a ke HPN th a rit r rf H Heth Ly iy Ta! Ying) oH 144 Upeat ¥ = Selig : ese 4 2 SS 3 3 = - 4 it yr e . Shee eee ee Ve Bh aoa! BE are ins fee phen ob Peiihy omits er Me 9 hs cer eer endpete pipte = tsi eee » Perel FON? 8 hme ray predend BO ee Aeteg ie Th bat eit } atthe aay bate) paryeiee ay i nety? wets bby’ bepors ey *4 phibe beh tery he DCEN when PO pHs heh 1} tpipbe baby byl payeute + se ‘Sead baad dod r4 +r eye > f ’ aed ov 4 ‘tt ody bs it hp boinee +4 \ tik a mye NE THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. CONDUCTED BY SIR OLIVER JOSEPH LODGE, D.Sc., LL.D., F.R.S. SIR JOSEPH JOHN THOMSON, O.M., M.A., Sc.D., LL.D., F.R.S. JOHN JOLY, MA. D:Sc., B.R:S., F-G:S. RICHARD TAUNTON FRANCIS eae AND WILLIAM FRANCIS, F.I.8. - “Nee aranearum sane textus ideo melior quia ex Sse fila gignunt, nec noster vilior quia ex alienis libamus ut apes.” Jost. Lips. Polit. lib.i.cap.1 Not. Pp I VOL. XUI.—SIXTH SERIES. JANUARY—JUNIH 1921. ERO - . LONDON: TAYLOR AND KRANCIS, RED LION COURT, FLEWT STREET, SOLD BY SMITH AND SON, GLASGOW ;—HIODGES, FIGGIS, AND CO., DUBLIN — AND VEUVE J. BOYVHAU, PARIS, “‘Meditatiunis est perscrutari occulta; contemplationis est admirari perspicua.... Admiratio generat queestionem, queestio investigationem, investigatio inventionem.”— Hugo de S. Victore. ——“ Cur spirent venti, cur terra dehiscat, Cur mare turgescat, pelago cur tantus amaror, Cur caput obscura Phoebus ferrugine condat, Quid toties diros cogat flagrare cometas, Quid pariat nubes, veniant cur fulmina ccelo, Quo micet igne Iris, superos quis conciat orbes Tam vario motu.” J. B. Pinelli ad Mazonium, cr ALERE Ss FLAMMAM, CONTENTS OF VOL. XLI. (SIXTH SERLES), NUMBER CCXLI.—JANUARY 1921. Mr. W. N. Bond on the Properties of Plastic Crystals of Pommoninim Nitrate. Celates E. & Il.) Sow oe te Mr. D. L. Hammick on Surface Energy, Latent Heat, and CURDS STS MUL e es re le RN aorta da ea Dr. N. W. McLachlan on the Effective Inductance, Effective Resistance, and Self-Capacity of Magneto Windings Mr. E. H. Darnley on the Transverse Vibrations of Beams aud the Whirling of Shafts supported at Intermediate sO S peat Att ste ye ticnreal aerials Ae agessharerdle Wire sack es Prof. F. Slate on Electronic Energy and Relativity ........ DE. a Bateman on an Electromagnetic Theory of Radiation. Dr. H. Stanley Allen on the Angular Momentum and some Related Properties CHUNG lew Tecmo “Aca bu woes oe Mr. L. Simons on the Beta-Ray Emission from Thin Films of the Elements exposed to Rontgen Rays. (Plate I11.)s. Prot. Alex. M°Aulay on the Inertial Frame given by a FR MET DOES PACE Le Wiad Melee oars ese ase oe Sir George Greenhill onthe Newton-Hinstein Planetary Orbit. Prof. F. Y. Edgeworth on the Genesis of the Law of Error. Proceedings of ‘the Geological Society :— Mr. G. W. Lamplugh on some Features of the Pleistocene Glaciationsol Knaland = teen a ees. oe. Oe Mr. H. Crunden Sargent on the Lower Carboniferous Cheniohioruations on Derbysinive.. 20.4 o0... s+. so: NUMBER CCXLIT.—FEBRUARY. Miss D. Wrinch on an Asymptotic Formula for the Hyper- SECM AO TAU IOLY GyVN (2) 8 Grain SG ee ee eae Miss D. Wrinch on a Generalized Hypergeometric Function NUMA OMEMIMCLOES AWM etre acai te ee eS ke Dr. R. A. Houstoun and Miss M. A. Dunlop: A Statistical Survey of the Colour Vision of 1000 Students .......... Page iL All 33 1V CONTENTS OF VOL. XLI.—SIXTH SERIES. Dr. J. R. Airey on Bessel Functions of small Fractional Order and their application to problems of Elastic Stability . Mr. H. Carrington on the Determination of Values of Young’s Modulus and Poisscn’s Ratio by the Method of Flexures. Prof. Nibal Karan Sethi on Talbot’s Bands and the Colour- Sequence in the Spectrum =2..2.7........+.. == Prof. W. B. Morton and Mr. T. C. Tobin on Times of Descent under Grayity, suggested by a Proposition of Galileo’s Dr. J. S$. G. Thomas on the Thermal Effect produced by a slow Current of Air flowing past a Series of fine heated Platinum Wires, and its Application to the Construction of Hot-Wire Anemometers of great Sensitivity, especially ap- plicable to the Investigation of Slow Rates of Flow of Gases. Miss L. M. Swain on the Period of Vibration of the Gravest Mode of a Thin Rod, in the form of a Truncated Wedge, when im Rotation about its Base ..:........=. see eee Dr. Megh Nad Saha on the Problems of Temperature Ra- diation jot Gases... (baper ©.) 23) ee.) an, Prof. A. H. Compton on Possible Magnetic Polarity of Free Klectnons: . 2. .... ..... eee 809 Mr. H. Fricke and Prof. T. Lyman on the Spectrum of Helium in the tixtreme Wiltra- Violet. 0.7... /... 4.5 814 Mr. J. H. J. Poole on the possibility of separating Mercury intoits Isotopic Horms by Centrilugine. (222-6. =m 818 Mr. F. W. Hill and Dr. J. B. Jeffery on the Gravitational Field of a Particle on Hinstein’s Theory................ 823 Dr. Balth. van der Pol on Systems with ‘‘ Propagated Coupling”; with a Note by A. W. Porter and R. E. Gibbs. 826 Proceedings of the Geological Society :— Mr. H. 8. Warren on a Natural ‘“ Eolith” Factory beneath the hanet Sand... cose. eo. ees > eee 828 CONTENTS OF VOL. XLI.——SIXTH SERIES, NUMBER CCXLVI.—JUNE. Mr. R. T. Lattey on the Dielectric Constants of Electrolytic SP SI RTICTNE NER ego oe ek ne rR eareeCin e Mr. H. Carrington on the Moduli of Rigidity for Spruce. ree MIENG) Menten tars oe Sapeens hee at ie ies an we ee ys SS Prof. C. V. Raman and Prof. Bhabonath Banerji on the Colours of Mixed Plates.—Part II. .................. Mr. L. St. C. Broughall on some Dimensions of the Atom .. Mr. Wilson Taylor on the Coalescence of Liquid Spheres— Pelee miat Diameters) e250. o ee wee ic as Ole whe see as , Mr. T. Smith on the Accuracy of the Internally Focussing " Mielescope ii TacheOmetry. coisa els eb ame wae es 6 Mr. A. H. Davis on the Heat Loss by Convection from Wires in a Stream of Air, and its Relation to the Mechanical PCCM RES ie a ecard tics Gla apee oA) adel awa as abated Dr. C. Davison on the Annual and Diurnal Periodicity of MAKES IM OAPAN cy oak 4s eM eee vee es ele es Mr. A. G. Shenstone on the Effect of an Electric Current on puemiznonosblectric Witect <0. asic. ees ed eek os Prof. Frank Horton and Miss A.C. Davies on Critical Electron Velocities for the Production of Luminosity in Atmospheric a OMA Oe oe Sides ary ahd sbi O's Maa log eth we ol Sir Oliver Lodge on Hther, Light, and Matter ............ Sir Oliver Lodge on the Einstein Spectral Shift .......... Notices respecting New Books :— Transactions of the Bose Research Institute, Calcutta, Bo see ay Sc7 Te en ee eS Busy aa tae Sooo e's Mr. lL. Southerns’s An Outline of Physics............ Major P. A. MacMahon’s Introduction to Combinatory BAUM GISi tors sr aT lope wmie ce ies on Ses es cute eS ot Profs. EK. T. Whittaker and G. N. Watson’s Modern PACA VASE Ree ican e sian a re vanies Weutram yA ik Olas 2, Miss Ida Freund’s The Experimental Basis of Chemistry. Proceedings of the Geological Society :— . Dr. C.G. Knott on Earthquake Waves and the Elasticity imine Miah even. Neon tee Wig a tees paMR tae wore Wied wel v5 we «a Mr. Olaf Holtedahl on the Scandinavian Mountain TRAD) OEMs Games Misa © 6c 0 CCI een errr Dr. E. Greenly on an Aolian Pleistocene Deposit at Slovecione(Somerset)heraac eine oss e hs ak ee Dr. T. O. Bosworth on the Structure and Stratigraphy of the Tertiary Deposits in North-Western Peru; and on the Geology of the Quaternary Period on a Part of tinoemacine @OastOh PeCrUa).. och. acc csv cece ss 949, vil Page 947 948 948 950 PLATES. I. & II. Lllustrative of Mr. W. N. Bond’s Paper on the Properties of Plastic Crystals of Ammonium Nitrate. IIT. Illustrative of Mr. L. Simons’s Paper on the Beta-Ray Emission from Thin Films of the Elements exposed to Rontgen Rays. IV. & V. Illustrative of Profs. C. V. Raman and Bhabonath Banerji’s Paper on the Colours of Mixed Plates. VI. Illustrative of Prof. H. Nagaoka’s Paper on the Magnetic Field of Circular Currents. VII. Illustrative of Mr. R. V. Southwell’s Paper on a Graphical Method for determining the Frequencies of Lateral Vibration, or Whirling Speeds, for a Rod of Non-Uniform Cross-Section. VIII. Illustrative of Mr. G. Stead’s Paper on the Design of Soft Thermionic Valves. IX. Illustrative of Mr. H. Carrington’s Paper on the Moduli of Rigidity for Spruce. THE LONDON, KDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES.) _ —— JANUARY 1931. > I. The Properties of Plastic Crystals of Ammonium Nitrate. by WON. Bonn, Se., A.h.C.Sc., A.lnst.P* Plates E & I7) | aaa paper consists of a description of some experiments performed with crystals of Ammonium Nitrate that exhibit plasticity. The paper may be divided into three parts. In the first, an account is given of experiments on the bending of these crystals; in the second, an account of observations on bent crystals with polarized light. The third consists of a discussion of the results, and of the internal’ _ changes accompanying bending. Part I. The specimens originally used were fine need|e-like crystals, produced by Messrs. Brunner, Mond and Company. — Later some Ammonium Nitrate of French manufacture, trom Messrs. Baird and Tatlock, was used for re-crystallizing. The long needle-like crystals can be bent and twisted in the fingers (see Pl. I. figs. 1 & 2). If this be done ca:efully and not too rapidly, small loops of diameters of the order of half a centimetre or less can be made. ‘Too rapid bending * Communicated by Professor A. W. Porter, I.R.S. Phil. Mag. 5S. 6. Vol. 41. No. 241. Jan. 1921 B 2 Mr. W. N. Bond on the Properties of results in fracture at the part most strained, the erystal being thus both plastic and brittle. The thinnest needles seem to bend better when fresh from the solution they have been erystallized from. In connexion with this, it may be noted that, according to Kleinhanns*, rock salt is bent more easily when immersed in liquids that dissolve it. The following account of the crystalline forms of Am- monium Nitrate is based on that of Grotht. At ordinary temperatures, it crystallizes from an aqueous solution as rhombohedral and pseudo-tetragonal crystals. These change on warming, as first noticed by Lehmann f, to a second form, which is apparently also rhombic ; then to an optically uniaxal form, and finally toa single-refracting fourth form. This fourth form is stable at 124°C. ; the third from that temperature to 83°C.; the second from 83° C. to 32° C., and the first from 32° C. downwards. Wallerant, however, says that the third form is not, as Lehmann supposed, trigonal, but tetragonal and optically positive. This passes into the more strongly double-refract- ing second form mentioned above, of which the prisms have their length parallel to the second axis of the tetragonal crystals. By adding traces of potassium nitrate to the molten sample, the 32° transition point is lowered, so that the second form still remains stable at ordinary temperatures. We are then able to produce twinning lamelle. The transi- tion from the third (tetragonal) form to the less dense second form cannot be produced if the former is under pressure; the change to the first (rhombic) form (which has the greater density) takes place instead. This first modification is stable at normal temperature ; according to Wallerant, if cooled to —16°C. it undergoes a change to another form, less dense, uniaxal, optically posi- tive, with double refraction somewhat weaker than that of the third, tetragonal, form. Wallerant concludes that the form below —16° C. is the same as that stable from 124°C. iO Bo- CO. The tetragonal form arising from the cubic at 124°C. remains stable even at the lowest temperature if a little CsNO3 is added ; but changes into the rhombic form under pressure. At 20°C.a very slight pressure is sufficient. IE the amount of CsNQs is very slight, the crystal changes back * K. KJeinhanns, Phys. Zettschr. xv. pp. 862-3863, April lst, 1914. + Groth, Chemische Krystallographie, Zweiter Teil, p. 66. t O. Lehmann, Zezisehr, f. Krystal. 1877, 1. p. 106. Plastic Crystals of Ammonium Nitrate. 3 atter removing the pressure. (This is not the case if there is a rather gr eater proportion of CsNQO3.) The usual or “first” form consists of thin pseudo-tetra- gonul prisms without end planes (but Grossner also found a tabular form). This crystal is a strongly double- refracting biaxal erystal, with 2H=59° 30’ (Grossner), and the ‘dispersion of the axes is weak (r xe 4°54 55 2:09 ” 5°85 56 "34 ” 6:03. 1-,; 093 _,, ohaiiven | Wey 0s IEAM ie ce ae eee pley as over 32 Pe 4°13 ” 0338 ” 6°20 9 704 ” erasers Fes ee « SA OTe ae ay ‘LOGa Siretehine escent ce 2°38 fe over 409 _,, ee 6-0 x 102 Viscosity of Ice...... 1 sa77 << 1012 a4 (Deeley & Parr, 1913.) In the above table, X is what Trouton* terms the co- efficient of viscous traction, and is, according to him, three times the viscosity. It will be seen that the values of the pseudo-viscosity vary considerably with the stress. _If we assume the viscosity of ice to be as given by Deeley and Parr +, the above table shows that the rate of bending of the crystals of ammonium nitrate under stresses of the order of those employ ed in these experiments is about equal to that observed in glacier ice under similar conditions. Fig. 10 Fig. 10 shows the forces that are acting on one half of the crystal when bent by the second method. At any section of * Prof. Trouton, Coeff. of Viscous Traction, etc., Proc. R. Soc. A, vol. Ixxvil. 1906. + R. M. Deeley & P. H. Parr, Phil. Mag. xxvi. pp. 85-111, July 1913, and xxvii. pp. 183-176, Jan, 1914. Plastic Crystals of Ammonium Nitrate. tt the crystal AA, if we consider the forces acting on the right hand side, we see that the unknown frictional force T does not take any appreciable part in the bending. Further, the moment of the force N is almost directly proportional to the distance measured along the crystal, since with even con- siderable bending each half of the crystal is almost straight. (See fig. 12 0.): For a elastic bending of a uniform beam we have :— te Me ~= FP where p= radius of curvature of beam. Pp M = applied bending moment at the point, E = Young’s modulus for the beam, I = moment of inertia of a transverse sec- tion (constant). Hor simple viscous flow, the equation for calculating bending may be written ee dt p Ne ie Value of Ke or Sy Thus for moderate deflexions, the curved shapes for elastic and simple viscous bending are of approximately the same form, being identical for very small amounts of bending. It should be noticed th 1at,.as the erystal bends, its ends slide over the pegs inwards. Thus the parts now forming the ends of the curve have not been subject to bending for the full time. ‘’hey should thus be less bent than the corre- sponding elastic curve On comparing these (see figs. 12a and 12 6), we find that in either case the part concerned is sensibly straight, so that no appreciable alteration is caused by this ‘sliding over. 12 Mr. W. N. Bond on the Properties of This fig. '2 shows for comparison the forms of the following specimens when subjected to a central load: (a) the elastic curves (theoretical), (6) the curved forms of some bent crystals, (c) the curve of a celluloid strip, and (d) some curves obtained with strips of lead. Fie. 12: a b € ad It will be seen that the celluloid is more curved at the central parts than the corresponding elastic curves. This is due, partly at least, to there being a plastic bend of the celluloid at the central portion (as was noticed on removing the load). In the case of the lead, two of the curves shown have almost equal depression. Of these, the one with the sharper bend was bent more rapidly than the less pointed one (i.e. a larger load was applied). The specimens were made from the same sheet and of the same dimensions. In the case of more rapid bending where the stress was every- where greater than in the slower case, the bending is more pronounced where the bending moment is greatest. Thus the rate of bending increases more rapidly than in direct proportion to the stress. Thisis in agreement with the results obtained by Andrade*. If we do n»t consider his initial * Andrade, “On the Viscous flow of Metals,” Proc. R. Soc. A, vol. Ixxxiv. 1910; & vol. xc. 1914. Plastic Crystals of Ammonium Nitrate. 13 i Lee : “8” flow, he showed that for lead 1 (or g in his notation) increases very rapidly as the stress reaches a certain value (see fig. 11). : Since the curves of the beut crystals are more curved at the centre than the corresponding elastic curves, which approximate to the simple viscous curves, it is evident that also in the case of the crystals, the rate of bending increases more rapidly than in direct proportion to the stress or bending moment. This agrees with the results of the tests on bending crystals, described earlier in the paper. When bending has taken place, the inner edge of the erystal seems to have suffered 2 compression, and the outer an extension, near the bend (just as in the case of elastic bending), as observation indicates that there is no longitudinal slip detectable at the ends. Part IJ. Let us now turn to the optical properties of the beat crys- tals. Thin crystals were bent into loops by the fingers. These loops were then ground on a smooth stone or fine ground glass in oil (since water dissolves the crystal). After both faces of the loop had been ground, it was examined between crossed nicols. If no interference effects were noticed, it was bent toa smaller radius. Then, if thin enough and ‘smoothly enough ground, it was ed. in Canada balsam dissolved in xy Tol. Straight crystals of ammonium nitrate show straight extinction, when examined between crossed nicols. Then, if the crystals were bent, we might expect extinction hes consisting of four dark fearisyverse, bands, equally a round the circular loop of the crystal. BietetoeCel I shows the result that was obtained. It will be seen that the transverse dark bands are not radial, but slope in this case at about 45° to the radius. If the crystal i is of the full orthorhombic syminetry, there is no reason to expect the slope to bein one sense more than in the other. And it is found that in some slides the sense chan ges. The sense is probably determined by the exact method of bending, though in what way has not been ascertained. Local irregularities may produce sudden changes in sense, as shown in fig. 14 (Pl. IT.), which shows three positions of the same slide, obtained by a small rotation between each figure and the Hess The sense of the line changes, and then ie returns to its original direction. 14 Mr. W. N. Bond on the Properties o) The angle of slope is, however, as nearly as can be estimated, of constant value. When the sense is changing, an intermediate portion may show a double dark extinction- line or cross. Or the sense may change half-way radially across the crystal, causing a “ V ” shaped band. To a first approximation, the band is at 45° to the radius of the are formed by the crystal. It is difficult to measure this angle accurately, and it is possible that it is not quite constant. On the other hand, it will be seen, on looking at the fioures in the following table, that for all radii of curvature and widths of crystal examined, the slope is not greatly removed from 45°. TABLE II. Radius of Curva- Width of Radius Slope of Extine- ture. Crystal. Width ° tion-line in degrees, ity Zeal 81 44 30 2:0 iL 64 iy 18 9°45 60 10 1°6 6°25 53 795 2 375 55 to 64 10 2 5 50 Td 16 4-69 40 If these measurements be used to plot the slope of the gare 4 Lkerobins extinction-line against Width? 2° Connexion between them can be found. Some of the crystals were bent in sucha direction that the plane containing the optic axes was at right angles to the plane of bending. The plane containing the optic axes is parallel to the principal section of one of the nicols at all points where extinction is occurring on any one dark extinction-line AB (fig. 15). Thus the angle between this plane at any point and the radius to that. point would vary with the distance of the point from the inner edge of the curve. Let @ be the slope of the dark line as indicated in fig.15. Then the change in direction of the plane containing the optic axes on passing radially from B to © is equal to the change in direction on passing from A to C, and is thus also equal to the angle subtended by the extinction-line at the centre of Plastie C ‘ystals of Ammonium Nitrate. 