where Q is the molecular heat of dilution, T the absolute temperature, and R the “ gas constant.” Let us consider now the case of an ideal dilute solution. As the concentration diminishes, the quantity Q/n tends to a zero limit. This follows, of course, from simple kinetic considerations. Hence we have p.m a 1? P1 Ty * Numbered equations quoted from Dr. Tinker’s paper will be dis- tinguished in this commurication by the numbers given them there. Other equations will be distinguished, when necessary, by letters. Theory of the Ideal Dilute Solution. 523 which, since it holds to the first order of small quantities, mav be written T— 7! 1 7, Now equation [5] may be written 7 — Ty’ n (2) 2 ! Piss Pie / me lee Vee so that for a dilute solution we have Pr ae me ee (* as bs + ") A) Pu N Ve ae b, ; : i ; : instead of the equation Pi 1 | ee eee jaa (B) which expresses Raoult’s law. We see, therefore, that Dr. Tinker’s theory, when applied to the case of a dilute solution, gives a law for the lowering of the partial pressure of the solvent quite different from Raoult’s law. In his paper, however, Dr. Tinker deduces Raoult’s law from his theory. This deduction involves an obvious mathematical error. On pages 433 and 434 he endeavours to prove that for a dilute solution 11! 3s N P 71 N-+n Considering the imaginary case ofa litre of a decinormal aqueous solution, for which V.—6, +¢€=10(V,—0,), Dr. Tinker states :— ‘We have ae N=55°5 us B Vos re near TA, rox maven Af Vie )= 60 “PPNOM and N+” = Boone = = A N I The error in counting 7 = N is thus = on unity, or less than 2 per cent.” mI te & Dr. Tinker here overlooks the fact that deviation from the laws of the ideal dilute solution depends upon the deviation of (7,—7y')/m,' from the value n/N, not on the deviation of 7 '/7, from unity. In the above imaginary case, we have 7—7, _10n, y >] 71! N so that the deviation from Raoult’s law, instead of being 2 per cent. is 900 per cent. 524 Dr. S. A. Shorter on the Kinetic A similar error occurs in the deduction of the expression for the osmotic pressure of a dilute solution. Dr. Tinker’s general expression is P—ipP?= cE ice 1 Ne (t= = Pika \ Vid Gy N N Vi-}, Be | from which he deduces, for a dilute solution, the expression PY,—RT~ +Q, eT] which is erroneous, since the logarithm, when n/N is small, is equal, not to n/N, but to nN (pF); N\ V;-8 7’ so that in the imaginary case considered above, the value of the osmotic pressure given by Dr. Tinker’s theory is 10 times that given by the “Gas law.” An error in the manipulation of small quantities also occurs in the case of the equation ne=ni{(V,;—6;) = (V,—b,)} — (N+7) (V,—Vy’). fe [7} considering the application of this equation to the case of a dilute solution, Dr. Tinker states :— ‘‘In the case of dilute solutions (in which V,’=V,) equation [7] indicates that the total volume change ne=n4 (Vi b:)—(Vo—05) te There will thus be an expansion or contraction on mixing according as (V,—D,) is greater or less than (V,—6,).” Now equation [7] defines mathematically the quantity V,’ in terms of the volume change. the concentration of the solution, and the molecular ‘free spaces” of the pure components. From it we see that as n/N is decreased V,— Vj becomes a small quantity of the first order, and that instead of equation (C) holding, we have (N+2)(Vi-Vi') e—1 (Vi —5;) er (V,—}») $=Lt _riIN=0 The falsity of equation (C) is evident if we consider the application of it, first to a dilute solution of a liquid A in a liquid B, and then to a dilute solution of Bin A. It follows at once that if one solution is formed with expansion, the other is formed with contraction, the volume changes per erm.-mol. of solute being arithmetically equal in the two Theory of the Ideal Dilute Solution. 525 cases. It need hardly be stated that ample experimental evidence exists to disprove this deduction. ‘Thus dilute solutions of both alcohol in water, and water in alcohol, are formed with contraction of total volume. In conclusion a few remarks on this important question of the kinetic explanation of Raoult’s law will not be out of place. The fundamental difficulty is to reconcile the varied volume relationships shown by different ideal solutions with the uniformity exhibited in their compliance with Raoult’s law. Thus in the case of a dilute solution of alcohol in water con- taining one molecule of alcohol in a thousand molecules of water, the number of water molecules per unit volume is reduced *3 per cent. by the addition of the alcohol, while the corresponding reduction in the case of a similar solution of water in alcohol is only about -023 per cent. Yet in both cases the reduction of the partial vapour pressure of the solventis‘l per cent. Dr. Tinker’s theory, in its application to the ideal dilute solution, amounts to the statement that this discrepancy between the proportional reduction in the volume concentration of the solvent and the reduction in the molar fraction, disappears if instead of considering the total volumes of solvent and solute, we consider the “ free space ”’ ineach. Itis evident from equations (A) and (B) that this explanation necessitates the universal validity of the relation* eS (Vi;—6,)— (V2 — bg), which, as we have seen, cannot be generally true. The failure of Dr. Tinker’s theory is due to the fact that it takes no account of the change in the intermolecular forces produced by the addition of the solute. The fulfilment of Raoult’s law is due to the combined action of two separate fastors (1) the diminution of the volume concentration of the solvent molecules, (2) the alteration of the intermolecular forces acting on the solvent molecules. Both these factors vary very much in different solutions, but must somehow be mutually adjusted so as to produce the fulfilment of Raoult’s law. I hope, in a future communication, to show how this exact adjustment is brought about. lam 9 High Kilburn, York. Yours faithfully, Sept. 11, 1917. S. A. SHORTER. * Dr. Tinker does not appear to have noticed that this relation (p.435 of his paper) and the assumption of the possibility of a relation such as (V2—b, Fe)=10(V1—8,) (p. 434) are inconsistent with each other. Fal. Mag. 8; 6. Vol. 34, No. 204. Dec. 1917. . 2-P itl [ 526 J To the Editors of the Philosophical Magazine. The University, Birmi 5 GENTLEMEN ,— oF Senienved ioe 1 THANK you for your courtesy in allowing me to reply to Dr. Shorter’s criticism of my paper. Dr. Shorter seems to imply that I attempted to give a rigid mathematical deduction of Raoult’s law from my general theory, whereas all I did was to indicate certain conditions under which the law should hold good. These conditions were explicitly stated as being that the addition of the solute shall cause no change in the molecular volume of the solvent (p. 433), and that the solution shall have no heat of dilution (p. 437). Dr. Shorter overlooks these limi- tations which I was careful to impose, and devotes undue attention to the obviously extreme and indefensible numerical illustration given on p. 434. I stated in the paper that this illustration was extreme: unfortunately it was too extreme to stand the general application which Dr. Shorter gives to it. The trend of Dr. Shorter’s criticism and development of my theory is scarcely valid since at the outset of his argument he deliberately omits as of no account the heat of dilution, which I myself was at pains to include, knowing that in many cases it is by no means negligible. This omission of Dr. Shorter’s accounts for some ef his misunderstandings of my paper. Had Dr. Shorter considered it more fully he would have discovered that the heat of dilution is actually a measure of the self-same vital factor which he says I have omitted, viz. the effect on the vapour pressure of a change in the intermolecular forces acting on the solvent molecules. As is well known, the Dieterici work factor A of my paper is determined almost entirely by the magnitude of these forces, so that the heat of dilution, which is equal to the change which A undergoes when the solution is formed (pp. 435-6 et seq.), is also determined by the change in the intermolecular forces caused by the addition of the solute. Under these circum- stances Dr. Shorter is scarcely entitled to state categorically that the theory breaks down because of the supposed omission to consider the effect on the vapour pressure of changes in the intrinsic pressure. The omission is in Dr. Shorter’s development of my general theory, and not in mine. The general expression for the lowering in the vapour pressure to which my treatment leads is Kinetic Theory of the Ideal Dilute Solution. 