15 curvature of the crystal (if « is constant for this part of the crystal). | Pie oto: a | Planes of the nicols, nl R Or 6, this change in diréction in crossing the crystal radially from B to C, is given by = tan o. (A pproximately.) Experiments were carried ont with the slide shown in fig. 13 (Pl. I.) in order to determine the direction of the plane containing the optic axes at different points. The “eyes” were observed by employing convergent polarized light and an anxiliary lens in the microscope tube which enabled the back focal plane of the objective to be viewed. Fig. l6a Fig. 16, s Pe es a) “itn 6) as 23:4 5 6 7 8 ems. a. b. shows the direction of the plane at different points. This was determined by moving the slide over some graph paper to the desired positio.s ascertained by means of a seale drawing of the slide prepared previously. A high-power 16 Mr. W. N. Bond on the Properties of objective was used throughout, in order to examine small regions only of the crystal. It will be seen that, to a first approximation at any rate, the plane containing the axes keeps tangential to the curve. That is to say, that its direction relative to the crystal is little changed by bending. Fig. 166 is a composite diagram showing the form of the dark bands exhibited by the specimen (PI. I. fig. 13) in different successive orientations, the portion shaded being irregular. (I orsome of the positions of the extinetion-lines, the corresponding direction of the planes of the nicols is indicated by a small cross.) An experiment was next undertaken to see in what way the plane changes direction as we pass across the crystal. The microscope stage was slightly rotated till the two branches of the dark brush observable in convergent polarized light united to forma cross. The reading on the stage rotation scale was then taken. ‘This was repeated for different points lying on a transverse line AA, fig. 16} (crossing the bent crystal twice). The results indicate a value of the change in direction of the plane that is in moderate agreement with the predicted results. The differences are probably due to the dark line not having a quite constant value of « at different adjacent points along the length of the crystal, or to the difficulty in making observations exactly along the desired line AA. These would both affect the predicted results. The measured results are plotted in fig. 17. The MICROSCOPE STAGE SCALE points lie along straight lines, which is to be expected, since the dark extinction-lines are straight. Also the left-hand side yields an almost horizontal line, corresponding to the dark line sloping very little on this side (see Pl. 1. fig. 13). But the slope of the extinction-line on the right-hand side, Plastic Crystals of Ammonium Nitrate. ify as deduced from these results, is about 20°. A microscope having both rotating and travelling stage was not available; so in this, as in the previous experiment, the position of the slide had to be determined by fine graph paper. The angle between the apparent optical axes (2 (2B) was measured in these and other experiments, and showed no change from the normal value that could not be accounted for by experimental error. Parr III. In considering the internal changes that accompany bending, it is clear from the optical results that the crystal- line structure is by no means destroyed, nor is it greatly altered. This result may be contrasted with those obtained by Terada* in the ease of rock salt. The crystals were bent between two parallel cylinders. whilst being heated (at tem- peratures of at least 150° C.), one axis of the crystal being parallel to that of the cylinders. When examined by the X-ray method, the crystal was found to have been changed into a number of lengths having the space lattice unaltered, with the spaces between these lengths filled by mixtures of broken crystal. When the bending was carried out at 100°C., the original structure was quite destroyed. Later work fats been Mone by Carmakf on this subject. The bending is probably plastic in its nature. It seems difficult to imagine a purely viscous flow taking place in a erystal. Unless the atoms could migrate to cause this change in shape, the space lattice would have to be distorted to an indefinite extent (in the case of a longitudinal pull, the spaces in one direction becoming longer and longer). Nor is it possible to regard the structure as a set of small crystals originally arranged at random in a matrix and later becoming alignedf. The slight change in the direction of the plane containing the optic axes, on crossing the width of the bent crys tal, corresponding to the slope ‘of the extinction- lines, might be supposed due to the effect of a residual strain, super posed on the other normal optic properties. Since the axes of the optical ellipsoid are coincident with the ea overs axes we see by symmetry that this strain would have to be o blique * T. Terada, Mathematico-Physical Soc., Tokyo, Proce. vil. pp. 290-291, May 1914. + P. Carmak, Phys. Zeits. xvii. p. 556, Nov. 15, 1916. { Andrade, loc. cit., where a full econ of the distinction between plasticity and viscosity is given. Phil. Mag. 8. 6. Vol. 41. No. 241. Jan. 1921. C 18 Mr. W. N. Bond on the Properties of to the axes of the crystal, or no rotational change of the plane containing the optic axes would result. This strain would also have to vary gradually in amount from the inner to the outer curved faces. It is, however, desirable, in the case of a crystal, to attempt to explain the results in terms of changes in the space lattice. Some internal change may take place. This seems, perhaps, more likely, since am- monium nitrate crystallizes in different forms at different temperatures. Also its molecule is fairly complex, giving a complex lattice and opportunity for internal change. The optical results would suggest that the internal change might consist of a rotation. If the space lattice is subject to slight forces, we should expect a slight deformation to result, which would disappear when the forces were removed. If the ferces were such that the crystal was bent into an are of a circle, we should have to suppose that the distance between adjacent atoms was smaller at the inner curved face than at the outer. This would appear an impossible configuration for equilibrium, when the external forces are removed. When a long crystal is bent (as in fig. 1, Pl. I.) near its centre, the ends remain almost unaftected. Thus a sharp “V” shape may be reached by the crystal. If simple slipping is assumed, in order to account for the bending shown in figs. 1, 13, and 14 (Pls. I. & H.), the slip must be supposed parallel to the length of the crystal. The change on bending would then exactly correspond to folding a book, as supposed by McConnel*. . This means that slip would be supposed to occur along the whole length in order to produce the sharp and purely local bend at the centre. This seems exceedingly unlikely; and attempts to determine whether there has been slip at the ends, by ob- serving the relative positions of marks on the surface of the crystal both before and after bending, indicate that probably the slip is not appreciable. If such slip does not occur, it is necessary to conclude that the spacing of the lattice is greater at the outside of the curve than at the inside, unless - some atoms have migrated across the crystal from the inner to the outer edge. If we assume that simple slip has not occurred, but that the lattice spacing is greater at the outer edge, we have to explain how there can be stability after the bending forces have been removed. We have also to explain the observed slight rotation of the plane containing the optic axes, as we * McConnel, Proc. R. Soc. vol. xlvili. p. 259 (1890); and vol. xlix. p. 328 (1891). Plastie Crystals of Ammonium Nitrate. 1, eross the bent crystal radially. This, it should be remem- bered, would also have to be ac counted for 1f we assumed that the plasticity was due to slip. When a erystal of ammonium nitrate is bent, probably causing the atomic spacing to be greater on the ute edge of the curve, we may suppose that a rotational change occurs in the structure of the crystal near the bent portion, causing the atoms to keep in equilibrium at their new distance apart. This rotational change would have to vary steadily in extent from the inner to the outer curved faces, as the spans would vary continuously. Since refraction apparently de- pends on the atomic structure, this rotational change would probably cause a change in the direction of the plane containing the optic axes, which would also vary steadily from the inner to the outer curved faces. If the rota- tional change were proportional to the fractional] elongation or contraction, the slope of the extinction-line would be almost independent of the curvature, as appears to be the ease (provided the bending is enough to render the line quite distinct). Such a rotation or orientation of the atomic groups would explain the very rapid bending which takes placé at first when a load is applied. Thus the hypothesis of a rotation in the atomic groups seems, on the whole, the most plausible one ; and it is in fair agreement with all the observations. On the other hand, it is still possible that some slight longitudinal slip may oceur (though attempts to detect it failed). But even if this were the case, the optical properties show, almost beyond doubt, that a rotation forms at least part of the change accom- panying the plastic bending of these crystals. In conclusion, 1 wish to thank Dr. Porter of University x College, eons for his kindness and the help that he gave me whilst I was carrying out this investigation, w hich consists of a continuation and extension of an unpublished preliminary investigation which he had made. SUMMARY. - When Ammonium Nitrate was crystallized from an aqueous ‘solution at laboratory temper atures, long needle-like crystals formed, these being often complex in cross-section. Some- times ie it or tabular crystals formed instead. It was found that both had 2 E about 59° 30’ as given by Groth ; also that r T = coefficient of cubical expansion. OK: 1 ee) or IS =) 1s s({—2a+ = il Loe 1 0a i =has{ - 247 ait x ). From (ii.) and (iii.) above, we have Any aS es bpv D Fi ry ve pune bye ue je ye 1 Op 1 oo zeae (esha) Tine oY Oe vee ae (P= pat In short, if ap is independent of volume (and hence pres- sure) we expect to find 1 == —= 2r18, e ane (iv.) 24 Mr. D. L. Hammick on Surface Energy, whereas if ap is a function of v, : Sie 6o 1 By substituting in (iv.) and (vi.) the value ses) for ys (of (ii1.)), we get : 1 Eee e oa —2, fe (vil. ) ee O00) ee and a= a Sls a ie In order to test the above conclusions, data are needed for surface energy, latent heat, s specific gravity, and compressi- bility. For the latter, Tyrer’ s values (Trans. Chem. Soe. 1914, p. 2534) for pressures between 1-2 atmospheres have been used ; his paper contains data for the evaluation of a, the coefficient of expansion. Tyrer’s results extend over the range 0° C.—boiling-point ; hence the latent heat data avail- able are limited. Those ged are due to Young unless other- wise stated. ‘The surface energy data have been taken from Jaeger (Zeit. Anorg. Chem. ci. pp. 1-314 (1917)) and from Ramsay, Shields, and Aston’s results. It is noticeable that though Jaeger’s results, obtained by the “ gas bubble” method, agree very well with the Rams say values at low emperatures, marked divergencies occur at higher tempera- tures with the more volatile liquids. In most cases therefore the two sets of results have been plotted and those values for the surface energy used that lie on the straight line (or smoothed curve) that runs through Jaeger’s low temperature and Ramsay's high temperature values. | In the following Tables, (iv.) and (vi.) are applied to some liquids and (vil.) and (vil. to others. The number of liquids available is necessarily limited by the paucity of data. In the column headed ¢ in each table is recorded either the ratio —“ =e or ae It will be seen from (iv.), d (vi.), (vil.), (wili.) that the ratio e should be equal to : if a,p=f(v) and equal to —2 if ap f(v). In the cases where d, the molecular diameter, is required, it has been galoalaned from the latent heat at the boiling-point from the relation AY A al} — Ope é a * The sign of p will clearly be negative. Latent Heat, and Compressibility. TABLES. Ether. “se | : eee 6x10" djs p ae eee oa se ae dynes X em.? cals, ec. ergsxcm2. — ? e | : —h,s a —30 108(extr.) 72°4(extr.) 28:5 a = 304 Pepa iios -,, 68d, Nie roster O 130) 202, ' y= 4-18 153 646 19:2) 0658 Ojlol) 241 436 10 1703 61-7 ws OOS, ONS | Sei 4:48 20 189 59-9 Ljalen O58.) 0) 162 2-10 4-68 Seee2 11-8 56-2 2:01 35 - 224 (extr.) 52:9 2:02 Benzene. ae 5 . BJ e f. Sisere d,s *, (. —p. a. eee Fas 54 81-95 92-2 30:6 0,485 0,107 Soy 872 20 95°65 88-4 98:4 0,469 0,120 2°83 3°90 ae 2 111-4 88:2 25:6: pe nOro2Il 0,125 2-60 417 60-1300 76:3 23:0) 2) 105580 OMB | 241 4°43 Ree 1432 73°5 BIG 70,617 OMe 227) 4-60 80. 156-5 70-0 20:4 0,653 ‘0,138 Dalla 4-73 | * Krom Young (Kaye & Laby) and from Griffiths &jMarshall (Phil. stag. [5| xlvi. p. 1 (1896). Oarbon Tetrachloride. : TT: (Gs Ne Sue 0. —p. a. —20 81-O(extr.) 83:2 old 0,426 05112 (extr.)| One O-2 91 20 105:0 74:8 40 124°5 70°3 66 Si sch 60 1475 66°3 208055297) Osta 80 172-4 62°1 ISi7 05086) 05135 IG:48 505676 ~ “0,148 100 * Calculated from As = OB d=435x 1072 cm. d 1 Paso Ge =\,s —355 —3s0 ool 3°05 2:73 ae 2rd 4:00 2°24 4:28 473 -. } Bakker (Zect, Phys. Chem, Ixxxvi, p. 129 (1914)) gives © =8372 at 107 €. 26 Carbon Bisulphide. Mr. D. L. Hammick on Surface Hnergy, i fs i " pI Pr: I’. De NGS p- 0. ae — a ay a —20 71-0 (extr.) 122-4 364. 0,381 0,112 270 ee 0 Sls 1149 336 0,413 0,114 2-56 3°62 10 875 110°1 29:2) BI 0,116 2:48 371 20 93-8 105°9 31:0) 0447 0,119 9°41 3°76 30 100°6 100-8 295 0,470 0,121 2:36 3°88 40 107-9 96-0 981 0,494 0,123 oa 4:01 50 1163 Of =) 2677 0.5208 ONG 226 4:13 * Reenault. Chloroform. i a | : Ba ? ed is B. \\8 0. 0. a. — age ec —20 72O(extr.) 1075 32:3 0,443 -0,115(extr.) ‘09 —3°85 0 85:9 953 285° -0,479 -0,121 2-92 3-95 20 101-2 896 268 -0,534 0,127 2-64 4-20 30 110-0 868 253 -0,565 0,130 2-51 4-34 50 180°7 Sil 275 - 01635 =. OSs 2:26 4-60 _ 60. 1425 78:2 215 70. 157-O(extr.) 753 202 Ethyl Acetate. 1 : BI p iM p. NaS: of. Sa a. sae ae —20 82(extr) 975* 25°38 -0,457 0,193 |. —2-99.5 eae 0 963 89°7 Se a 0,127 Dh Sake 10 1050 859° 92:2 0,531 0,131 | 2-65 seme 30 1253 79:1 200." 0,590 01139 | ae 4-24 50 149°8 2:7 175 0,674 0,148 | 2-20 4-56 60 1642 69°7 16:3 0,724 0,152 2-09 476 70 .181:4 66:9 0,156 | 1:97 80 187°5(extr.) 64:1 0,152 1-99 * From vapour-pressure at —20° C. Tt Bakker (doc. cit.). + Ramsay & Shields. Latent Heat, and Compressibility. Ethylene Dichloride. z B. p — 20 63 (extr,) 37:2 O LOL 341 10 73°2 32°6 20 80°7 31°2 40 93'8 28:4 60 109-7 25°6 80 1296 23°2 90 140 (extr.) 222 d=459x 10-8 cm. (from p=23°7, Hihyl Lodide. bo ~I eae 0. a, | = bs . : ore ‘0,391 0,106 (extr.) Egon 3-67 00429 Oil eo 0 3:85: 0,446 0,114 (312 391 0,466 °0,116 304 402 0,512 0,121 2°88 4:23 0,568 _ 0,127 272 4-47 0,627 0,133 | 254 4-72 | 2°46 a p. p pe a. — 20 76 (extr.) 32°5 0,339 -00109(extr.) Os 83:6 30°4 0,363 0,112 10 92°4 29-2 0,377 0,115 20 99°5 28:1 0,892 0,118 40 1159 25°9 0,426 0,123 60 135°7 23:7 0,465 0,128 tO AUK} 22°6 0.468 i OFuol 80 157-5 (extr.) 21:5 0,513 0,134 Jd—AN9X10—%em, (from p—22°5) Chlor-benzene. T ise 0 — 20 60(extr.) 38:0 —10 63:2 36°8 0 67:0 30°7 20 75:2 33°3 40 85:0 31:0 60 96°3 28°6 80 §=109°3 26°3 100° 127 (extr.) 24:0 vl 0,322 0,382 0,842 0,366 0,394 0,427 0,464 v="864, 4,=64 (Regnault) at b.p.). zB p é= +. ee d —2°83 —d3 11 2°69 3°24: 2°59 3°28 2°50 3°32 2°33 3°46 217 3°65 2°10 3°73 2:06 3°83 nS —42-4 (Andrews) at b.p.). 0,918 0,940 "0,949 ‘0,972 ‘0,100 ‘0,108 ‘0,106 —3'50 3°53 3°60 \.-- oti 394 414 4°38 d=4-37 x 10-8 em. (from p=2U'2 (mean) at b.p, and A, =67'06). 28 Mr. D. L. Hammick on Surface Energy, Toluene. eee Z i Bo ny i Gi Pp p. a = _ 6p re d —20 72°5 (extr.) 342 0,353 00100 —3:05 —3°93 ) 79°3 31°6 ‘0,380 Os0Sm—n 3:03 3°69 20 90:8 291 "0,412 O07 = | 2°86 3°85 40 1047 2671 ‘0,450 OS) 2D 3°98 60 1216 24°3 ‘0,494 ‘0,119 2°52 4°15 80 61405 22:2 | 2°43 100 160:5(extr.) 20:2 2°33 d=4:53 X10—-8 cm. (from p=19°5, A,=79-0 at b.p.). Nitro-benzene. eae eal ops [a 0, Us a. C= Boel p. ; 6 a d 0) 44:7 44:3 0,273 0.83 —3:29 —329 10 47°3 43°1 0,279 ‘0,834 3:24 83°34 20 50:6 41°8 ‘0,288 0,837 3:16 3°44 3 52°8 40:6 0,297 ‘0,847 3'08 3:50 40 55°6 39°5 0,305 ‘0,856 3°00 3°56 50 58°4 (extr.) 38°3 0,314 ‘0,863 2°95 3°64 60 G14"; 37°7 0,319 ‘0,869 2°86 aiGy d=4:16 X10 Siem: (Gromyo— 21527 y= ls2-at bape It is apparent from the above tables that the ratio 1 1 iene. [es GL ere | : €= So tends to approximate to =, at lower tempera- : d tures, and to the value —2 at higher temperatures (near the boiling-point in many cases). The course of the variation of e with temperature is shown for a few examples in the accompanying figure. For each liquid two curves have been drawn. ‘The upper curve in each example shows the varia- tion of the ratio C (vertical axis) with temperature (horizontal Latent Heat, and Compressibility. 29 axis) ; the lower curve represents the variation of e with temperature. In the case of nitrobenzene, the two curves touch at 0° C., at which temperature those for ethyl iodide are also very close. With ethyl iodide and acetate, chloro- form and ether, the « curve touches the —2 axis in the neighbourhood of the boiling-points. ETHER ETHYL 100/0E ---- NITRO BENZENE -30 =/20) -10 O 10 20 30 40 50 60 70 80 ——-> Temperature. It would thus appear that from (roughly) the boiling- point downwards it becomes progressively necessary to regard “a” as a function of the volume. Latent Heat and Heat of Expansion. It will be noticed that at 0° C. the value of ¢ is roughly —3 (mean value for the liquids dealt with is —2-94, ex- cluding ether). In other words, at 0° C. we have if By —3A4s approximately, or —— —d3A1. 30 Mr. D. L. Hammick on Surface Energy, If vp is the pee volume at the absolute zero, we have approximately v=vo(l+eT'), where «=coefficient of expan- sion and T=absolute temperature. Hence vol bara) ely BI ELA) U (1 Ss al’) eae v Or meee. _ avy e v5 : y iL : v a J (3 18 at)= ee, Now the ratio of critical volume to volume at absolute een : ; zero, —, is a constant and approximately 4. Putting the 9 ° Ve ratio — =n, we have - 7 gr and from above: JeyAal ' = gt gt)a—3x as. - . a eae : Ve ca dae ihe) Taio. 7 — 7 has been evaluated for 21 liquids (using Young’s data for v,). Excluding chlor-benzene (n=2°0) au acetonitrile (n=3°'4), the values of n range between 3°1 and 2°8, the mean value being 3:0. We have, therefore, tia) (1K.) above : +(3 -- oe = —4),s. But we have found a7 = —d),s approximately (at 0° C.). Hence T'=—)js approximately at 0°C. .° (x.) Say] BR ~ Latent Heat, and Compressibility. 31 Now =at = is the ‘“‘ work of expansion” or the heat that must be withdrawn from a body in compressing it iso- thermally by unit volume. In other words, we should expect to find that at 0° C., the work of expansion of a liquid should be roughly equal to the latent heat per unit volume. ‘That this is the case was discovered empirically by Lewis (Zeit. Phys. Chem. |xxviil. p. 24 (1911)). Surface Energy and Compressibility. In conclusion, it may be pointed out that since, as shown 1 ; : S above, B =—ed,s (in mechanical units), we have, from (ii1.) : ee. OP ( ee: a : or Be Sem er a CXS) Since d is approximately the same for the common organic liquids (about 45x 1078 cm.) and € at 0° C. is not very ditferent from —3, we may expect to find the produet fof) approximately constant at 0° C. and equal to The mean value for 8p from the data used above is *252 x 1078 at O° C. It is, of course, well known that the product Bp is by no means constant at other temperatures ; this follows at once from (xi.), since € varies with temperature and dif- ferently for different liquids. it is to be expected that at higher temperatures, where the ratio e¢ has become —2, Be would again become approximately constant at the value 4°5 12 and Matthews (Zeit. Phys. Chem. Ixi. p. 449) and particu- larly Tyrer (Zeit. Phys. Chem. Ixxxvil. p. 169), who found x 1078=:39x 10-8. The empirical relations of Richards 2 oy o respectively that the expressions Sp*°? and ~),,.. are approxi- mately constant over the ordinary temperature ranges, 382 Surface Energy, Latent Heat, and Compressibility. obviously take into account the variation of e with temperature. Summary. 2 (1) If the van der Waals constant ‘‘a” is independent of volume, it is shown that the compressibility @ of a liquid - should be connected with the internal latent heat by the relation . = — 2158. 