527 = Q Q — wl $-) eRT 4+(eRT—1), {vide equation [| 10]}* and not Dr. Shorter’s similar but incomplete expression (A) in which the heat of dilution (Q) is left out of account. But it is evident that in cases where Raoult’s law is obeyed, and where the heat of dilution is very small, as at very great dilutions, the theory leads to the result that the factor es in the above equation must be practically equal to Ps ee unity. When this is true, the expansion (e) in the combined volumes of the solvent and the solute, due to mixing one molecule of the solute with a large volume of the solvent, would be given by the expression = (Vs ~ bi) sae Vo Ba). But this latter expression only holds good for infinitely dilute solutions, or in the case of stronger solutions which both obey Raoult’s law and have no heat of dilution. As minor points I may mention further that in discussing the volume changes undergone during the mixing of two liquids, Dr. Shorter seems to have misunderstood my defi- nition of the quantity e, which is the excess of the volume of the solution over the sum of the volumes of the two pure components. Moreover, I stated at the outset of the paper that I only dealt with solutions in which it is commonly accepted that there is no molecular association, and not with such solutions as alcohol in water, with which Dr. Shorter illustrates his discussion. lam, Yours faithfully, FRANK TINKER. * This expression can be brought into a form which contains quantities which are all directly measurable by combining it with the relationship established between the free space and the coefficient of compressibility on p. 447 of my paper. The equation then becomes / > Q Q Beet % rr ae +(eRT -1), Pu N 2 Ree where f; and £ are the coefficients of compressibility of the pure solvent and solute respectively. Sinilarly the expression giving the expansion on mixing one molecule of the solute with a large volume of the solvent becomes e=RT(6,—f2). f 528 | LI. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from p. 460. ] June 6th, 1917.—Dr. Alfred Harker, F.R.S., President, in the Chair. — following communications were read :— 1. ‘On the Geology of the Old Radnor District, with special reference to an Algal Development in the Woolhope Limestone.’ By Edmund Johnston Garwood, 8c.D., F.R.S., F.G.S., and - Hdith Goodyear, B.Sc. The district comprises an inher of Archean grits and Woolhope Limestone forming an elongated dome bounded by Wenlock Shale. It was regarded by Murchison and the Geological Survey as con- sisting of Mayhill Sandstone succeeded conformably by Woolhope Limestone, and they attributed the unfossiliferous character of the sandstone and the abnormal facies of the limestone to alteration by igneous intrusions. Dr. Callaway, in 1900, first suggested that the so-called ‘Mayhill Sandstone’ was of Archean age, and re- corded an unconformity at the base of the limestone. The authors confirm Callaway’s views, and give evidence for correlating these Archean rocks with Prof. Lapworth’s ‘ Bayston Group’ of the Longmyndian. The unconformable relation of the limestone to the Archzean is established in several portions of the district ; while a study of the trilobite and brachiopod fauna of the limestone and included shale confirms the Wenlock age of the deposit. ‘The most interesting fact brought out by a study of the limestone is the important part played in its formation by the calcareous alga Solenopora (of which a new species is described), the deposit constituting by far