5 5 2 2 : (xil.) IL aa E ~ SEEMS, 2. ny ee eee sea) rey 1 4 foes where «= P (p= ~ oP : coefficient of variation of surface a pol energy with temperature : eye os coefficient of cubical expansion). vol’ Tt is found, for the limited number of liquids for which data are available, that (xii.) is approached at higher temperatures (the relation holding as such for some liquids at the boiling- point), and that (xiil.) becomes progressively applicable at low temperatures. (2) Lewis’s empirical approximation, that heat of com- pression is very nearly equal to latent heat of vaporization per unit volume at 0° C., is deduced. (3) The approximate constancy of the product of com- pressibility and surface energy at 0° C. is deduced, and the observed lack of constancy at other temperatures accounted for. The Chemical Laboratory, The College, Winchester. [ser ot —~ or) = TIL. On the fective Inductance, Effective Resistance, and Self- ~ Capacity of Magneto W “inddings. By N. W. McLacuuan, DSc. Eng, M1IEE. (From the National Physical Laboratory.) * SYNOPSIS. (1) Introduction. (2) Measurement of the Self-Inductance and Effective Resistance of the primary winding by Undamped Oscillations. (8) Measurement of the Self-Inductance and Effective Resistance of the secondary winding by Undemped Oscillations. (4) Measurement of the Seli-Capacity of and Dielectric Loss in Secondary winding. (5) Bridge method of Measuring the Self-Capacity of Secondary winding. (6) Results of Experiments :-— (a) Armature in Housing. (6) Armature out of Housing. (c) Kffect of Polarization. (d) Kffect of Armature Brass End-plates. (e) Effective Permeability. (f) Self-Capacity of Secondary. (g) Dielectric Loss in Secondary. (7) Calculation of Knergy Loss prior to the passage of the spark. (8) itiotiatve method of calculating the Energy Loss. (9) Comparison of methods used in (7) and (8). (10) Comparison of Loss with armature in and out of Housing. (11) The effect of an Air-core Inductance in the Primary circuit on the Damping. (12) SelfResonance of the Secondary winding. (13) Effective Inductance and the Effective Resistance of iron- cored coils under Damped Oscillations. (14) Measurement of Effective Inductance and Effective Re- sistance of iron-cored coils under Damped Oscillations. (1) Introduction. : the Phil. Mag. March 1919 Dr. N. Campbell published ay results of experiments on the high-tension magneto. p. 286 of this paper we find that a method of obtaining s self-inductances of the primary and secondary windings is described. ‘This yields the results given in fig. 3 and on p- 293. From these it appears that the inductances of the primary and secondary windings for damped oscillations remain constant over a certain 1 range of frequency up to about 2500 ~ per sec. This is shown to hold a for a Thomson-Bennett rotating ee magneto and foraB.T.-H. polar inductor magneto. Now the inductance of a magneto * Communicated by the Author. Phil. Mag. 8. 6. Vol. 41. No. 241. Jan. 1921. D 34 Dr. Mehachlan on Effective Inductance, liffective winding of fixed dimensions depends chiefly on the magnetic quality ; and behaviour of the iron. Provided the current i Is constant and the oscillations are damped or undamped sine waves, we may write approximately that L=A,u,, where, for a given position of the rotor, L is the inductance, Ay is a constant, and pe. is the effective permeability of the magnetic circuit, including air-gaps. The value of Ao is appr oximately constant for the range of frequencies in a magneto say from 2x10? ~ to 10*~, but the value of p, varies with the fre- quency. Itis well known that the apparent permeability of iron (#2) found from measurements with a coil wound on an iron core in which there is no air-gap or in which the de- magnetizing coefficient and leakage are very small, decreases with the frequency whether the oscillations are damped or undamped. The magneto circuit is one in which the magnetie * path is not completely wound with copper, especially in the inductor type, and one in which there are several air-gaps. Although the air-gaps are only a small proportion of the length “of. the magnetic circuit, they are equivalent to a len oth of iron pa times that of the gap. ‘Theair-gaps do not, hoe ever, alter the gener al behaviour of the iren qualitatively. This can easily be shown approximately neglecting leakage as follows :— Let 7 2?=mean length of magnetic cireuit containing alr-gaps, “assumed equal to length of mag- netizing coll. 1, = mean length of iron of circuit. A = cross-sectional area of circuit assumed constant. IN = totalitl axe B = N/A = apparent flux density under alternating current in cireuit having air-gaps. H = apparent magnetizing force under alternating current. ‘B/E =a. Pay le = apparent permeability * and effective perme- ability under alternating current. Then co ANE NN Hl 7 ie oe or Hl id Lj —pa(l—h) Bo bea or B % tial a) eee * When magnetic circuit is of uniform cross-section and there is no alr-gap. Resistance, and Self-Capacity of Magneto Windings. 35 Taking ug == 500 (j= Orcs dength of air-gap (—4, = 0°03 em., we have SOE Be 10 + (500 x 003) = 200. Ata higher frequency assume ie — 00: then GR, PE) Bs The effect of the air-gaps is clearly greater at the lower frequency owing to the higher apparent permeability of the iron, or in other words the air-gap at the lower frequency is equivalent to a greater length of iron than at the higher frequency. As the value of p, decreases, the proportionate effect of the air-gaps decreases also. Thus the etfect of the air-gaps in decreasing the effective permeability of the magnetic circuit is greatest when the current is growing in the primary before break, since wa is greatest then. The gaps cause a diminution in the self-inductance of the primary winding. In the above case it is of interest to observe that little advantage would be gained by using an iron core In which the value of pz was 1000, since this would only increase p, to 250; 7. e., 100 per*cent. increase In fa produces only 25 per cent. increase in p.. On the other hand, increasing fa trom LCO to 200 increases pe trom (4 so 120,47. é..02 per ‘cent. increase. When the air-gap in the magnetic circuit 1s proportionately large there is a considerable leakage and the flux density in the gap is far from uniform. In this case we may write approximately Me Maier Oia. > + - (4) where “gq” is a variable parameter depending on the frequency. When /, is small compared with gpa(l—/,), the expression for the effective permeability reduces to pe Ui(l—l)g}. Since 7 and 1, are constant it Follows that pe is approximately constant and independent of the frequency, provided g is constant. The latter condition will obtain at acoustic frequencies. This case corresponds ap- proximately to that of a magneto in which the armature is D2 386 Dr. McLachlan on Effective Inductance, Hifective removed from the housing. It is found by experiment that the above conclusion is correct. (Nee fig. 10, curve 3.) It is clear that the air-gaps cause the effective permeability to attain an approxim fately constant value at a much lower frequency than would otherwise be the case. Moreever, the primary and secondary inductances will diminish with the frequency until they are nearly constant. This is an un- desirable feature for high peak-voltage in the magneto. Hence the air-gaps drone Ibs medweal ko A minimum, and the apparent permeability of the iron non the higher frequencies—be as large as possible. This latter condition ean be fulfilled by using very fine liminations—-z. e. as thin ns conditions of economical manufacture will allow. It is also necessary that the brand of iron used should lave a low remanence, smill hysteresis, and high resistivity to ensure small loss. There is one point to be noted, however, In connexion with fine laminations. It ean Bee shown * that with a given brand of iron, for a given total cross-sectional area of the magnetic airentt. aren tron: inculaen between sheets, there is a certain Jntceness which gives optimum peak-voltage, and, therefore, maximum secondary energy £0,V,.?. It is well known that at frequencies of the order of 50 ~ per sec. the primary inductance of a magneto depends on the position of the armature with reference to a pole-shoes. This can be accounted for in two ways: (1) the variation in length of the iron and air-g@aps completing the magnetic circuit through the armature core, 2. e. the variation in reluctance ; (2) the variation in polarization of the iron due. to the flux from the magnet. An increase in either the. reluctance of the magnetic circuit of the armature or the Pola Hou entails a decrease in the inductance. This also- applies to the secondary inductance, since both primary and secondary are wound on the same core. In the inductor type magneto. is polarization and reluctance have minimum values. just before br eak, when the timing lever is set at full advance. In this position the greater part of the magnetic circuit through ‘the primary is unpolarized, since the “flux through the primary winding is zero, and the circuit is completed across the inductor. We should expect, therefore, the in- ductance to be a maximum in this position, and this is. confirmed by experiment. In any other position of the: timing lever, the magnetic cirenit through the primary winding is completed partially, at least, through the magnet, and the reluctance of the circuit is greater owing to the * See McLachlan, ‘ Wireless Year Book,’ 1918, p. 898. Resistance, and Self-Capacity of Magneto Windings. 37 additional air-gaps introduced. In the rotating armature magneto the maximum inductance also occurs before advance break, for in this position the polarization and reluctance of the armature magnetic circuit have minimum values. Owing to diminution in the value of pw, at high frequencies, it follows from above that the inductance would not be ex- pected to vary so much with the angular position of the armature. ‘This also is corroborated by experiment. When the magneto is in operation and connected to a peak-voltage measuring apparatus, the damping in_ the secondary circuit is very large owing to the high effective resistance of the iron under ecu tiens of audio frequency. The effective inductance and effective resistance in this case must be viewed ina different manner from that when the oscillations are undamped, as will be shown later. The measurements described hereafter have been- carried out with undamped oscillations as accurately as experi- mental conditions would ea None of the methods was susceptible of a high degree of accuracy. Data were required which showed “the © order of magnitude” of dif- ferent quantities and which exhibited the general behaviour of the magneto under certain experimental conditions. These conditions are different from those which obtain in practice. The chief object of the research was to obtain some know- ledge regarding the loss due to iron, etc. Using the values of the Eoomicicn(s of the circuits cine obtained, it is possible to get an estimate of the efficiency of the ‘magneto. A B.T-H. polar inductor magneto was used in all experiments, as it 1s well suited to re search work. (2) Measurement of the Self-Inductance and Kffective Re- sistance of the Primary of a Magneto by Undamped Oscillations. Before describing the method of measurement adopted it is necessary to define the quantities it is intended to obtain. If a sinusoidal current is passed through an iron-cored coil, the voltage on the coil is nearly sinusoidal provided the frequency i Is ‘fairly high and the magnetizing force low. At low frequencies, e. g. 50 —~ per sec., the v oltage wave form is not sinusoidal * owing to the variation in the permeability of the iron with the current. The sinusoidal voltage can be split up into two components R.I and owL,1 * See Journ. I. E. E. June 1915. 38 Dr. MeLachlan on Effective Inductance, Effective in quadrature, as shown in fig.]. ‘These represent, respec- tively, the ohmic drop due to the effective resistance caused Fig. 1.—Vector diagram illustrating definition of effective inductance and effective resistance. Rel chiefly by iron loss but also by dielectric loss and eddies in: metal fittings, and the drop due to the effective inductance. The apparatus used is shown diagrammatically in fig. 2. A triode is employed to generate oscillations in cireuit (1). Fig, 2.—Diagrammaiic arrangement of apparatus used to determine the primary inductance and effective resistance. Valve Generater oJ i i t i Lrsvsaness| if re @ loz Loceneng f Primary of Magneo A second circuit contains an inductance coupled to the inductance in cireuit (1), a variable condenser, variable inductance, and the primary of the magneto under test. It is necessary to -remove the secondary winding and to arrange that the total inductance in circuit 2 (apart from that of the magneto) is as small as convenient. This is essential so that a small change in inductance will produce a perceptible change in the current measured on the Duddell milliammeter A, since the high effective resistance of the magneto causes the resonance curve to be rather flat-topped. The variable inductance is adjusted so that cireuit (2) is brought into resonance. The resonance point is found by obtaining the inductances on either side of the maximum current for which the current has equal values. The Resistance, and Self-Capacity of Magneto Windings. 39 mean of the two values of the inductance is taken as the inductance at resonance, and the variometer is set accordingly. The magneto is then disconnected and the indnctance again varied until resonance is obtained. . The difference in the readings is the effective inductance of the primary winding of the magneto. The effective resistance is found in the well-known manner by obtaining (a) the current at resonance, (6) the current when the circuit is out of resonance. The latter condition can be obtained by varying either the condenser or the inductance. In these experiments the latter, which had a constant resistance, was varied. The difference in current should be appreciable to obtain accurate results. It is important that the current in the primary oscillating circuit should be the same for cases (a) and (6). If it varies, a correction can be made, San the two values are known Let I, = current at resonance with magneto in circuit. I, = current when circuit is out of resonance. L = variation in inductance to put circuit out of resonance. @ = pulsatance = 2a. Ra = effective resistance of primary winding at fre- quency f. r. = effective rexistance of circuit without primary of magneto. Then n+ Ra = ol (74 a) = es) a (3) r. is found in a similar manner to that of 7, + R,. A more accurate method, if the apparatus is available, is to employ a bridge as used by Hund *. In this way it is possible to find the ‘inductance and resistance for the funda- mental and various harmonies. This method should be useful where currents of the same magnitude as the maximum primary oscillatory current under working conditions are employed, since in this case there would be harmonics owing to the apparent permeability of the iron not being constant over the range of values of H employed. In the present experiments, it was found that the inductance and effective resistance varied slightly with the current at the lower frequencies when the armature was situated in the housing. Throughout the experiments, the root mean square ‘current used in measuring the inductance was about * Tlectrician, vol. xxv. p. 78 (1915). 40 Dr. MeLachlan on Hffective Inductance, Lifective 40 milliamperes. In finding the effective resistance it is ne- ae to vary the current. This causes a variation in R. -at the lower trequene ies. The valve obtained for Rey increased with decrease in the current ee the coil when the circuit was thrown out of resonance—2. e., R., increased with decrease in J, for a given value of i. Owing to time limitations, the inductances with currents larger ho 40 milli- amperes were not ascer tained. ‘The primary current in au oe is much greater than this (see Proc. Phys. Soe. Dec. 1919). Since the peak-y oltage is propor tional to a current eee in the primary from "0-2 25 amp. to £amp., 1 is very probable that the inductances are constant ioe this range. (3) Measurement of the Selj-Inductance and Kfective Resistance of the Secondary by Undamped Oscillations. The methods of measurement adopted for the primary winding are inapplicable to the secondary, owing to the magnitudes of the quantities to be obtained. ‘The effective resistance of the secondary is of the order of 5 x 10* ohms at a frequency of 1700 ~ per sec., and it is quite clear that resonance could not be obtained on a milliammeter with this resistance in circuit (2) of fig. 2 without the aid of extremely large currents in cireuit (1). The above qt uantities can be tound, however, by modifying the resonating circuit in fig. 2. Fig, 3,—Diagrammatic arrangement of apparatus used to determine the secondar y inductance and effective resistance. Valve Generstar Secondary ef Magneto in housing Veriable Condenser Electrostatic Volt meter The arrangement of the modified circuit is shown diagram- matically in fig. 3. The magneto ee is used to obtain a coupling between circuits (1) and (2). _Im this case) dhe aluminium end plates are removed and the primary is on open circuit. The variable condenser is adjusted until the voltmeter-reading is a maximum. Under this condition, circuit (2) is in resonance. Knowing the frequency of the Resistance, and Self-Capacity of Magneto Windings. 41 oscillations in this circuit, the effective inductance can be calculated from the expression where C is the self-capacity™ of the secondary winding plus the capacity cf the variable congas plus the capacity of the voltmeter for the. re: ading shown thereon, it is evident that this method cannot be ie for frequencies exceeding that at which the coil has its first self-resonance. In fact the self-resonance cannot be obtained by direct mea- surement owing to the capacity of the voltmeter. Although it is not possible to measure the secondary inductance at all frequencies by the preceding method, its value can be calculated Approx taper) from that of the primary, and the results at low frequencies can be compared with those found experimentally. The agreement is about 9 per cent, at frequencies between 1500 ~ and 2500 ~, the calculated values being in defect. Let L, = Inductance of primary winding, out of housing, with air core (measured with secondary re- moved). L, = Inductance of secondary winding, out of housing, with air core. A; = Cross-sectional area of iron core. A, = Cross-sectional area of primary with air core (mean). A, = Cross-sectional area of secondary witli air core (mean). Ly = EKffective inductance of primary in housing. L = Effective in iuctance of secondary in housing. e = Effective permeability of magnetic circuit including air gaps. Then L,, = Additional inductance due to iron+inductance with air core = by ety =e Ui! (ie RG ee 9) where 1G Srey Ba Similar] i: imilarly [Lip Jd (eae er (2) where IS5 = PSypeue * This can be measured, as shown below. 9) i r +1 1 $ Ve ‘ > > 2? e 42 Dr. McLachlan on Effective Inductance, Lifective If all the factors in (3) but », are known, then p, can be calculated. Substituting this value in (4) it is possible to calculate Ls. | The mean areas of the primary and secondary windings used in formule (3) and (4) are obtained by measuring the self-inductances experimentally when the iron core is removed. The inductance of the primary is found after the secondary has been unwound. The value of a, the equivalent mean radias in Lorenz’s equation L=an’?Q*, is then ascertained by plotting an “a” and “L” curve and reading off the value of ya corresponding to the value of “LL” found by experiment, or by trial and error. The above formule are only intended to apply when the magnetic cireuit is nearly closed and_ the leakage is not very large. Approximate overall dimensions of the armature windings are given in fie. 3a. The armature used for these measurements was slightly different from that mentioned in section 4. eS Fie. 3a. Section AB. Secondary, (4 Primary Primary = 160 turns. Secondary = 10* turns. Primary res. (d.c.} = 0°55 w. Secondary res. (d.c.) == 1900 a The methcd of measuring the effective resistance 1s to find the voltage at resonance, reduce the capacity so that the circuit is thrown out of resonance, and again find the voltage. In this case as in that of the primary winding, the effective resistance includes iron, dielectric, and stray losses in the metal framework. | Let V, = voltmeter reading at resonance. C, = condenser reading at resonance including volt-. meter. I, = current reading at resonance. V.= voltmeter reading when circuit is out ‘of me= sonance. C, = condenser reading out of resonance including voltmeter. J, = current reading when circuit 1s out of resonance. @ = pulsatance. R,, = effective resistance of secondary. * Bulletin Bureau of Standards, vol. viii. No. 1. Resistance, and Selj-Capacity of Magneto Windings. 