the most striking development of algal lime- stone yet recorded from British rocks. The limestone represents a reet-facies of the normal Woolhope Limestone, being largely com- posed of bryozoa and calcareous alge. Corals, although present, play only a subordinate part. The reef appears to have grown round a subsiding peninsula of Archzan rocks, which evidently then formed the south-western continuation of the Longmynd range. The same reef-facies is also found to occur at Nash Scar, 3 miles away to the north-east, where it rests on the Upper Llandovery Sandstone. The sudden change to the normal type of Woolhope Limestone at Corton, near Presteign, appears to mark the northern limit of this lagoon phase. The paper concludes with an account of the movements that have taken place in the district, to which its general Caledonian trend is due. 2. ‘Correlation of Jurassic Chronology.’ By 8.8. Buckman F.G.S8. 529 INDEX ro VOL. XXXIV. ATR-COAL gas flame, on the origin of the line spectrum emitted hy iron vapour in the, 221. Airey (Dr. J. R.) on the numerical calculation of the roots of the Bessel function J,,(2:) and its first derivate Jn'(x), 189. Aluminium, on the absorption of X rays in, 168. Anderson (Prof. A.) on the foco- metry of lens-combinations, 76; On some properties of the nul point of thin axial pencils of light refracted through a symmetrical optical system, 174. Ashworth (Dr. J. R.) on the ap- plication of van der Waals’ equation of state to magnetism, os. Astronomical consequences of the electrical theory of matter, on, 81, 163, 321, 517. Atmosphere, on periodic convection currents in the, 112, 449. Atomic frequency and atomic num- ber, on, 478. Audibility factor of a shunted tele- phone, on the, 184. - Barkla (Prof. C. G.) on the absorp- tion and scattering of X-rays and the characteristic radiations of J series, 270. Barton (Prof. I. H.) on vibrations under variable couplings, 246. Bateman (Dr. H.) on some funda- mental concepts of electrical theory, 405. Batteries, on high-potential, for supplying small currents, 327. Bazzoni (Dr. C. B.) on the limiting frequency in the spectra of helium, hydrogen, and mercury in the extreme ultra-violet, 285. Berkeley (the Earl of), on osmotic equilibrium, 33, 459. Bessel function Jn(x) and its first derivate, on. the roots of the, 189. Biedermann (H. A.) on the energy in the electromagnetic field, 142. Books, new:—W.H.& L. W. Brage’s X-Rays and Crystal Structure, 1651. Browning (Miss H. M.) on vibra- tions under variable couplings, 246. Bruins (Dr. EH.) on the application of van der Waals’ equation of state to magnetism, 380. Bullet, on the motion of a, 382. Burton (T. H.) on the microscopic material of the Bunter pebble beds, 460. Chapman (Dr. 8.) on the partial separation by thermal diffusion of gases of equal molecular weight, 146. Chaundy (T.) on a method of line- coordinates for investigating the aberrations of a symmetrical optical system, 496. Chemical affinity, on the nature of, _in the combustion of organic com- pounds, 66. i 530 INDEX. Collodion films, on the colours dif- fusely reflected from, spread on metal surfaces, 423. Coloured flames of high luminosity, on the production of, 248. Combustion of organic compounds, on the nature of chemical affinity in the, 66. Convection currents, on periodic, in the atmosphere, 112, 449. Coolidge tube, on ‘the penetrating power of the X radiation from a, 158. Correlation coefficient, on the varia- tion of the multiple, 205. Coupled vibrations, on, 246, 510. Currents, on high-potential batteries for supplying small, 527. Dey (A.) on the maintenance of vibrations by a periodic field of force, 129. . Diffusion, on the partial separation by thermal, of gases of equal molecular weight, 146. Dixey (F.) on the cuarboniferous limestone series of the South Wales coalfield, 382. e, N, and related constants, on a new determination of, 1. Eddington (Prof. A. 8.) on astro- nomical consequences of the elec- trical theory of matter, 163, 321. Electrical conductivity of mica, on the temperature variation of