43: Then it can be as 2 ie R.» 1s constant, = isp a legiNe eeeee aa 6). ele ae (at 7__ | 2): 0 Since L = wC,V, . ; ll => aC, V5 We obtain wa Ce Cs Com q 2 a eee a oC, 4 (Gave=cave) : In this case the effective resistance of the remainder of the circuit is negligible. In carrying out experiments with the armature in the housing, it was found that the inductance and resistance varied with the current for the range of frequencies obtain- able, viz. 1000 to 2500 ~ per sec. The inductances were measured with a current of about 6x 10~* ampere, since this produced approximately the same ampere-turns as the current used in the measurement of the primary cireuit. Owing to the necessity for vary ing the current to obtain the effective resistance, the results are not very accurate, but merely serve to indicate the order of magnitude. Using the above formula for R,», its value increased slightly with decrease in CoV2, when ©,V, was constant. Measurements of the inductance and effective resistance for various conditions have been made. These will be dealt with in discussing the experimental results. (4) Measurement of Self-Capacity of and Dielectric Loss in Secondary winding. The cireuit is arr anged as shown in fig. 4. Cireuit 2 1s loosely coupled to ore 1, and brought to resonance Fig, 4.—Diagram showing apparatus for determination of self- capacity ior secondary winding by substitution method. Valve eames tor Magneto Secondary by varying either C, or L when the magneto is completely disconnected. This latter precaution must be observed, owing to the capacity effect due to the housing. The 44 Dr. McLachlan on Effective Inductance, Hffective condenser reading is observed (Cy ee be small, L iarge, and the frequency fairly high); also the reading of the miulliammeter. The magneto is ten connected in parallel with C,, and the latter is varied until resonance is again obtained. The difference in the condenser readings (the condenser should have an open scale) is the self-capacity of the coil. It was found that the self-capacity did not vary much with the frequency (2x 10* to 8x 10+ ~), but it was less when the armature was out of the housing than when it was in, as one would naturally expect. When the armature was out of the housing, the iron core, which was a little longer than the armature coil, made little difference in the self- -capacity. With the armature in the housing and the primary winding not connected thereto, the self capacity was less than “that with the primary connected. to the housing but greater than that cut of the housing. Theory of method.—To simplify matters we shall assume the secondary coil with parallel condenser to be approxi- mately represented by an inductance L having resistance R in parallel with a condenser C equal to the self-capacity of the coil, plus the condenser in the secondary circuit. Neglecting ‘dielectric loss. the admittances of the two paths between A and B are respectively (see fig. 5) 1 Re +)@ Lis Fig. 5.—Equivalent (assumed) circuit of secondary coil for determination of self-capacity. L2. F2. a eae i Self Capacity t Aaded Capecity and ja. The combined impedance between A and B is therefore 1 (= oe Hosa lata +joaCRs _ (Re+jo le)(A—joC Ro) A? + @?C?R,? ; where A = (l—o@’L,C) re ee —w? 11,0) Ro + @? LoC Rs +/@ Le(1— w? L2C) —jo CR»? @ * Le?Co? + w? C? RY? , Resistance, and Self-Capacity of Magneto Windings. 45 Tf w°L2C is large compared with unity, we get 7, & Ro gw C(R? +o? Lp?) wo C?(Re?+@7 Lis?) — @? C?( Re? + w? Lp”) ae Rs ee ae ; aL ie w2C?(R2 + foe) ao es R TG ? z r (8) *» 2. é@., the combination is equivalent to the condenser © in series with a non-inductive resistance R' = = as shown in noe (O. Fig. 6.- Urea: which behaves in the same manner as that of fig. when certain conditions (stated in text) are fulfilled. if This is due to the fact that by far the greater portion of the current passes into the coil as displacement current and not as a circulating current passing through the coil. In order to obtain this condition, it 1s essential that w?LoC should be large compared with unity. Since C must be small so that a foal difference in C* affects the ine of the circuit, w* and therefore f must be correspondingly large. In carrying out experiments using the above method, it was found that the resonance currents were different according as the magneto or a condenser of equal capacity was connected in circuit. It was greater in the latter ease. This difference must be due to either copper, iron, dielectric loss, or all of them. In order to investigate this point, the resonance currents were measured (a) with the armature in the housing, (b) with the armature out of the housing, (1) with the iron core in place, (2) with the iron core withdrawn. The current was smaller in case (a) than in case (>). In the latter case it was greater with the iron than without it. The frequency was so high that o°C?R,’+7L,*) was much greater than Ry except when the iron was removed. Thus, since Ly, is much less: without * Tf the capacity in parallel with the coil is large compared with the self-capacity, the distribution of current in the coil will be fairly uniform, 46 Dr. MeLachlan on Effective Inductance, Effective the iron core than with it*, the value of R' in expression (8) was not negligible. By increasing Cand keeping @ constant, there was no measurable difference with and without the iron, whether the latter was very thick or very thin. Thus the reduction in current is in this cage brought about by dielectric and copper loss. The dielectric loss can be represented in the usual manner by a high resistance in parallel with the condenser. ‘This is ‘equivalent to a series resistance of small magnitude. Since the current passing through the inductance (fig. 5) is extremely small, it can be neglected. The conditions that R, w?C2( vn + w? Lis”) is negligible, and that w’?L2C is much greater than unity, must lie eo ed. Under these conditions the coil with the condenser in parallel is, therefore, equivalent to the circuit of fig. 7. fe —LRepresentation of secondary winding as a condenser shunted by a high resistance, when certain conditions (specified in text) are fulfilled. Fig. 7 ty When w?C "Re i is large compared with unity, the arrange- ment of fig. 7 is equiv: alent to a condenser C in series with a resistance of magnitude R= a g ees : q w?C7Ra where R, represents the resistance of the dielectric to alter- nating currents under the special conditions in the magneto Fig. 8.—Cireuit equivalent to that of tig. 7 5 G w=CF Rd. | | * By calculation and measurement, Ls without the iron was of the order of 3 henries for the coil used in these experiments. Resistance, and Self-Capacity of Magneto Windings. 47 or 1/Ra is the conductance. This of course includes dielectric hysteresis. In conducting experiments on di- electric loss the armature should be taken out of the housing and the iron removed. ‘The primary circuit should be open and the condenser absent. The separation of the dielectric and copper losses is dealt with later. (5) Bridge Method of measuring Self-Capacity. ‘The self-capacity at frequencies greater than 20,000 ~ per second can be found by using a bridge of the form shown in fig. 9, instead of the substitution method already outlined. Fig. 9.—Diagram showing bridge connexions for the measurement of self-capacity of secondary winding. fron Cored Differential Sy Trensformer — Aa S Oetector Va/ve Circuit The cirenit of fig. 9 consists of an iron-cored differential transformer having two primaries with equal numbers of turns and a secondary. The prima aries are connected in opposition. The coil ‘under test is connected in parallel with a variable condenser in one arm, and a variable non-inductive resistance and variable condenser in the other. When baiance is obtained we have C)4+(C,=C, pro- eee 0 lin(C,+ C2) is very large compared with unity, w Lis ian and Re/@?C?(@? Le? + Ry?) is very small: also i 1 Gs 2 =e al Ro, where Rois the ohmic resistance of the copper of the secondary winding. Thus the dielectric resistance for the particular conditions which obtain in the magneto can be found. Since ©, is known, C, is found by subtraction. In order to obtain accurate results, the value of C, should not be too small compared with ©). At high frequencies C, could be zero. The detector may be ai telephone* with interrupter, tikker or other device to provide R * The telephone might be troublesome owing to the noise produced when the current passes through the magneto winding, also the noise from the valve generating the oscillations. 48 Dr. Mclachlan on Effective Inductance, Hfective audible frequency, ora valve amplifier may be used in con- junction with a rectifying valve, smoothing out condenser,and a microammeter. Care must be taken, however, with regard to self-resonance effects in the (ifeven eal irar atone and the valve-amplifier circuit. The current in the bridge can be measured by using a transformer and amplifying if the current is small, or by a thermoammeter re eading to 100 milli-- amperes, if the current is lar ge enough. (6) Lesults of Haperiments. (a) Armature in Housiny. The values of the coefficients of the primary and secondary circuits obtained with the armature in the housing are ane in figs. 10 to14. It will be seen that the inductances decrease rapidly until the frequency is 15,000 ~ per second. Beyond that value the decrease in induotmes | is much more gradual, ee to the fact that the effective permeability of the nagnetic circuit 1s approximately constant. This is in ne oe with the argument at the beginning of the paper. The inductance, pee at low frequencies, depends on the position of the rotor , being greatest just before advance break and least after retard break. The difference in the values after advance and after retard break was only a small percentage. The same holds good with reference to the effective resistance. This variation is due to the change in the reluctance and polarization of the magnetic circuit with the position of the rotor. As the frequency increases the difference between the inductances and resistances for various rotor positions decreases, owing to diminution in the apparent permeability of the iron. The ratio of the secondary to the primary inductance is shown in Tabie L., and it will be seen that the variation with frequency up to 5000~ is small. Api ls Showing Ratio Lys/La. Armature in housing ; rotor position after full advance break. Values of lio obtained by calculation. i Le2/Lei L,/ AGF ee DEEP SII, with Mou core. with air core. iad 3 z sane rAd | le BUS 4-4 x 108 1510] — ee 0-234 10-3 oo , 4 oO 5) ey 4°35 xs 10 3 4:66 - 15 3 4°76 ; A .QR 20 > 4°85, 30 A Dye Resistance, and Self-Capacity of Magneto Windings. 49 Tt will be observed from fig. 11 that the rate of increase of the primary effective resistance diminishes slightly with the frequency, although it is nearly constant up to 10,000 ~ per second. This is dug, of course, to the behaviour of the-iron Fig, 10.—Curves showing variation in primary.inductance, under different conditions, with frequency. (Millihenries.) Primary Inductance Frequency — per second. Curve 1 = Primary Inductance with iron unpolarized, 7.e. magnet removed, after full advance. Coil with brass end-plates, and in housing, Curve 2 = As in Curve | but with magnet in place, 7. e. iron polarized. Curve 3 = Primary Inductance with armature removed from housing. Points thus: () = As in Curve 1, but before advance break. resulting from the screening effect of eddy currents. ‘The power Eactor of the primary winding 1s illustrated by the curve of fig. 13. It increases rapidly until 7=5000 ~ and afterwards at a much slower rate, until at f=l5 5,000 ~ it is Phil. Mag. 8. 6. Vol. 41. No. 241. Jan. 1921. EK Oo SiGe Tayo (Siero ceo 25 Zu eee of Primary (Ohms). Resistance Effective 50 Dr. McLachlan on Effective Inductance, Hfective nearly constant. This is again attributable to eddy currents and to the effect of the air-gaps. A similar curve is obtained in testing an iron circuit completely wound and without air- gaps. In this latter case the power factor of iron sheets of the = Fig. 11.—Curves showing variation in primary effective resistance, under different conditions, with frequency. 300 mM on oO 100 Frequency —~_ per _— second. Curve 1 = Effective Resistance of Primary with iron unpolarized, 7. é. magnet removed after full advance. Coil with brass end-plates, and in housing. . Curve 2 = As in Curve 1 but with magnet in place, ¢. e. iron polarized. Curve 3 = Effective Resistance with armature removed from housing. Intercept on vertical axis = res. of primary at zero frequency 0°55 w. same thickness as those in the magneto is greater at 10,000 ~ than that in fig. 13. The power factor of the primary winding cannot be considered as “‘ small.” Its magnitude is due to (1) the increase in loss and (2) the decrease in inductance with frequency. ‘The use of thin sheets of iron of high permeability, high resistivity, and low hysteresis loss would improve matters considerably. (Henries). Secondary Inductance Le» Resistance, and Self-Capacity of Magneto Windings. 51 (b) Armature out of Housing. The various coefficients obtained with the armature out of the housing are plotted in figs. 10,11,14. From the curves it is evident that the results agree closely with those fore- shadowed in the remarks at the beginning of the paper: There Fig. 12.—Curve showing variation in secondary self-inductance as obtained by calculation, using the curve of fig. 16. Frequency —~ per _ second. Out of housing, secondary inductance = 9:7 henries from 1200 ~ to 3000 ~. XC = Experimental points. © = Points obtained by calculation from fig. 16. Armature with brass end-plates ; in housing. Rotor position = after full advance break. is little variation in the inductance of either primary or secondary with frequency. The variation of the former 1s greater than the latter, since it is wound nearer to the iron core and leakage effectsaresmaller. The effective resistance is reduced very considerably and it increases at a more rapid rate than the frequency. ‘This is due to the absence of large E 2 Cos 9 = Power Factor of Primary Winding with Secondary removed. 52. Dr. McLachlan on Ljfective Inductance, Effective masses of solid iron from the magnetic circuit, and can be verified by using a solid core or a core of thick strips. The effective resistance was also measured, using cores of 0°38 mm. stalloy and 0:06 mm. pure iron, of the same dimensions as that supplied with the magneto. Both cores Fig, 18.—Variation of Power factor of primary winding with frequency. Frequency —~ per second. Armature with brass end-plates ; in housing. Rotor position = after full advance break. gave slightly smaller values of the effective resistance than the ordinary core. Stalloy was the best of the three cores, owing to its high resistivity. It may seem peculiar that stalloy 0°38 mm. thick should give a lower effective resist- ance than pure iron of 4 the thickness, since a reduction in eddy-current loss would be anticipated. This, however, is due to the fact that for the same ampere-turns or magnetizing force due to the oscillatory current, the. flux density, and therefore the loss, 1s greater with the thinner material: With equal flux densities the loss in the 0°06 mm. iron would be Jess than that in the 0°38 mm. iron. Resistance, and Self-Capacity of Magneto Windings. 53 (ce) Lgect of Polarization.- In Phil. Trans. A, vol. 184 (1893), Ewing and Klaassen showed when a sample of iron was subjected to a magnetizing foree produced by a constant current, and therefore causing polarization, that the hysteresis loss resulting from a super- posed force varying between definite limits, decreased with Fig. 14.—Variation in effective resistance of secondary with frequeucy. 30x10% @ 6 25 AN Dv fs > a 20 |-20 = 3 D S 2 Seles - : : r= < CS 2 @ 2 9 ce 10 Lo © S = S s & i y Fl 5x10 }5x102 ar Aousimg 2000 300) O- 2006 4006 6000 8000 10000 Frequency —~ per _ second. Curve 1 = Effective resistance of secondary in housing, with magnet aud brass end-plates; after full advance break. Curve 2 = Effective resistance with armature removed from housing. Intercept on vertical axis = res. of secondary at zero frequency =1900w. increase in the polarizing force. This is due to the constraint of the molecular magnets and the consequent diminution in the variation of the flux density. Owing to the latter effect the eddy-current loss is also diminished. In the magneto the effect of polarization is clearly exhibited by curves (1) and (2) in figs. 10 and 11*. These were * During the experiments the rotor was moved slightly beyond its position at advance break in order to show the polarization effect more clearly. 54 Dr. McLachlan on Effective Inductance, Effective obtained using a definite current, 7. e. the variation in H was nearly constant. The reduction in the hysteresis and eddy- current loss results in a diminution in the effective resistance. The decreased variation in the flux density causes a diminution in the inductance. The same ‘effects were observed in con- nexion with the secondary winding. Owing to the fact that the iron formsa greater proportion of the mean cross-sectional area of the primary than of the secondary, the polarization effect was more marked in the first case than in the second. Just before advance break the polarization is nearly zero and the influence of the magnet should, therefore, be zero. This was found to be true, since any variation in ‘induetames was within the limits of experimental error. The points obtained are. shown in fig. 10thus:—. It is evident that they follow nearly the same curve as those obtained w ‘ith the magnet removed. ‘The slight difference is due to the slightly altered configuration of ie magnetic circuit before and Nee break. In the absence of polarization, the variation of inductance with the position of the rotor is considerably reduced. In order to determine the influence of the metal of the magnet, apart from polarization, a spare magnet was demagnetized and fitted to the magneto. There was an increase in the secondary inductance, when measured after advance break, of about 5 per cent. (d) Effect of Armature Brass End-Plates. The brass end-plates are fixed to the core as shown in fig. 15 by bending two iren plates, A A, at each end and riveting over to the end-plates, BB. Considering each end to be replaced by a single turn of thin wire of radius a, it is clear that no current would flow in the system, since the E. M.F.’s in the two coils at. any instant would be equal and opposite. Owing, however, to the fact that there are lar ge masses of metal above and on each side of the slots in the end- plates, eddy currents are induced therein and the circuits of ress currents act as tertiary circuits when the magneto is in operation, and also when undamped oscillations are passed through the primary or the secondary winding. The effect of the end- pla ates can be studied by means of an endless coil of copper wire having several turns of 18 s.w. .g. The end- plates are remov ed ; ‘the primary of the armature with core is taken from its housing and connected with the circuit of fig. 2. The circuit is brought to resonance, and the coil is then slipped into the position occupied by one of the end- plates. It is found that ‘the effective external inductance Resistance, and Self-Capacity of Magneto Windings. 