the, - —— field, on the distribution of the active deposit of radium in an, 429. theory of matter, on astro- nomical consequences of the, 81, 163, 321, 517; on some funda- mental concepts of the, 405. Electrodynamic field, on the flux of energy in the, 385. Electromagnetic field, on the energy in the, 142. Electronic charge of a gram-mole- cule, on a new determination of the, l. frequency and atomic number, on, 488. Electrons, on the motion of, through gases, 3d. Energy, on the flux of, in the electrodynamic field, 385; in the electromagnetic field, 142, Evaporation from a circular water surface, on, 308. Ferguson (Dr. A.) on evaporation from a circular water surface, 308. Field of force, on the maintenance of vibrations by a periodic, 129. Flames, on the production of col- oured, of high luminosity, 248. Flicker photometer speed, on hue difference and, 99. Focometry of lens-combinations, on the, 76. Garwood (Dr. E. J.) on the geology of the Old Radnor district, 528. Gas atoms, on the frequency of the shortest vibrations emitted by, 285. Gases, on the motion of ions and electrons through, 33; on the partial separation of thermal dif- fusion of, 146. ji Geological Society, proceedings of the, 151, 382, 460, 528. Goodyear (Miss E.) on the geology of the Old Radnor district, 528. Harker (Dr. A.) on igneous action in Britain, 151. Helium, on the limiting frequency in the spectrum of, in the extreme ultra-violet, 286. Hemsalech (G. A.) on the origin of the line spectrum emitted by iron vapour in the explosion region of the air-coal gas flame, 22]; on the production of coloured flames of high luminosity, 243. Horton (Prof. F.) on high potential batteries for supplying small cur- rents, 827 ; on the application of thermionic currents to the theory of ionization by collision, 461. Hue ditference and flicker photo- meter speed, on, 99. Hydrogen, on the limiting frequency in the spectrum of, in the extreme ultra-violet, 285. Inertia, on electrical high-speed, 81, 163, 321. fonization by collision, on the ap- plication of thermionic currents to the study of, 461. Ionizing potential of sodium vapour, on the, 176. Ions, on the motiod of, through gases, 3d. INDEX. Tron vapour, on the origin of the line spectrum emitted by, 221. Isserlis (Dr. L.) on the variation of ‘the multiple correlation coeffi- cient in samples drawn from an infinite population with normal distribution, 205. Ives (Dr. H. E.) on hue difference and flicker photometer speed, 99. J series, on the characteristic radia- tions of, 270. Jeffreys (H.) on periodic convection currents in the atmosphere, 112, 449, Kinetic theory of solutions, on the, 158, 521. Lead, on the absorption of X rays in, 158. Lens-combinations, on the focometry of, 76. Light, on the nul point of thin axial pencils of, 174. Line-coordinates, on a method of, for investigating the aberrations of a symmetrical optical system, 496. Liquid, on the pressure developed in a, during the collapse of a spherical cavity, 94. Livens (G. H.) on the flux of energy in the electrodynamice field, 385. Lodge (Sir O.) on astronomical consequences of the. electrical theory of matter, 81, 517. Magnetism, on the application of van der Waals’ equation to, 380. Mercury, on the motion of the peri- helion of, 81, 163, 321, 517. -——, on the limiting frequency in the spectrum of, in the extreme ultra-violet, 285. Metal surfaces, on the colours dif- fusely reflected from collodion films spread on, 423. Mica, on the temperature variation of the electrical conductivity of, 195, Millikan (Prof. R. A.) on a new determination of e, N, and related constants, 1. Multiple correlation coefficient, on the variation of the, 205. Nul point of thin axial pencils of _ light, on some properties of the, 174, | Okano (S.) on the ionizing poten- tial of sodium vapour, 177, 531 Optical system, on the nul point of light refracted through a, 174; on a method of line-coordinates for investigating the aberrations of a, 496. Organic compounds, on the nature of chemical affinity in the com- bustion of, 66. Osmotic equilibrium, on the theory of, 31, 459. Pendulums, on double-cord and cord- and-lath, 246. Perihelia of certain planets, on the motion of the, 81, 168, 321, 517. Periodic law, on the curves of the, Photometry, notes on colour, 99. Physical constants, on a new deter- mination of some, 1. Plummer (Prof. H.C.) on the action of coupled circuits and mechanical analogies, 510. Poisson’s equation, on the failure of, for certain volume distributions, 138. van der Pol (Dr. B.) on the audibility factor of a shunted telephone, 184. Poole (H. H.) on the temperature variation of the electrical conduc- tivity of mica, 195. Population, on the variation of the multiple correlation coefficient in samples drawn from an infinite, with normal distribution, 205. Prasad (Prof. G.) on the failure of Poisson’s equation for certain volume distributions, 138. Prescott (J.) on the motion of a spinning projectile, 332. Projectile, on the motion of a spin- ning, 332. Radium, on the distribution of the active deposit of, in an electric field, 429. Raman (Prof. C. V.) on the main- tenance of vibrations by a periodic field of force, 129. Ratner (8.) on the distribution of the active deposit of radium in an electric field, 429. Rayleigh (Lord) on the pressure developed in a liquid during the collapse of a spherical cavity, 94; on the colours diffusely reflected from some collodion films spread on metal surfaces, 423. Te \ ae INDEX. Richardson (Prof. O. W.) on the limiting frequency in the spectra of helium, hydrogen, and mercury in the extreme ultra-violet, 285. Rutherford (Sir E.) on the pene- trating power of the X radiation from a Coolidge tube, 153. Shorter (Dr. 8. A.) on osmotic equi- librium, 31; on the kinetic theory of the ideal dilute solution, 521. Sibly (Dr. T. F.) on the carboni- ferous limestone series of the South Wales coalfield, 382. Sodium vapour, on the ionizing potential of, 176. Solutions, on the kinetic theory of dilute, 521. Specific heat, on the thermodynamic cycles with variable, 168. Spectra of helium, hydrogen, and mercury in the extreme ultra- violet, on the, 285. Spectrum, on the origin of the line, emitted by iron vapour in the air-coal gas Hame, 221. Spherical cavity, on the pressure developed in a liquid during the collapse of a, 94. Spinning projectile, on the motion of a, 332. Steel wire, on the maintenance of vibrations in a, by. a periodic field of force, 129. Telephone, on the alii by factor. of a shunted, 184. Thermal diffusion, on the partial separation of gases by, 146. efficiency, on an expression for, 168. Thermionic currents, on the applica- tion of, to the study of ionization by collision, 461. Thermodynamic cycles with variable specific heat, on, 168. Thomas (Miss N.) on evaporation from a circular water surface, 308. Thornton (Prof. W. M.) on the nature of chemical affinity in the combustion of organic compounds, 66 ; on the curves of the periodic law, 70. Tinker (Dr. F.).on the kinetic theory of dilute solutions, 521. Ultra - violet spectra of helium, hydrogen, and mercury, on the, 285. Vibrations, on the maintenance of, by a periodic field of force, 129 ; on, under variable couplings, 246, 510. Volume distributions, on the failure of Poisson’s equation for certain, 138. van der Waals’ equation, on the application of, to magnetism, 380. Walker (W. J.) on thermodynamic cycles with variable specific heat, 168. Water surface, on evaporation from a circular, 308. Wellisch (HK. M.) on the motion of ions and electrons through gases, 33. White (Miss M. P.) on the absorp- tion and scattering of X-rays and the characteristic radiations of J series, 270. Wireless signals, on the measure- ment of the strength of, 184. X radiation, on the penetrating power of the, from a Coolidge tube, 155. X rays, on the absorption and scat- tering of, 270. Wood (Prof. R. W.) on the ionizing potential of sodium vapour, 177. END OF THE THIRTY-FOURTH VOLUME, Printed by Taytor and Francis, Red Lion Court, Fleet Street. ia ee eh ke “a G+b WAL Bi, nae EY eal ot, ad LS = seit ace? noel, —— | le 3 9088 Ml . ‘l ———— SS