55 and effective resistance of the primary are decreased. Both of these effects are due to the demagnetizing influence of the current induced in the short-circuited coil. The “decrease ”’ in effective resistance is due to the decreased iron loss caused Fig. 15,—-(a) Pictorial representation of armature before primary and secondary are wound. (6) Ulustrating electric circuits completed by the brass end-plates and the iron plates riveted thereto. b a lates - by the smaller variation in flux density resulting from} the demagnetizing action of the coil—i. e., transformer action. With an air-cored coil the external inductance decreases, but the copper loss increases owing to ohmic loss due to the currents in the secondary exploring coil. Thus with the iron core in place, although the total effective resistance Cron 56 Dr. Mclachlan on Ejective Inductance, Effective loss + copper loss) decreases, the portion due to copper loss increases. These effects are more e onounced the greater the coupling between the exploring coil and the primary winding. In the actual case of the magneto the effects are not nearly so marked as those obtained with the exploring coil, owing to the brass end-plates being slotted. At 10,000 ~ the diminution in the inductance and effective resistance was of the order-of 5 per cent. with the armature in or out of housing *. The effect of the aluminium end- plates carrying the outer ball races was examined, but it was extremely small. The influence of the other non-magnetic fittings surrounding the iron circuit was not examined owing to lack of time. It will be shown later that the dielectric loss is proportionately small. It is probable, therefore, that the greatest loss is due to.the iron, although the eddy currents in the non-magnetic fittings in which the laminations are fixed must contribute to the total loss. One method of investigating the latter source of loss would be to build a magneto with the laminations held in non- metal fittings, and compare the results with those obtained for the standard machine. (‘are wouid have to be taken to ensure small dielectric loss in the non-metal fittings. Tests could be conducted with the rotor at rest or in motion. (e) fective Permeability. The ettective permeability of the complete magnetic circuit, not merely the iron per se, has been defined by “equation (3). It is an index of the usefulness of the iron in increasing the flux through the windings. The curve of fig. 16 shows the variation with frequency. Its general shape is in accord- ance with the deductions on pp. 35, 39. On comparison with a> static permeability curve, it will be seen that pu, is ereatly reduced*owing to the air-gaps which are of paramount importance at low frequencies. This will be clear on in- spection of curves (1) and (2) of fig. 10. Since the inductances decrease with increase in polarization, it is evident that p, decreases too. (£) Self-Capacity of Secondary winding. The self-capacity of the secondary was measured at fre- quencies from 2x 104~ to 7x104~ per second by the * Froma peak-voltage test on full advance and retard the effect of the brass end-plates was negligible. Ifthe slots in the end-plates are closed, the peak-voltage is roduced considerably. Effective Permeability of Magnetic Circuit (je). Resistance, and Selj-Capacity of Magneto Windings. 57 substitution method outlined herein. Any variations were within the limits of error of observation and experiment. Fig. 16.—Curve showing variation in Effective Permeability of magnetic circuit with frequency. 5x!0° 0 (5 20 25 30 35x 105 Frequency —~ _ per __ second. Rotor position = after full advance break. ; Armature with brass end-plates. In housing. The results obtained with two different armatures for the polar inductor magneto are tabulated below. TasueE II. Self-Capacity of Secondary winding of B.T.-H. polar inductor Magneto. Armature (1) without brass end-plates. Frequency Self-capacity _ (~ per sec.). (Picofarads). Experimental conditions. 3°5 x 107 6d Armature in housing. Loose end of primary connected to housing. a YD) As above, but end of primary free. i 50 Armature out of housing, with iron core inserted, oo 104 50 Ditto. 4 50 Ditto, but iron core removed. Armature (2) with or without brass end-plates. 3°5 x 104 50 In housing with primary connected thereto, 40 Out of housing with iron core. $2) 58 Dr. Mchachlan on “fective Inductance, Hffective (g) Dielectric Loss. The problem of obtaining an accurate measurement of the dielectric loss in the magneto is rather difficult owing to the peculiar conditions therein. This is partly due to the fact that in using the coil asa condenser, the charging current has to traverse the copper winding. This geome? in an ohmic loss, which is rather indefinite owing to the distribution of the current. All that can be done in the present instance is to obtain a rough approximation, and to show that even if the ohmic effects are included, the loss is not large compared with that due to the iron. When the iron core and the brass end-plates are removed from the armature coil and the inductive reactance is very Sseet => oC,’ the coil acts as a condenser, and the circulating current passing through the winding can be neglected in comparison with the charging current. Thus, when the coil is connected as shown in fig. 4, the loss which occurs is that indicated above. The current at resonance is obtained using the coil, asbefore. The coil is removed and the resonance current of equal magnitude obtained by inserting a non-inductive resistance in series with the thermoammeter A. The total loss in the coil is equal to the loss in this resistance. In separating out the dielectric loss the procedure given below is adopted. much greater than the capacity reactance, 7.e. wl Let I be the current at resonance. Let R., be the resistance inserted to give the same resonance current as the coil. Total loss in Coil = Dielectric loss + Ohmie loss due to charging current : eS = Piney Assuming charging current uniformly distributed in the winding, the Ohmic loss=(Charging current)? x res. of winding = (wC,K)’R, Ee PR(3 i: where C, is the self-capacity of the coil and C is the total capacity in circuit. Hquivalent series resistance = (=) pie (charging current) Resistance, and Self-Capacity of Magneto Windings. 59 In an experiment using’ a certain armature coil, without brass end-plates or other metal parts, the following data were obtained :— Wa—=20 27 LOS per sec. : 80x 1052 amp. i, = 2.2 ohms. R, = 2200 ohms (without iron core). C = 2100 picofarads. ©, =)50 ys Thus equivalent series resistance for CRISS current Yeqg=1:25 ohm, and therefore the series resistance equi- valent to dielectric loss, R’.,~=Rg—7eg=1'25 ohm. Equi- valent dielectric resistance for particular case of magneto secondary winding ll I ; 1 Ras SOR, == 1192 0 mang Meseat. jo 2x 10° —..and V max: = 140 VOlts== 4° 7 x 10-3 watt. Assuming loss « f/V® max.*, the amount dissipated during one quarter period prior to the passage of the spark when V max.=10* volts, is 1°57 x 107‘ joule, 7.e. only 6 per cent. of $C,V.”. Inclusion of the loss arising from the charging cient in the winding increases this percentage to 7. (7) Calculation of Energy Loss in Primary and Secondary of Magneto prior to the passage of the spark. In calculating the energy loss in a magneto, we are only concerned with the energy which is lost during the time which elapses between the breaking of the primary current and the passage of the spark. The ee method of calculation is approximate, but it serves to show the order of magnitude of the loss. Let V2 = secondary voltage when spark passes. C2 = self-capacity of “secondary + capacity of leads to spark gap. R.. = effective resistance of secondary. Igms = root mean square of current. dV» dt ~ * See G, E. Bairsto, Proc. Roy. Soc. A. vol. xcvi. pp. 863-882 (1920). Then current in secondary at any instant is 2=C, 60 Dr. McLachlan on Effective Inductance, Ejfective Assuming for simplicity that the voltage follows a sine law ans zero to its first maximum, we Lave ole to=@C2,V2cos@t and Ipus = The loss in joules is W=Ipns? R.2 x t, where t=time for the voltage to attain its maximum value. The energy stored in the condenser when the voltage isa maximum =3C2V,’, and this is dissipated ae in the spark. Hence total energy in secondary is W+4C.2V.*. Thus oe 30, V2? efficiency SOW CO Nee The maximum or peak voltage which would have occurred if there had been no loss, is obtained from the relationship LOW? = W432 - Ww $ 2 or V’s [2 = + Ve Ae Cs As an example take the following data :— V2 = 10* volts. Cy = 05 picofarads, including leads. 2 x 10° ohms. 6 x 10° ~ (mean value fer the two oscillations after break) *. = [ —™ Then Bad W = lems? Rot w°C,” V2? Reot/2 =al 6 0 ee oute: Actual energy obtained ) — LoVe from magneto in electrostatic form. 2°8 x 10-? joule. Treating the primary circuit in a similar manner and assuming as a first approximation that the ratio of the peak voltages is equal to the ratio of the turns, we find the loss is 13x107* joule. Hence, neglecting the loss in the primary condenser, the total loss prior to the passage of the spark is 3-1 x 107? joule. If this energy had been transferred to the secondary, the * This frequency is assumed for the pend prior to sparking. Its magnitude is “suggested by the curve of fig. 2 in Dr. Campbell’s paper (loc. cvt.). Resistance, and Selj-Capacity of Magneto Windings. 61 total electrostatic energy would have been 5:9 x 1073 joule, and the peak voltage 14,600 volts. Thus the effect of the energy loss in the primary and secondary SLC) is to reduce the peak voltage 32 per cent. (8) Second method of calculating Energy Loss. (a) The energy loss in a magneto can be approached from a different point of view using “the formule given by Taylor- Jones*. He has shown that when (1) there is no loss in the primary and secondary circuits (the whole of the energy in the primary would not necessarily be transformed into the spark in the secondary if there were no loss), (2) the two sets of oscillations which occur after break are damped sine waves, wand (3) the interruption of the primary circuit is perfect, the peak voltage is given by the expression Wo = = See to U sin d. : ° c > (9) Taking L..=M=h ¥ Le L,.2, this expression can be written ) L Using Vo= [ee L, er ampere ; v Lee ‘ La | A/ OF P P broken in the primary circuit. . (10) U is given by the expression i [u+s—2{(1—h)u}?]” La, where u=—~—’. hieC, (11) eee 2or @ is given by Lei where ng and n, are the frequencies of the two oscillations, and 1»/n; lies between 1 and 5. If n/n, is greater than 5, the expression: for ¢@ is found by multiplying by some power of 2. The ratio nofrry is obtained from the relation (2) = stut [(stu)?—4(1—k2)u]? ny stu—[(stu)?—4(1—h)u]? * Phil, Mag. vol. xxxvi. Aug. 1918. 62. Dr. McLachlan on Hfective Inductance, Effective From f=2000 ~ to f=10,000 ~ the value of L/L remains approximately constant. (See Table I.) Since C,/Ce is constant, it follows that u is constant. We will also assume that &? is constant. For a certain B.T.-H. inductor type magneto we have Cy = 55 picofarads (including leads). Cr 02s microkarade UT Taking s as unity, the following values of n2/n, are found for different values of k? (see Table III.). TasiE III. N»/N1. lig in) 0°84 d3 08 34 0:7 It is very probable, judging from the experiments of Taylor-Jones, that the value of 4? lies between 0°7 and 0°84. Assuming its value to be 0°8, the foregoing expression can be utilized to calculate gd. Thus we find that sind=0°93, and from (11) the value of Uis 1. In order to obtain V, we must substitute in (10). The individual magnitudes of n, and nz not being known, a difficulty arises with regard to the values to be assigned to L,, and L,.2, since both of the latter vary with the frequencies m, and ng. As a first approxi- mation we will take the values of L.; and Lo at a frequency of 6000 ~ per second. This gives I,,;=3°14 x 10-3 henry and L..=14'4 henries. Substituting the above values in (10) we find that the peak voltage in the absence of loss is V2=12,600 volts for a current of 2 amperes. With 2 amperes broken in the primary at full advance, the peak voltage was about 10,000. Thus due to the losses, (a) the reduction in peak voltage is 21 per cent., (b) the loss in energy is 60 per cent. of the actual energy given by 10,V2?=2°8x 107% joule. The reduction ins peak voltage found from the calculation in the preceding section worked out at 33 per cent., and the loss was 110 per cent. of 3C)V2?. Consider the conversion of the electromagnetic energy in the primary winding into electrostatic energy in the secondary. Apart from the loss. which occurs in both cireuits, we must take the coupling and other factors into account. Ina magneto or an induction-coil with a primary Resistance, and Selj-Capacity of Magneto Windings. 638 winding only, all the electromagnetic energy is dissipated in iron, copper, and stray losses. ‘The efficiency of con- version is zero. Suppose the secondary to have the same number of turns as the primary. ‘The secondary peak voltage and the self-capacity are extremely small, and there- fore the electrostatic energy represented by the expression 40 V.? is very small. As the number of secondary turns is increased, the electrostatic energy is also increased (apart from losses) and with it the efficiency of conversion, until it reaches a maximum value. With a given number of primary and secondary turns and a fixed primary condenser, Taylor- Jones has shown that for a certain induction-coil the peak voltage and, therefore, the electrostatic energy prior to the passage of the spark has an optimum value, when the coupling is about 0°57. This is due to the phase relationships of the two sets of oscillations, 2.e., their combined effect in enhancing the peak potential is a maximum for this degree of coupling. In the magneto it is not practicable to vary the coupling by withdrawing the primary core, but it can be varied by adding external air-core ind neHanee to the secondary. ‘Experiments by Morgan ™* show that the peak voltage can be increased in this manner. If at the moment at which the spark commenced, the energy of the primary was zero, the electrostatic energy obtained from the magneto per se would be the maximum possible. By reducing the losses this maximum would increase until finally the electromagnetic energy in the primary would be wholly converted into electr ostatic energ oy in the secondary There is another method of increasing the secondary electrostatic energy apart from that of altering the coupling. If a condenser is connected in parallel with the secondary, the product 3C2V.? increases with the capacity up to a certain point. For example, with a total capacity of 600 picofarads the peak voltage of a magneto was reduced from 10,000 to 5,000 at a certain speed. ‘The self-capacity of the magneto was 65 picofarads, so that the energy was increased 150 per cent. Unfortunately, however, the peak voltage, which appears to be the prime factor in ignition, was decreased 50 per cent. Since the primary current was the same in both cases, it is clear that the conversion efficiency was also increased in the same proportion as the energy. Returning to the example cited in section (7), the initial energy in the primary winding was not taken into account, * Morgan, ‘ Electric Spark Ignition’ (Crosby Lockwood). 64 - Dr. McLachlan on Hfective Inductance, Effective This can be done in the following way. The initial primary energy can be conveniently expressed in the form 4LI’, where L is some factor, usually termed the sel(ndaeeee and I is the current at break. It is difficult to give a pertectly definite meaning to L owing to the peculiar con- ditions which obtain in the magneto. The usual proceedure in measuring the value of L is to pass an alternating current through the primary. The conditions which then obtain, however, are very different from those just before the primary circuit is broken. It is impossible to state whether the value of L found by means of an alternating current is equal to that in the expression LI’. L in this expression would probably be found most accurately if the actual energy in the primary for given values of the current and a definite position of the timing lever could be ascertained. In the absence of data with respect to L, we will take the value as found by measurements with low frequency alternating current. Here again there is a discrepancy since the inductance varies with the current, so that we can only approximate by taking the “ainelene: of the armature at the point of break for a current whose maximum value is equal to the current broken in the primary. Knowing L, and |, at break *, also the peak voltage, we can approach the energy pre éilem and determine the overall conversion efficiency from primary to secondary, 7. e., the ratio $C.V2?/ZLaI,?. Imnethe Baie magneto used for these tests, L,, =6°5 x 10% henry at full advance break and I,=2 amperes when V.= 104 volts. Thus $L,,],7=12°5 x 10-? joule. From the calculation in the preceding section we have Energy lost in primary and secondary prior to spas +energy in spark = 3:1 10-24 2:8 L0n2— 5:9 105" joules Hence the energy unaccounted for is (12°5—5'9) 10°? = 6:6 x 105° joule. This is unconverted into electrostatic energy prior to the passage of the spark. Part of it is transferred after the spark commences, the remainder being dissipated in the primary and secondary windings. Thus we may, fol- lowing Morgan (loc. cit.), consider “that the energy, apart from that derived from rotation, is supplied from the * McLachlan, Proc. Phys. Soc. Dec. 1919. Resistance, and Self-Capacity of Magneto Windings. 65 primary in the electrostatic form up to the time of sparking and thereafter in the inductive form as a current in the secondary circuit. It is the former component which determines the ignition of an explosive mixture. ra) The overall conversion efficiency of the magneto is 280 57 22 per cent. poe now the figures obtained by using the formule of Tayl or-Jones, we find that the secondary energy, if there were no loss, is 4°44 107? joule, whereas the former method wave 09x 10%. (6) dffect of Variation in Inductance on Peak Voltage and Energy Loss. Apart from the reduction in peak voltage due to dis- sipation of energy in the primary and secondary windings, there is also a reduction caused by the diminution in IG inductance. It has already been shown that ue is nearly constant. The value of k does not undergo any very pronounced variation up to /=6000~; it decreases with 7, owing to the reduction in apparent permeability of the iron, but the rate of decrease is not very large. Fou simplicity & will be taken as constant. Moreover U sing is constant. Thus at full advance the ratio of the peak voltage oe unvarying inductance to that in the case cited Dice where /= 6000 ~, is approximately equal to Vv L..; to steady currents WV Li; at f=6000 ~ Oo = —=_ LAA. =/ er aw Hence the diminution in inductance causes a drop in the theoretical peak voltage of 31 per cent., provided of course the value of ny/n,; due to increase in “ %” does not increase beyond 5 and so alter the phase relations of the two oscillations which oceur after break. Substituting the direet-current or low-frequency values of the coefficients in (10), we find that the peak voltage is 18,200. In. practice it is 10,000, 2. e. 55 per cent. ol the ideal. . 2 af fa : ‘ \ hn Mag. s.o. Vols 41. No: 241. Jan. 1921. I 66 Dr. Mehachlan on Effective Inductance, [ifective Finally, if all the primary energy at break were trans- ferred to the secondary in the form $C,V,”, the peak voltage - would be 21,300. (9) Comparison of foregoing methods of calculating Loss. We are now in a position to compare the values of the loss in a magueto obtained by the two methods of com- putation. The first method yields 3°1x 1073 joule and the second 1'°6 x 107° joule. It is evident, therefore, that one or both of the methods or the assumed frequency is in error. It is.possible of course to select a frequency for which the methods yield identical results. In the experiment with the short-circuited exploring coil (see 6d) it was shown that the effective resistance of the primary winding was decreased owing to mutualaction. Now in a magneto under working conditions there is a mutual action between the primary “and secondary currents. The oscillations of lower frequency are almost in phase, whereas those of higher frequency are almost in opposition. Thus the magnetization will be increased in the first case but decreased in the second. This affects the effective resist- ances accordingly, and introduces an indefiniteness with regard to chee noutide under working conditions. It | appears, therefore, that without a knowledge of the precise conditions, the data obtained in the experiments is not of much value in the prediction of the loss. Owing to the variation with frequency the requisite values for substitution in expression (10) are unknown and can only be assumed. Hence on this basis both methods of computation are unreliable. The interaction of primary and secondary suggests a third way in which the problem of ascertaining the loss can be attacked. ‘The effective resistance of the primary is found by current variation as before, but with the secondary winding in position. In this way the reaction* of the secondary is obtained. The results are exhibited in fig. 17, which has the same appearance as the usual type of resonance curve. This curve may be termed a resistance-resonance curve, since the peak value of the resistance occurs at approximately the same frequency as that at which the secondary coil has its first self-resonance. If the mean frequency of the oscillations in the magneto is assumed, we can calculate the total primary and secondary * The reaction under damped oscillations is different from that obtained with undamped oscillations. Effective Resistance of Primary with Secondary in place (Ohms). 300 200 100 | Resistance, and Self-Capacity of Magneto Windings. 67 loss prior to sparking by method 1. The difference is ee the altered value of the resistance. In this way the loss works out at 5 x 10~° joule, a figure which is larger than either of those obtained from the previous paleouieiste ae. The high value is evidently a result of choosing a frequency in the neighbourhood of the self-resonance of the secondary coll. Fig. 17.—Curve showing effective resistance of primary winding at various frequencies, measured with secondary in position. a ) 2xl0° 4 6 8 19 12 14 Frequency — per second. Resonance of Secondary Winding occurs when f = 5500 ~ per sec. Armature in housing. Rotor position = after full advance break. With damped oscillations in the magneto the conditions are not the same as those outlined above. However, some experiments have been carried out which seem to indicate that a phenomenon similar to that described already oecurs in the magneto when certain conditions are fulfilled. This will be treated. separately below. Since no consistency was discovered among the three methods of estimating the loss. it was imperative to elucidate the matter from another point 2 68 Dr. McLachlan on Effective Inductance, Lifective of view. In Table LV. figures are given showing the ratio of the first maximum and minimum values of the voltage wave when the magneto is connected to a peak-voltage mea- suring apparatus ~ * li we assume an oscillation of the form Se sin wt, it is possible to caleulate the loss due to damping. Thus for the m agneto used in these experiments. X=0°92 @/7, and the ratio of the actual peak voltage to that when A=Ois 1:1°5. The value of X depends of course on the effective resistance and inductance under damped oscillations. It is increased by increase of the former and diminution of the latter. Hence X includes these two. effects and is not merely a measure of the total energy loss alone. The. energy loss calenlated from the above figures is 3,0 <1 Ope joule, a yalue which is of the same order of magnitude as that obtained by method 1. It will be evident from the foregoing computations that little advantage would have been gained by obtaining the various soothes of the circuits to a high deoree of accuracy. The above results amply demonstrate that in its present form the magneto (at least the particular type used in this research) is an inefficient apparatus for con- verting electromagnetic energy into electrostatic energy of high peak potential. The same conclusion was formulated by Taylor-Jones using a rotating armature type magneto, who attacked ae problem from a slightly different aspect. (10) Comparison of Loss with Armature in and out of Housing. With the armature out of the housing we shouid expect the loss prior to the passage of the spark to be a good deal smaller than that which occurs in the housing when the iron circuit is almost complete. Some approximate data are given in Table LV. illustrating this point. Ist maximum of voltage wave Ist minimum of voltage wave The damping = the ratio The values of A are calculated from equation (10) using the low-frequency inductances. It is of interest to observe that, using the same primary condenser, the iron only increases the peak voltage about threefold. The conversion efficiency, however, is trebled. * For method of measurement see McLachlan, Proc. Phys. Soc. Feb 1920. Resistance, and Selj-Capacity of Magneto Windings. 69 Taste LV. Rotor position = after full advance break. Conversion Peak Volts per ampere Se ene amiatrerenne : = I l Ratio B/A. | y Damping broken in primary. ° te Pani ORM Ara Teil? | Theory. | Exper iment. | A Bb. | | wees: Le Sas eee ao | Se gree hau = Se &0 eb = Be = aS BO eS Chel a) R . L . iv | 2 isa | . a0 = aD = ap = S Seah eee =| = 2 ‘= oO e S S S ) ol = 4 on = | ° | = foal D ra m = wo so esl whe D pis = — =) ts a G4 | me = = ra Se) ) 2) e) ‘e) c | = ~ o fo) 4 Qe ee S) = Oo ho 753 s Oo 9100 54°0 | 5000 3900 055 0-72 24 o2 29) 1 1-2 Wampime with aim-cored wimabure: ...6.. eee cess ence aeees = J Peak voltage per ampere broken in primary with air-core = 1250 Froin these figures it will be evident that the iron loss is a greater proportion of the initial primary energy with the armature in the housing than out of it. The most striking contrast is that of the damping, which is halved by removal Sree of the armature from the housing. It is of interest to note that with an air core, the damping is not much greater than that with the iron core. This indicates that the iron loss with the armature out of the housing is not very large. (11) The Hfect of an Avr-Core Inductance in the Primary Circuit on the Damping. Prof. Taylor-Jones * and Dr. Campbell f have carried out experiments on the peak potential obtainable from a magneto when an air-core inductance is put in series with the primary winding and a definite direct current supplied from an external source is broken. The secondary peak voltage under these circumstances varies with the primary capacity, and for a certain capacity it has a minimum value. This can be predicted from the theory of Taylor-Jones. It is the purpose in what follows to describe the effect on the damping of varying the inductance when the primary * Loc. cet. + Loe: cit. Voltage. Peak Secondary 70 Dr. McLachlan on Effective Inductance, Hifective capacity is fixed. There are two cases which arise: (a) the armature out of the housing, (b) the armature in the housing. The measurement of the damping, 7. e. the ratio of the amplitude of the first positive to that of the first negative oscillation, is conducted in the manner described ina former paper *. It should be mentioned that both the wave form and the separate frequencies of the two oscillations alter with the added inductance. Weareat the moment, however, only interested in the above ratio. Fig. 18.—Showing the effect on the Peak Voltage and the Damping of adding air-core inductance to the primary winding. A current of 1 ampere broken by using a separate contact- breaker. 6000 4500 Added Inductance (Millihenries). Armature out of housing, with core of stalloy sheets 0°38 mm. thick. Current broken in primary = 1:0 ampere. Some experiments of the above nature resulted in the curves of fig. 18. It is seen at once that not only do * Mclachlan, Proc. Phys. Soc. Feb. 1919. pam me re B30 Resistance, and Self-Capacity of Magneto Windings. 71 the peak voltage and the damping attain maximum values, but these occur at the same value of the added inductance. In the absence of wave forms, it is impossible to explain the phenomenon fully; but there is a similarity between the results and those of fig. 17. It appears, therefore, that when the phase relation of the two oscillations is such that the peak voltage is a maximum the effect on the iron of the core is to magnetize it in such a way that the damping is a maximum. Although the peak voltage is increased, it will be found that the conversion efficiency is practical! y the same with an added inductance of 2°2 millihenries as it is without this inductance. Similar curves were obtained with the armature in the housing, but the damping was much higher. By increasing the inductance sufficiently, more than one peak was obtained. Oneot the mostimportant factors in ignition witha sparking plug is the “rate of rise of voltage.’ Although the peak voltage fora given current is augmented by adding uncoupied inductance to the primary, it dee not tollow that this would be beneficial, since it is possible that it would cause a reduction in the rate of rise of voltage. Dr. Campbell has shown that, apart from the foregoing consideration, it would be impracticable to add uncoupled inductance to the primary of a magneto*. General Conclusions cn Losses. (1) The evidence with regard to iron loss caused by hysteresis and eddy-currents is such that its occurrence reduces the efficiency of the magneto considerably. (2) The reduction in the effective permeability of the magnetic circuit caused (a) by air- gaps, (4) by the screening effect of eddy currents, (c) by polarization due to tlie magnet especially at retard break, gives rise to an appreciable drop in peak voltage. (3) Apart from modifications arising from Tee in the coefficient of coupling of the two circuits, the detri- mental effects commented on in (1) and (2) can vie partially remedied by the use of thin laminations of the form specified at the beginning of the paper. (12) Self-Resonance of Secondary Winding. Owing to the large effective inductance and self-capacity of the secondary coil, its first resonance occurs at a fre- quency well within the acoustic range. Two methods ot * Loe. cit. 2 Dr. McLachlan on Lyfective Inductance, Effective obtaining the self-resonance were used, one of which has eleady. been outlined. The other method of obtaining the ralue of the resonant frequency is a combination of experi- ment and calculation. The inductance can be measured up to a frequency of 2500 ~ per second, and the self- capacity is measured by the method described above in (4). The value of the latter when the coil is shunted by a com- paratively large condenser (say three or four times the self-capacity of the coil) is assumed to be the same as that at frequencies in the fava bourhood of the self- resonance. Beyond f=2500 ~ the inductance is obtained by the method of calculation outlined previously in (3). Fig. 19.—Curve used to obtain self-resonance of secondary windine. A ; ne i a ee i 1000 =. «2000 3000 4000 5000 6000 Frequency —~ _ per _— second. xX = Resonance point. Ff = 5900 ~ per sec. Armature in housing; brass end-plates in place. Rotor position after full advance break. Thus, since values of Land C, are known for frequencies oui 2,500 ~ to 35,000 ~, the corresponding values of w’LeC: can be found. Plotting w?L.C. against frequency, the curve shown in fig. 19 is obtained. The self-resonance frequency occurs when w?L..02=1, 7. e. where the curve meets the horizontal line AB. Resistance, and Self-Capacity of Magneto Windings. 73 Referring once more to the curve of fig. 17, its utility is not confined merely to the magneto. The method of procedure can be adopted to gain some knowledge of the _self-resonance of any irom-cored coil. Since the resonance curve (obtained by varying the inductance in circuit (2) of fig. 2) is extremely flat-topped in the neighbourhood of the self. resonant frequency of the coil, the values of the primary effective resistance can be obtaimed much more accurately by a bridge method. It is highly probable that the curve obtained voli show subsidiary peaks indicating minor resonance points at frequencies greater than the first self-resonance. Some evidence of this was obtained during the experiments, since the apparent inductance of the primary was alternately positive and negative. It does not toilow, of course, that with an iron-cored coil the frequeucies at which these peaks occurred would bear any definite relation to one another. The self-resonant frequency obtained by the methods given in this section and in section (5) differs by 7 per cent. This is due doubtless to the calculated value of lL.» being low, since it was 5 per cent. in defect at f= 2500. As a matter of interest, the self-resonance of the secondary without an iron core was ‘investigated using the apparatus of fig. 2, and the value so obtained agreed with that calculated fom separate inductance and capacity measurements to about 1 per cent. (13) fective Inductance and Resistance under Damped Oscillations. Consider an oscillatory circuit such as that shown in fig. 20, in which the coil has an iron core. When the Fig. 20.—Oscillatory circuit with iron-cored inductance and condenser, Le. Re. condenser discharges there will be oscillations, provided ane . e >» . ] he the usual well-known condition is satisfied, 2. e. LC > 1L2" Je 4e 74. Dr. McLachlan on Effective Inductance, Effective Now the apparent permeability of the iron varies with the current flowing through the coil, and this causes a variation in the effective resistance, the periodic time, and also the shape of the voltage and current waves. It is essential, therefore, that in order to treat the subject at all, some assumptions must be made with regard to the above variables. The simplest method is to assume the per- meability and the effective resistance to be constant, with a given capacity in cireuit, from which it follows that the oscillation is of the form e— sin (@f + @). Let eee: L, = effective inductance, assumed constant. R, = effective resistance, assumed constant. C = capacity of condenser. Np ee Lp ve = voltage on condenser ai any izstant. * » inductance s | 99 he ,, resistance . r 7 = current at any instant. V j= maximum voltage on condenser. Then it can be shown in the usual manner that the Instantaneous voltage on condenser, 69) t= Vo(* 22 em sin (@t-+ 0): .. °. > =eeeneles Instantaneous voltage on resistance = vycR.(**2 ee qe ‘sin (wt + 7) . es Instantaneous voltage on inductance ay, = li, (di/dt) = V,Cu, (te sin (w@t—@); (5). where ’ he ae l Re = tan! o/X. and) 0. Regen Now the vector sum of the e.m.f.’s round a closed cireuit is zero: hence we can represent the e.m.f.’s of the circuit of fig. 20 at any instant as shown in fig. 21. Resistance, and Self-Capacity of Magneto Windings. 75 As the time increases from the instant the condenser discharges, the vectors are to be imagined to rotate with angular. speed w and shrink in accordance with the damping factor e~-. They also preserve the same relative angular BE cements as those shown in fig. 21. Fig. 21.—Vector diagram showing phase relations of voltages in circuit of fig. 20. VnL=V; in magnitude and the vector sum of the two is V;. When V; is zero or very small, 6=7/2 and Vi and V- are equal and opposite. It is clear that there is a difference in the definition of the terms L, and R,, if not in actual value, from that for undamped oscillations of the same frequency (see fig. 1). The experimental work to obtain the values of the above quantities for a magneto under actual conditions is extremely difficult, owing to the small amount of energy available and the very high damping. (14) Measurement of the Lfective Inductance and Lifective Resistance under Damped Oscillations. We will now examine a peculiar result obtained by Dr. Campbell and mentioned in his paper (see p. 385, loc, cit.). Using an oscillation method (damped oscillations) of determining the primary inductance, he obtained a value of 6°23x 107% henry. On inserting an air-cored coil of 1x10-? henry in series with the primary, the self- inductance was reduced to 5°78 x107°%. This is, of course, a physical impossibility since the total inductance should be 7:23x107° henry.. The apparent paradox can be 76. Dr. McLachlan on Hffective Inductance, Effective explained if we consider the effect of the iron loss on the natural period of the circuit. Consider the relation This ean be written = leew aS 0) 4 =(¢—b2)@. (> If we plot w? and x, the curve obtained is of the form shown in fig. 22, 2.¢. a parabola. Draw amy limegeee x, where x=1/L, Fig. 22.—Carve showing relation between w° and 1/L in an oscillatory circuit having resistance of appreciable magnitude such that it affects the natural period. Vob = YORE Va=CRe r=1/L, parallel to the a axis. This cuts the curve in two points A, B. ‘The value of ?, and therefore of w and f, is the same at A and B, but the value of w and therefore of L, is quite different. Thus, so long as 1/L.>4/CR.’ or CR < 4h, and the resistance and condenser are fixed, there are two values of L, which give the same frequency. When L,>CR.*/4 but Resistance, and Self-Capacity of Magneto Windings. 77 CR2/2, w? decreases with increase in inductance. Thus increasing the inductance will increase or decrease @ according as we are at B or A. Let L,= effective inductance of primary winding. L = inductance of air-core coil. C = primary capacity. @) = pulsatance for primary alone. @, = pulsatance for primary + air-core coil. Then from (10) we have a es Re: R. = 2b. -p-«"), . ° . . : a «lt "if z Peo bee of) 8) Since the values of w) and @, are not very different, the values of RK, from (17) and (18) are practically identical, provided the max. currents and the damping are equal i in both cases”. Hence, equating (17) and (18) and squaring both sides, we aphan L ; rule Ley meet) = (Let 2bL-+ l(a 07°"): from which we get a — Lo, 112 "@4 “Gel = Lo,2) (o°—0,°) L. = (19) In the case mentioned above, the capacity employed in determining the inductance is not stated, but we will assume this to be 1 microfarad. Neo! ecting thie effective o) resistance as Dr. Campbell did, we have 2 2 W, — Wy AL One Lc for the primary winding =a Coon lila) ale ol SCO re 1 OC eee ee LOL pillar ain-core coil) Pe Cee iy... ) == TRC ORES * The maximum current should (if possible) be such that the inductance, resistance, and periodic time do not vary appreciably as the oscillations die away. 78 ~=Dr. McLachlan on Effective Inductance, Effective Substituting in (19) for L, C, @p, @1, we find d= Z Seo? henry. Substituting in (14) for R, we obtain R= 10) ohms 7. @huseat ais quite evident that the effective resistance cannot be disregarded. If is taken as 0:1 iicro- farad, L, still has the same value but R, is now 240 ohms. The variation in R, is a result of the assumption regarding the value of C, or of the fact that the correct value of @ is not known. If the value used in the experiments were known, we should be able to find the correct value of R,. From the preceding analysis we can immediately suggest a method of obtaining R. and L, experimentally. The method is to find » or / using the primary winding, preferably with the secondary remov ed, and a eoneenen. The latter can be varied to vary 7. 7 1s then found when an air-cored coil, of such a value aa the former is not increased more than about LO per cent., is put in series. L, and R, are obtained by substitution in the formule given above. In order to secure accuracy the measurement of 7 requires careful attention*. The above rests on the assumption that the oscillations are damped sine waves, and that there is no variation of L, and R, due to the slight ehange in frequency. If we assume R,x@w and L,«1/o, we can obtain formule for L, and R. of the same nature as those already given. ‘This refinement is hardly justifiable. It is possible to examine these experiments “from another point of view. We have Poe eae or 4m’ CL? —4h,.4+ CR? = 0. —. ieee @” Assuming that R and L are constant, we have dag? Uo e —4L,+ Cok? ae 0, 40°C, b7—-4L. + Clee — 0, where Cy and C; are two values of the primary capacity as used in experiments in the above paper (loc. cit.). Solving these equations we obtain mt Re 7 A= ) 9 be ape eee hl-F * See Taylor-Jones and Campbell (doce. citt.). Resistance, and Sel/-Capacity of Magneto Windings. 79 If T? is directly proportional to C we have aN Gp CG In 2 ie SS eal) and since R, is not zero L, is infinite, which is ee ccble This, however, does not occur actually, since the line through the points obiained by plotting Cand T? does not pass through the origin. Eicon (20) we obtain |? C iy ee (1 ave "16m Le Thus if J?/C is constant we must have approximately (neglecting the intercept on the ‘I? axis) = R27? ols, (1 e SS! = 8 constant = slope of line in fig. 3 of above paper. Thus if L, is assumed constant and found from the slope of the line, as in Dr. Campbell’s experiments, its value is 22 approximately +a) times too large. Using the values of R,, w, and L, obtained previously, we find that ihemabove: tactor is 2°6. Hence the value of ,, viz. 6°23 x 107% henry as found by Dr. Campbell, is 2°6 times too large. It appears, therefore, that in finding the inductance of a magneto by methods in which damped oscillations are employed, the effective resistance of the winding cannot be neglected. The effective inductance and the effective resistance can also be obtained by varying the capacity in series with the iron-cored cell, instead of varying the inductance. Using the same symbols as before, we obtain a com Het 29 Oy, == LG, Abe ° e . . ° - (22) i R? or = Ch be 5 . ° e ° e (23) From (16) and (17) we find that Oo -O, 1 80 Inductance, Resistance, and Self-Capacity of Magneios. The difference in capacity Cy—C, should be small, so that R, and L, do not change ap ypreciably, and the maximum current dole be the same in each case. The frequency corresponding o to the values of L, aud R, may be taken as (fo+fr)/2, t.e. the mean of the two frequencies. It is not suggested that either of the methods outlined will give particularly accurate results. They will at least indicate the order of magnitude. in measuring inductances and resistances of iron-cored coils, the conditions are invariably complex, especially at low frequencies where the apparent permeability is susceptible to pronounced variation, and also at high frequencies if the magnetizing forces are large. More accurate results could probably be obtained by using the method shown in fig. 23. Damped Fig. 23.—Diagram of apparatus used for bridge method of determining sel-inductance and effective resistance under damped oscillations. Battery La Ne mmps li 1 ve Contact . Brea kerS* G sie oscillations are obtained using a battery and a contact- breaker. R and L are adjusted to obtain a_ balance. Then R=R, and L=IL., and the frequency of the system is given by Iron Cored Oifferentiral Transformer 2 Detector By varying C, different oe. ean be obtained. If the detector is selective e, the effect of harmonics can be examined. In place of fe battery and condenser, a source of undamped waves can be used to determine R, and L, instead of the methods used herein, as has been stated before. November 1919. Eee | IV. The Transverse Vibrations of Beams and the Whirling of Shafts pep onied at Intermediate Points. By EH. Rk. DarnLey * WEE HIS paper investigates the equations giving the periods of the transverse vibrations of uniform beams and the whirling speeds of uniform shafts, when the beams or shafts are simply supported at any number of “intermediate points (§$ 4-6). The remainder of the paper is confined to the case ane the ends also are simply sup- ported. New functions denoted by @ and ¥ are tabulated and graphed. The use of these tables and graphs considerably extends the class of cases in which numerical solutions can be obtained, and suggests that some of Dunkerley’s s results need revision (§§$7-10). A general theorem is given relating to symmetrical arrangements of the supports (§ 11). This theorem is similar to one relating to the critical loading of a strut, recently published by Cowley and Levy f. ‘The question of the distribution of the supports to give the vreatest value of the slowest period or whirling speed is discussed ; and it is shown that this period or speed isa maximum oe the supports are equidistant (§ 13). The general theory of the vibrations of beains of one section or bay has been treated by Rayleich in chapter VIII. of his book on Sound, and by Love in chapters XII. and XX. of his book on Elasticity. The former gives a detailed account of the vibrations of a beam in the six possible cases of terminal conditions, arising according as either or both of the ends are supported, clamped, or free, with very exact numerical solutions fer the periods, and a study of the shape of the vibrating beam and of the position of the nodes and loops in certain of the graver modes of vibration. The latter gives an account of the flexural vibrations of a circular cylinder from the point of view of the mathematical theory of elasticity, leading to a complicated frequency equation containing Bessel’s functions, which reduces, when the radius of the cylinder is supposed ‘small, to the equation derived from irchhoff’s theory of thin rods or beams, which is used by Rayleigh and in this paper. "The 4 theory of the transverse vibrations of thin beams is analytically identical with that of the whirling of shafts, regarding which reference may in particular be made to the con) papers by Dunkerley and Chree, quoted in the footnote. * Communicated by Prof. A. N. Whitehead, F.R.S. + Proc. Roy. Soe. vol. xciv. p. 405 (1918). Pil. Mag. 8. 6. Vol. 41. No. 241. Jan. 1921. G 82 Mr. HE. R. Darnley on the Transverse Vibrations In this connexion it has found: important practieal appli- cations. Dunkerley* and Chree } do not apply the strict Kirchhoff theory to beams or shafts of more than two bays or sections, but they deal with three bays and with many eases of loaded shafts by means of an approximate method suegested by Rayleigh, in which the form of the vibrating shaft 1s assumed to be geaned by various types of algebraic formule. § 2. For anaccount of the metho: of forming the equations of motion reference may be made to the, books by Rayleigh and Love. Let m be the mass of the beam per unit length, i its modulus of elasticity, I the moment of inertia of the cross-section about a diameter. Take the origin at one end of the beam, measure & along the beam, and let y be the lateral displacement, supposed small. Then, neglecting the effect of the rotation of elements about an axis pe erpendicular to the beam, the equation of motion Is d+y Ly ae ae es which must be satisfied at all points along the beam, and the following are the conditions at the ends and at supported intermediate points. At a simply supported end, the ordinate and the bende a1 ; 7 =0. da At an end fixed in direction, the ordinate and inclination’ moment are zero, 2. e. y=O0, and - di are Zero, 2. €. y=0, and a! (0), dx At a free end, the bending moment and the shear are d?y d®y Le ; =0, and ls == 0) At a simply supported intermediate point the ordinate is zero, and both the inclination and the bending-moment are ay i ty 5 continuous, 2. e. y=, and “/ and —% are continuous. ( dix dx ZOLO, Is e. * Dunkerley, “ Whirling and Vibrations of Shafts,’ Phil. Trans. A. vol. elxxxv. pt. 1 (1894). + Chree, “ Whirling and Transverse Vibrations of Rotating Shafts,” Phil. Mae. May 1894. of Beams and the Whirling of Shafts. 83 § 3. If it is assumed that 7 is of the form wcos (pt +e), ‘we have . du LO ae TIE, (ites pem and, putting aa I dtu | SW ery) an The solution of this equation is u=aeos Kx+beosh Kv+csin Ke +d sinh Ka, where a, b,c, d are constants to be determined by the end conditions and the conditions at the intermediate supports. The differential equation for wis satisfied at all points of the beam, but the constants of the solution will be different for each bay. § 4. Consider the case of a continuous beam of n bays simply supported at the ends and at (n--1) intermediate points. Let J, l,...4 be the lengths of the bays. Take the origin successively at the left-hand end of each bay. The equation giving the value of u for the rth bay will be of the form u=a,(cos Ka— cosh Kw) +c,sin Ke +d, sinh Ka, since a,+b,=0, because uw=0 when v=0. The further eqaations expressing the facts that w=0 and ada’? dx? a,(eos Kl,— cosh KU,) +¢, sin Kl, +d, sinh Kl, =0 —a,(sin Kl, + sinh K/,) -+¢, cos Kl.+d, cosh Kl, ¢,41 +d;41- a,(cos Kl,+ cosh K1J,) +¢, sin Kl,—d,sinh Kl,=2a,41. . are continuous at the supports are of the form From (1) and (3) by addition and subtraction a,cos Kl, + ¢,sin Kl-=a, cosh Kl,—d, sinh Kl,.=a,44, whence, if sin KJ, is not zero, ¢,+d,=a,(coth KU.— cot K/,) —a,41(eosech K/,— cosee N/,). Let coth K/,,— cot Kl,=¢(Kl,)=¢,, cosech KI,,— cosec Ki,=y(K,.) =... 84 Mr. HE. R. Darnley on the Transverse Vibrations Then Cp + dy =O,07—Gr41Vr, +. and, similarly, if sin K/,,; is not zero ; rp + O41 = Ors 1Pr41— O21 . (7) By (4) as ui E cos" K1,. c, cos Kl.+d, cosh Kl,=a,+; cot K/ “sin KJ, zs cosh? KJ. —~— Arty coth KE = KG and by (2) and (7) - Ave e e cos? Kl, =c,cos Ki,+d,cosh Kl,= — a,416,— cosh? K/,. Qi sin Kd, s whence 2 r+i(b> ap hr+i) =r 49W 41 +a, { (sin Kis cos? Ki, sin KJ, +(sinh Rp "yh 0, ‘sinh Kl, aw, et Ar+1( Dy ok Dr+ 1) a6 Ap oWr 41 =. or § 5. If the end of the first bay be free, the equation for this bay will be u=a,(cos Kx+ cosh Kz) + ¢,(sin Kx+ sinh Ne 7? 3 since at the origin 2 = and ae 5 are zero. Seder ii dx? The conditions at the junction of the first two bays are a,(cos Kl, + cosh Kl,) +¢,(sin KZ,+ sink Kl,)=0, a,(— sin K/,+ sinh K/,) + ¢,(cos K/, + cosh Kl,) =e, + dg, a,(cos Kl, — cosh K1,) +¢,(sin Kd, — sinh K/,) = 2a,, whence ~ ¢ 2(1+ cosh Ki, cos Kl,)) _ (a+ ds) $y Waly Osta Ki, sin Kd, 358 X1, Say. But C9 + d,= doh 2 sP2— A3Po. Hence 1 \ am) So Oi imams OX pe Goth) a $6. Equations givi ceiving the periods or whirling speeds can pow be obtained as determinants. These determinants are of Beams and the Whirling of Shajts. 85 similar to continuants. ‘When expanded they contain only even powers of the w’s, so that the negative signs of the latter may be changed to positive. They are obtained on the hypothesis that sin K/, is not zero for any value of 7. The solutions when that hypothesis is not fulfilled can, how- ever, be obtained by considering the limiting ratios of the functions involved when they become infinite, and these solutions are important in the case where all the bays are equal. Case I1— Ends simply supported. The period equation is Res 1X (Gy t Die l,)= dit Ga, day ae Vo, Pe ae Ps, i. | | oO Va» dst bs | be 3 Us Pn—2, tn 25 () | abn 29 Oye a+ ope ly ere 1| : 0, Vn ly On—1 + Dy | which can be evaluated by the formula LX; vrs (OS ae ay Nene a tery Newie mal 0 Particular cases: two bays, $,+¢@.=0;3 three bays, (bi + 2) (bo +3) — Wo? =0 ; four bays, (1+ $2) (ho + h3) (3 + bs) — We? (b3 + hs) — W3?(b1 + hz) = 0. Case II.—One or both ends fixed in direction. The period equation may be derived from that in Case I. by regarding an end fixed in direction as the limit of an extra bay when the length of that bay is indefinitely diminished. When both ends are so fixed the equation is Dig Wri5) 0 : , 0) wb, Pi a Po; Wo w Wo, p» ath bs | pu Brat Ons 15 Nie 1s y) : Ne 1» On—1-- Hn Vv, ben) Wns Pr» 8&6 Mr. E. R. Darnley ou the Transverse Vibrations Case III.—One or both ends free. The equation for the periods is obtained from that in a orb, by — , or both. When there are two buys, one with a free end, otherwise simply suppor’ ted, the equation 1 Is difo=1 OF xX according to which end is fr ee. § 7. The general nature of the functions ¢(x) and (a) will be seen from the tables and graph at the end of the paper. (wv) is zero when w is zero, and has poles (or infinities) when z=7 or any multiple of 7. $(a2)=#2z# nearly when a, expressed in radians, is small. g(a) exceeds this value by Case I. by replacing ¢, by — 7 ; less than one per cent. up to v= y oF 60°. - Near z=7,, - 1 er Ai (uv) tends to the value re and passes through infinity from + to —. When z is greater than 7, ¢(#) tends to the value 1—cota, exceeding that value by less than *004 when =r. b(.0) vanishes when #=3'9266 radians, or 224° 56' 58". At this point it exceeds (1— cota) by 0008, Similarly, (2) =$(5) — (wz), and is nearly —4a@ when « is small. At e=7, W() passes through infinity from — to +, and W(v)=(2) near oe where both functions are nearly unity. F § 8. The case of two bays simply supported is - easily solved approximately from the figure. A more exact solution can then be obtained from the tables. The period equation is ¢$;+@.=0, and it is convenient to draw the graph of the function —¢d for values of the argu- ment from 180° to 225°. The problem then reduces itself to finding, by trial and error, a parallel to the axis of « which cuts the graphs of ¢ and —din points whose abscisse are in the ratio of the lengths of the bays. Rhus, —o6 feet. ae 5 feet, we see from the graph that Kip i is about 145°, corresponding to 6x 145 4°95 From the tables we find } (147) = ‘5518, $(196°) = —2°4853, and a closer result, say 1953°, “so be arrived at by double: interpolation. Taking K/,=1952°, the result in radians is 3°416. Ki degrees=2032°. of Beams and the Whirling of Shafts. 87 For the next higher frequency, it appears from the graph that Ki, is a little more than 200°. From the tables (206°) = —1:0488, and from a table of circular functions (274° 40’) =1— cot 274° 40’, =—1:0816. Interpolation gives 2052°, 2742°. ee +] car The third frequency is given approximately by K/,=2873°, Ki,= 3832". The actual frequencies are in the ratio of the squares of the corresponding angles, or 1 : 1°96 : 3°82. Dunkerley gives, on page 297 of his paper already quoted, solutions of the problem of two bays for various . Rael integral values of the ratio i: The author’s results in these Dy) “a 88 Mr. E.R. Darnley on the Transverse Vibrations cases:differ considerably from those of Dunkerley, as will be seen‘from the following table :— TABLE I. 7 = . = Slat sal 1 Least value of K/, in radians. Value of A : I | Dunkerley’s results,| Author’s results. i 3°695 | 1 36056 36923 j ; 2:5101 | 3-643 j 3-3989 | 3°565 The results in the last column, except the result 3°6923, have been obtained by double interpolation from the annexed tables. The result 3°6923 has been obtained in a more accurate way by the use of the large tables of hyperbolic functions issued by the Smithsonian Institution, and indicates the amount of error to be expected in the use of the short tables. It appears that the method of approximation adopted Dyce by Dunkerley is accurate only when | is very small. h Dunkerley’s experimental results in these cases differed from his theoretical results by percentages —2°4, +0°5, and + 6°2, or an average of 30 per cent. The new results would apparently give differences of +2°4, +7:1, and +7°6, or an average of 5°7 per cent. When the bays tend to equality, it will be seen from the graph that the least solution tends to 180°. § 9. The equation for three bays is (di 7 >) (dy ae ps) aa We”, and it will be easy to find the least value of K in any particular case. Take anexease Of 1, > ljueilg— a2) cea If any solution Apeisihs for which aa is less than 7, ¢,; will be positive, and ¢, will be rumenieally, greater than ro Hence, to satisfy the above equation, ¢; must be negative of Beams and the Wlharling of Shafts. 89 _ E . oT ‘ : and Kil, must lie between m7 and a0 approximately. A few trials show that K/,=202° is a close solution. When /,=/;, the equation becomes - dit po= ir or hh thea j and &(Kl,) +o Ki) =0 J © 2 The first equation corresponds to symmetrical vibrations. The second equation corresponds to skew symmetrical vibrations with a node at the centre of the middle bay, being the same as that for two bays of lengths 1,, 3). These are special cases of a general theorem which is elven in § Ll. For three bays, the symmetrical case gives the lower whirling speed. § 10. The equation for four bays simply supported is (b1+ $2) (f2+ $3) (pst bs) — Wo bs + $1) —W3"(bi + $2) =0. When 1, 12, 13, 14 are unequal, it appears that this equation could be solved by a somewhat tedious process of tabulation, in the form rn cae heey See oe unless (6;+¢2), (63+ 4) can vanish simultaneously. If, however, the beam is symmetrical about its middle point, so that 1,=1, and 1,=1;, the equation reduces to the two equations dit b2=0 and = ($i + b2)p2—Wx? = 0. The first of these equations is that for two bays simply supported, and the second is that for two bays fixed in direction at one extremity. This is a particular case of a general theorem which will now be given. § 11. Consider the case of 2n bays symmetrical about the amadie, poimes so that’ (,—=t5,. lg—lo,-1, etc. Then by a 90 Mr. HE. R. Darnley on the Transverse Vibrations property of reciprocal continuants, Lv2n =s i ayy a sl Deas es ai) and ON — Ao ey 0) is the period equation for n bays fixed in direction at one extremity. Thus the periods for the 2n bays are the same as those for the system divided at its middle point when (1) the middle point is simply supported and (2) the beam is fixed‘in direction at its middle point. In the first case the modes of vibration are skew sym- metricai about the middle point, and in the second case they are symmetrical. When the namber of bays is odd and equal. to 2n—1, bya. theorem in continuants Xap = Be SN INF 1s and the period equations are L\n as A Nea = 0. Taking the upper sign, a single determinant is obtained which differs from A, only in replacing the constituent (dni +.) by (Pratdet Yn). Since b.+y.=$(SKi,), this equation reduces to Nth, ES ee (tine Bile) =b and represents skew symmetrical vibrations, in which, of course, tlie middle point is a node. Taking the lower sign, the equation gives the symmetrical vibrations. It has been shown in special cases that the lowest period occurs among the symmetrical vibrations. § 12. The case when all the bays are equal, the supports remaining simple, requires to be considered separately. Let there be n bays each of length /, so that nl is the length of the whole beam. The results for one and two bays indicate that solutions may be expected in the neighbourhood of the poles. Now the limit of L when Kl=sz is (—1)*‘, so that the period equation can evidently be satisfied at all the poles, and the a e . r ve least of these solutions is K= oe This is the well-known ease in which the beam vibrates of Beams and the Whirling of Shajts. On in the same form as a string would vibrate. It will now be: shown that all the other solutions are greater than 7 The period equation reduces to "sin na Ldlaesthes (() sine where cosae= | 8 Hence other solutions are given by sinna=0, sin «<0, leading to o=w cos where s is an integer, not zero or a multiple of n. Sr n?’ When the argumentis less than 7, 7 is always numerically greater than unity. Thus these solutions are all greater ms than T° When the argument lies between mw and the point near O1 = from unity to zero. Hence the next gravest period is given by where 6=0, is negative and decreases numerically o=—v dos. Thus for three bays o=—v COs SSS i, or 3o(K1) = SKI). This solution is about 204°. These solutions tend to the limit 7 when n is large. § 13. The question what is the best distribution of supports so as to make the whirling speed of a shaft of given length as great as possible is of special interest to engineers. It will be shown that when all the supports are simple the whirling speed is a maximum when the supports are equidistant. Consider the function Coy == ZN sin tC peim I ton) sim KK C,,. The determinant A, is of the second degree in ¢' and ¥,, 92 Mr. E. R. Darnley on the Transverse Vibrations but the terms of the second degree are of the form of (6°—,"), and when Kl,->m the terms of the second degree in cosec Kl, are of the form cot? Kl,—cosec? Kl, or unity. Hence, when KI/,->7, A, 1s of the first degree in cosec K/, and /(K) remains finite. (IX) is thus a’continuous function without poles. Let Kl=a where nl=1,+i,+...+l,, and denote by JC&)z the value of f(K) when Ki,, KJ,,... KU, all tend to z, and by f(K)o the value of 7T(K) when & tends to zero. ea aE 7 (K), and f()) have opposite signs, the equation f/(K) = FIG will have a root between — and zero. l Let Ki=7+ a, 1 pestig 0h pe Ty, where aj, d)...d, are small and a,+ a,+ ... +a,=0. Let cosh r=, cosech r=y. Then, as far as the second order of smalk quantities, FES n—a1; Ao. BOL, | 24 | il if, x 2y+—+ y— —, O Oy (9 5 Hey a le! i Jae De ia Ag As U3 (3 } | ieee 0, y-—, 28+ —+-, l3 is Ag if 1} >= r vt —— [as a Gn-1 Un—1 An Denote by ,D, the value of this determinant if # and y were zero, and by ,D, the value of the similar determinants where only Gr, Gr4tiy...@ are retained. Then by a property of determinants eae Oy A Opt tatne - + hs eka Gene ; The terms of the first order of small quantities in /(K), are Qjd9...dn - Dn, and vanish since a,;+d.+...+a,=0. To find the terms of the second order denote summation from r=p to r=q (pq) by = P The determinant may be expanded by Laplace’s method in a manner analogous to that given by Muir for continuants (Proc. Roy. Soc. Edinb. 1874, p. 230). of Beams and the Whirling of Shafts. 93. The terms of the second order are thus found to be Ti n—1 1 A{Ag...Un | 22 Ss alt) S rt1Da— 2y ~ Dy gle . =} ] 2 Yr: n—1 =? > (a, +...4+4,)(Gppi+ sae + Gn} n—1 —2y Ss (ay 52 se + Ap-1)(Ap41 ate nies + Gy) , n—1 = —9Ir > (a,+...+a,)? al n—-1 +2y > {(ay +... +4y)? —(a, +... +4,)a,} n—1 = —2(x—1) - (a+...+4,)? n—] —2y > ape + = a,a, (all unequal suffixes) ; n—1 = —2(coth r— cosech 7) & (a,+...+4,)? 1 — cosech (a? + do? +... + dn”), and this quantity is always negative. Now let & tend to zero, then @(KI/,)=3K1, and y(KI/,) = —3 KI, nearly. The sign of f(K)o will depend on the sign of the determinant 1D,’= L+l,, are: 41, 0, = 5 56 lo + ls, — $s, : 1 —4tl,-4, (psa as kis With a notation similar to that adopted for D, the terms of ,D,,’ containing J,” or l, are #1D,—1 ° poe Oe +t D,-1 : pAD YA +,D,’ ° sli De Hence, if all the determinants in this expression are positive, ,;D,’ increases when /, increases. When all the l’s are zero ,D,’ is zero. Hence ,D,’ is positive if ,D,' is positive for all values of r and s for which ,D, is of lower order than ,D,/’. Now ,D,’, ,D;', .D,' are positive. Hence by induction ,D,’ is positive, and /(K)o is positive. Tables of the functions ¢ and w. (The values between 5° and 44°, which are omitted, may be obtained by sinple addition, ¢. g. ¢(12°)=¢(10°)+¢(2°). Owing to the manner in which the tables have been constructed, errors of one or two units may oceur in the last decimal place.) Deg. op. w. Deg. ib: wv. i +0°0116 —0°0058 88 + 1:0623 —U°5491 2 0:0235 00116 89 1:0762 0:5972 3 0:0549 0-0174 aru te) 8) 1-003 0°5654 4 00465 0:0232 91 1:1046 05739 5 0:0582 0:0290 © 92 11188 05822 10 O0-1164 0:0582 93 1-1333 0:5907 15 O'1745 0:0873 J+ 1°1480 05995 . 20 0-2327 0:1164 95 1°1628 06084 25 0:2909 0-1454 96 as 0°6175 30 0:5493 ORL Sr 97 11928 06267 35 0:4077 0-204] 98 12081 06361 40 04662 « 0:2335 39 1-2236 0°6456 42 0:4896 0:2453 100 1°2392 0:6553 44 0°5132 0:2572 101 12551 06653 45 0-5249 02631 102 12711 06754 46 0°5366 9-2690 103 1:2873 0°6856 47 05484 0:2750 104 1°3038 0:6961 48 0-5603 0:2810 105 13205 0:7069 Oe 0°5720 0:2870 106 13574 0-7179 50 0°5839 0:2930 107 13546 0°7291 51 0-5958 0:2991 108 IS Welk 0-7406 52 0-6077 0°3052 109 13898 0°7524 58 0:6196 0°3112 110 14079 0°7645 OF 06315 0°3172 Ill 1-4263 97769 Dd 06434 0°3233 112 1-4449 0-789) 56 06554 U°3294 113 14640 0 8026 57 06674 0°3356 114 14833 0°8159 58 06794 03418 115 15031 0'8296 59 06914 0:3479 116 15232 0°8438 60 0°7035 03542 117 1-5438 0°8584 61 0:7 156 0°3605 DS ee eH OAT 08734 62 0-7278 03668 WS) 1°5862 0-8888 63 07400 0:3732 120 16082 09047 64 07522 0-3796 121 163806 0:9211 65 0:7645 0'3860 | 122 16536 0-9580 66 0:7768 03924 - 123 16771 09454 67 0-7891 0:3989 124 17012 09734 68 08014 0°4054 125 1°7260 09920 69 0°8138 0-4119 126 17514 101338 70 0°8262 - 04185 127 17776 10315 VAL 0:8387 0:4252 128 18045 10523, i, 0:38513 04319 129 1-8322 10739 ae 0°8640 0:4387 130 1°8607 10962 We! 0:8767 04455 131 18901 11195 75 0:8895 0°4524 132 19295 11437 76 0:9023 0°4594 133 19519 1-1690 Feel O-9151 0:4665 134 19844 11953 78 09281 04736 135 20181 12229 79 09411 0:4808 136 20559 12516 80 0-9542 04880 137 20893 1 23h7 Sl 0°9674 0:4953 sy tss 2°1269 13181 82 0:9807 0:5027 139 21660 1°3460 83 9:9939 0:5102 140 2:2069 1:3807 84. 10074 0:5178 14] 2-2496 14172 85 10210 0°5255 142 2°2941 1-4554 86 1:0347 0°5333 143 2°3407 1°4957 87 +1:0484 —0°5412 144 + 2°3896 -- 15383 Deg Q. 145 +2-4409 146 24949 147 2°5518 148 26118 149 26754 150 2°77 427 io 23143 152 28906 153 29721 154 30596 155 3°1535 156 3°2547 1d7 3°3642 158 3°4832 159 3°6129 160 a 1050 16: 39114 162 40847 163 4:°2776 164 44939 165 4°7383 166 5D'0169 167 53374 168 5°7103 169 6°1500 170 66766 gel 73188 172 81204 173 9°1491 174 10°5190 175 12-434 176 15°305 Li7 20°085 178 29-640 179 08'294 180 0) isl —56°286 182 27-632 183 18-078 184 ts208 185 10°427 186 8 Dd114 1&7 71414 188 _ 61186 189 Dro lle! 190 46687 1G 4°1420 192 3°7022 198 J o292 194 3:0086 G5) 27298 196 24853 197 2°2688 198 20758 199 1:9023 200 17457 201 1:6033 202 14733 203 1°3541 204 1°2444 205 11429 206 10488 207 —0°9611 LEST9 14-243 19-016 28°564 Sie lek oO + 57°383 28°737 19°190 14-417 ii oa8 9-6447 82820 72616 64814 58315 OIA +8800 4°5145 4-P014 3°9301 36934 a 270 + 10001 + 1-O180 Dee. Q- Ww. 208 -- 08793 + 2-183! 209 0°8025 DAT, 210 0:°7307 2:0a12 Zl 0°6630 2°0919 BL, 0°5991 1°9365 Pris EyO Dac 18847 Pl4 04815 1°8361 215 0°4270 1°7903 216 O37b4 (405 YLT 0 3260 1:7069 218 (2790 1:6688 219 0:2540 1:6329 220 0-1908 1:5987 221 0:1495 1°5667 Daye (1097 1-5360 yr) 0:07 16 1:5072 224 — 00347 1:4796 OAD +0:0008 1:4536 226 0:0342 14298 227 00682 14054 228 0: 1002 aol 229 0:1314 1:3618 230 - 01615 13416 231 0:1908 Leas 232 0-2193 1:3039 Zao 0:2470 1:2865 2384 0:2740 1:2698 235 030038 1:2539 236 03260 E238 75) PBI 0:3510 12244 238 0°3785 1:2105 ee, 0°3995 1/1977 240 04231 11851 241 04461 vee 242 0:4687 1°1619 243 04909 11510 244 0°3127 1:1409 2-45 0°53840 11313 246 05551 1:1220 247 05758 ales 248 0:5963 11049 249 06164 1:0968 250 00-6363 10897 251 0°6560 1:0826 252 0 6754 1:0760 253 06946 11-0698 254 0°7136 1-V640 255 0°7322 1:0588 _ 256 0°7509 1:0537 257 0°7695 1-0490 258 0:7876 1:0447 259 0-8058 10405 260 0°8239 1-O0368 261 O-S418 10835 262 O 8596 L-O305 263 0:8773 1-0279 264 O:8950 10255 265 0-91 26 10234 266 0:9302 1-0217 267 0-94.77 L-O204 P68 0:9652 L-O192 269 0-9826 L-O185 96 Prof. F. Slate on Electronic ; Thus A(K) will have a root less than ” unless ay, Qa tse are all zero. l Hence the whirling speed is a maximum when all the bays are equal. ; In the case of two bays the restriction that the a’s are small may be removed, so that it can rigorously be proved that the createst possible whirling speed is reached when the bays are equal, but this theorem does not appear to have been proved for any greater number of bays. The introduction of the function 7(K) is due to Mr. Berry, and the author has extended its use to any number of bays. V. Electronic Energy and Relativity. By Prof. FREDERICK SLATE *. HE modus operandi characteristic of relativity proves to. be working in disguised agreement with what Newton’s equations also teach, when these are properly written to include variable inertia ft. The older problem of motion resisted proportionally to the square of speed has furnished the principal key in this reconciling dissection, after its equation of motion was put into a form inspired by a prospect of making it a model for electrons. That problem is surprisingly imerwoven witht our attempted analysis. It allows aspects to be rated normal under its general type that had once seemed exceptional. Hspecially is this true of vital relations for energy, where correspondences show that are of first importance to recognize. The equation of rectilinear motion, and the recasting of it - with the aid of terminal velocity (v,) referred to are Ga) P—R=m,"; (6b) p= (3 ae (Remko%] 4 a (1) In (6) that factor occurs directly, whose different powers run through all the calculations of elatie ity. Much can be made to grow out of its striking property that links an ar feneetical and an harmonic mean: if: « 9¢@ ge . s 2 2Leay+v y—vl” Way tv/\o,—0 vy —v oF * Communicated by the Author. + Slate, Phil. Mag. April 1920, p. 483; July 1920, p.31. This isa continuation of those papers, carrying on their plan and their notation consistently. They are cited as (1.), (I1.) To avoid wasteful repetition, free reference is made to results already recorded there. * Seer Ga) vege oy. Energy and Relatwity. 97 The first member is onthe track of the root-idea in Hinstein’s substitutions; while the equality leads to the convexions of sums and products, or of quantities and their logarithms, which control the mathematics at many points. Therefore it is not astonishing to find relativity at one with the present argument in building the force (P) and the equation of motion (1) into the foundations. Hinstein’s velocities actually secure for (P) invariance in form and magnitude throughout the group of frames (U), so long as their derivatives based on a common time-variable are ‘expressed *. Only at the transition to “local time” are the reduction factors combined with this invariant nucleus, which are disentangled and discussed in the previous papers. The task for physics is to trace the simpler elements here plainly indicated through the shifting patterns of a rather kaleidoscopic algebra. Retaining at first the general symbol (v,) for a terminal velocity, let us notice the scope of equation (1), some peculiar consequences of it, and its limitations. It implies a total (gross) flux of energy due to the propelling force (P), this being partly diverted from the kinetic energy of (m,) by passive resistances summed in (R). Complete diversion (and conversion) can be entertained as a limit. Secondly, remark how the mass (m,), ‘‘ weight-mass” and constant in (a), is replaced as a factor of the same accelera- tion by a variable effective inertia in (6). But the physics of the original form is not modified by this merely mathematical step. In the third place, since the trans- formation involves a terminal velocity essentially, it is intrinsically invalid outside the range (0, v,) thus marked off, whatever magnitude (v,) may have. Imaginary com- binations which occur beyond those logical mits argue nothing against greater speeds attainable under other physical conditions. Specifically, this holds for light-speed (c) 1£ that happens to enter as a terminal velocity. or an interval taken conveniently between rest and any velocity (v), the equation of work in the standard frame (1) leads through en (1(b)) to the value 9 9 (a Dera N04" ore ii nieie vdt = —.—log| ——., R aes 2 dt 2 hey —v? M1047 pic tal ee myPr ? og 3) OO — ——|00 : : o viene = dv 2 5 adv \ m =m ahi ral * See (IT.), note to eq. (24). Phil. Mag. 8. 6. Vol. 41. No. 241. Jan. 1921. H 98 Prof. FE. Slate on Electronic Any dimensions and magnitude are allowable for (z). The above expression covers any constant inertia (mM) 3 and its eliminations have brought it to include any propelling force (P), provided that the conditions preserve the same terminal velocity *. Writing next Prdt= | (m, ") v dt + on, dt 0 at Jo vA) mv mive -myv," V2 a ge my (Clg. ee fm mag SY, = a Uy we see the same cancellation of explicit kinetic energy operating as in the work equation for the “deformable electron” of Lorentz +. Therefore that circumstance does not by itself require electronic inertia to be wholly electro- magnetic. The differences d\\ dv aD ; = SS ye e 4 dt Ny ER OR (4) record the diverted (converted ) energy and the activity of (R). As a mathematical device, introduce now a variable inertia Me Net at depend son (v.m,), have for (@=—= vj (m’). Let it depend upon (@, m,), have for (= 1e magnitude (m,), anda rule of change making dim’ mie 1 (= 1, Dy" By Eee Joo l= \= 2 0 ae dv eu e AS, 2? 2 ye é Then work and activity restated by means of (m') become an ty — dim We myer? log ( ™ ie oes sa ); ( my v7 ae EE ! SN “(es eee Gil dt y Consequently, for arbitrary (7), v v Im! n | (P — ) —v} clits ] =(); V1 A/ vy — ue dt Vv~— oe * See (IL.}, comment on eq. (9). + See (1.), p. 485. Energy and Relativity. og The identifications 7;=c, m,=mp, bring the first of equations (7) into formal parallel with results adopted for the Lorentz electron and enlarged for legitimate frames in relativity, with (m=vy(v)m) replacing (m'). One basic proposition of our treatment is thereby corroborated ; but also the question is opened, in how far (m) . a physical quantity. Conceding the physics of equation (1 (6)) to be artificial, an admixture of fiction in the subsequent dynamics 1s unavoidable. Nor is it dangerous, unless pseudo-values cause misconception of the physical process *. Return to equation (3), and observe the possible general reading of this work as a kinetic energy, measurable by a changing multiple of (m,) at a standard speed (v,). The multiple is abet under all choices of (z); yet the influence of (z) elsewhere may leave its determination an important detail for each separate application resting on equations (1, 3,4). The vogue of the contractile electron hhas led to particularizing = aa (v) zy 2P=moy"(v) ae wll mM, /o—v? Y my dt WO OH CU ak TO gsi or (8) But the unique advantage of that choice remains a matter for deeper inquiry; at least one alternative will be con- ‘sidered presently, as well as the foundation on which decision must stand. The permanent factor (v) is a resource in making certain forms valid indifferently, whether inertia be constant or variable, after standardizing all momentum and_ kinetic energy at the terminal velocity by appropriate reductions. Define two ‘‘ reduced inertias”’ (4, w') corresponding to any (Mo, Vg) by PO=My_5 pv’ =mv,”; showing p= = (9) 1 ‘These reductions are similar to those of foree-moment and ‘moment of inertia in relation to axial distance. Further, it is evident how they expand the idea of equations (5, 6), and move toward making broadly representative what was invented more specially for the Lorentz electron. * See (IL), p. 36. The same thought recurs at equations (22, 27) ‘below. The reminder is in place, perhaps, that no pretension to close any such issue is here intended. On the contrary, the aim everywhere is to open some escape, where earlier conclusions might be fallaciously -confident. Ey 2 100 Prof. F. Slete on Electronic Then tanvential force and power are expressible gene- rally as d ds TAs ere | ap = — (Tints) =) == - ) MasLmE yy 5 — eee - ' T= ap miata) vy Te 7 i (Mmvo7) = vy di’ (10) Also activity enters into the forms, since pv, = myto, 1 ( dy" fee ‘); Lap Avy Val =V1\ V1 =, Oli ae ee 20 LN ele aamatie | rp ay, : dvs SUD =1,(Tm a) = ae ie 5 . (11): i The last member, vanishing for constant inertia, affords. some general measure for that rate of conversion—mechani- cal energy into other modes. ‘The first of equations (11) is the plain equivalent of es fon) dvs di (A OL a), (12) Qin Me Ove at aan ane) Ov, dt the partial derivative e being taken with (mg) stationary at its RE ie When (my) is constant, this partial becomes. total ; the principle of ws viva appears. Under a fictitious supposition of such constancy, the last member may be viewed as containing contribution from some (pseudo) potential to the work of other external force. For any epoch when (w=0) there are particular coincidences which, like equations (23) below, have bearing upon the use of rest-frames. An additional suggestion from the algebra associated with terminal velocity is to utilize a kinetic estimate of future action as potential energy, according to the conditions, for intervals beginning at rest and at (v ) me dv mye? ae m CT ep 1 tee Oi — — = =| 0 my dvu=— (v2 — ov) SUE i dt / 2 2 : as Then mv" Sell wu Fat ie OU; ee adv _m s nantes “OU; ds ee Bee os Me which prepares the way for a possible consolidation with electrostatic potential energy. Energy and Relativity. 1OL The specification that (P) should account completely for the influx of energy makes equations like es ds ‘one a total derivative and the other integrable. But after having recognized that forces (P) can present themselves through | artial derivatives of momentum, a discrimination about | ie: integral is seen to be needed there. Before clear physical! indications have fitted together acceptable values of momentum and of energy, tentative factorings of (P) place it in either class as a derivative of momentum. Note how the complete activity (vP) is compatible with these groupings and others: eae ( vy" a) = ( MO =) : = Ne a ak vy —v*/ dt ‘i ( MV} jie =| {my )( Vy =) BW 2 0 eee Note oneal eg ote, Oy tes : ‘ Se eee) The last member is guided bY equations (10) ; the third, fourth, and fifth members make GE} ira partial dletivative: The supplement to (P) changes with the possible pair (mz, V2) segregated ; that is, according to the more plausible selection of momentum (or quasi-momentum). This flexi- bility is added to that afforded by the zero-parenthesis and arbitrary factor in equations (7) and elsewhere. In effect, a liberal leeway in quantitative adjustment to experimental data is permitted, while equation (1) is retained as proto- type *. One turning-point is found to be the discovery of a “seale-factor ” (z) that meets some condition connectable with the energetics of equations (11) ; when (v,, m., T) are oD furnished otherwise : for example, making v2( my 72) =02{P— me) Mee tae) In determining the physical elements for a cycle of ess to which the sequence of equations (1) to (16) can be adapted, the unifying feature being the terminal velocity (v,), some margin of independent choice at equa- tions (1, 16) and at equations (3, 4, 7) can be foreseen. =a VPs aes... (10) * Compare previous remark on this matter, (II.), p. 39. 102 Prof. F. Slate on Electronic Rigid dynamics has used us to the couple, which affects (rotational) energy but not momentum. Moment of momen- tum there bridges the break; its factor (mw) appears in kinetic energy also. Both (@, v,) can be employed to standardize (the former at unit axial distance). That thought falls into line with equations (9, 10, 11, 12), and with several obtrusive analogies to symptoms of a hidden spin about the axis of advance, which the electron’s dynamics: reveals. The foregoing statements have been kept on general ground (for any (v,)), with a definite purpose of insisting upon the probably, unrealized range within which that is feasible ; but we take up finally the electron for which (v;=c). The dominant criterion on the~-physical side is yielded by a predetermined (or available) electromagnetic energy (M). ‘The Hinstein velocities have been brought under similar control by considerations based on activity *. The dynamical schemes will look towards devising a momentum (()’) reducible to consistency with (M) and its time-rate, when associated with the actual working-speed (v). The remark following equation (17) enlarges some- what the limits of that consistency; but the points of agreement to contrive and their companion conventions may be put thus : 1 : A'= a) =r"; VEnv; T= s (Q'); 2E/=0Q’; | I Cd Ioan dv adm le ae _— -——— iv! Sie aenpee = Cy eeee a4 2 OTS cee a eee | "30 a ta Adding an assumption about effective inertia (m ) to our knowledge of (v) fixes the other (mechanical) quantities ; hence any attempted adjustment of this sort stands or falls mainly on that as a cardinal choice. The last equation enforces the need, where (my) is variable, of upholding the distinction between (A‘) and power (dE!/dt) which equations (12) formulate. Turning first to Abraham’s electron, we shall illustrate how the plan outlined above can be adapted to his original development. That assigns a leading place to “ electromagnetic mass,’ as we know ; also it proves to conform to the present train of ideas, though as a whole they are not made explicit in it. Particularly for that. reason, the comparison is profitable. * See (I.), p. ART (II.), eq. (18), ete. Energy and Relativity. 103 Begin with Abraham’s value * M=2 moc" [clog = —1] ; and therefore gs 10 A gala Ria 2 fl Uae c c+v\] dv a -— (M)=T’=2mye E a log( ] . 2 oR 2 \ Ce dt | Hstablish a junction with equation (16) through its first or its last form; either shows (P) as the total derivative of an auxiliary momentum v0 ry : als = ( Pd =m'v=m, pet aa [Slog (S=")]. Gaal Z 6--vV ee 09) The last member being already standardized at (c), the plan of equations (9, 10, 11) gives 20 my = hha Tings (aa ) = 50 = log { - oD ome a 20 Neue en), (0) 2K = 5 log Fae PP (ae ee pee 42) (ze ie mac (+ Ne jam = = 5 On a= i oe = lo jo = Ce O06 emt NC a) dt dt Mark how the last member returns to one leading idea of equations (4, 6, 7). The scale-factor (¢'') that equalizes this rate of work- conversion with what equation (19) demands is then conditioned by dm" 3m i din'' i 2 / Mf 2 0\* ! D) ¢ Beno —— |) =v’: ig’: ( ric ea 21, = 3M | « ( a ) f 2m, J v’ at Len 0] Bai ae Reh 7). In the last equation, due choice of the arbitrary ratio (m/m,) has reduced one coefficient to unity. The corre- sponding activities in equations (6, 7, 22) exhibit, then, as sole essential distinction the effective inertia whose time- rate occurs in each one—a remarkably significant symmetry. Why should it include the Lorentz electron without par- ticularized values beyond (vy,=c; my=nw)? The answer * Translated into our notation, but unaltered otherwise. The quotation is from his ‘Theorie der LElektrizitit,’ vol. u. (1908). The chief equations referred to are (113, 113c¢, 118d, 117, 1l7a, 117 6). The numbering of equations is fortunately unchanged through a series of editions. est-mass (m,) coincides with the usage of Lorentz-Minkowski, allowing for “rational units,” ete. 104. Prof. F. Slate on Electronic ean be traced from the fact that this electron appropriates the advantage of the general relation in equations (5), in combination with equations Be Since («) vanishes with (7), (m'’) depends upon an excess above initial magnitude ; and beens the final result involves time-rate alone of (m'’), any special value (mp '') can be used in expressions like i 4 m!' = mg '+m,'’; giving correctly on = a 2S) dt dt The algebraic link with Abraham's treatment may now be closed by simple verifications. First, th at the “longitudinal mass’’ (m,) and the “transverse mass’? (m,) satisfy the relation z If mee = (m, +0 = (me) - os “en) = es) and thus stand’ squarely upon the completed second law of Newton. Secondly, that the separate time-rates of magnetic enerey and of electrostatic ener gy, as Abraham allots them, mateh exactly the bracketed segregation of activity In equa- tions (18), when (it, = my). ‘And fin: illy, that Abraham’s “Lagrange function” ((l) of his equation (113)) belongs to the | lan of equations (12), contributing, in fact, the par tank ee Ges there. A second auxiliary momentum, also derivable from equa- tion (16), shunts the calculations towards the Lorentz electron. Using (m’) consistently with equations (5), define | QQ,” = [may (@) | (ey) | = me’ = [my (v) |v. - (25) In respect to (Q,’”), (P) is a partial derivative ; from either factoring it proves that d / ! i Mae Bee Gian v ON oP(1+ 5) Stan eA lv = (0) Hae 2B) By simple reduction this yields the forms Mo Mm, dv dv mc+v' dv mdu , (v) a oe =m 5 eo oe my 2rdt dt 2 C=v kt 2 dt Since it is true that an me? ao m @+vr? dv ( m - Ae e—v dt 2° @—v? dt 2 dt dm din == 7) Pe —_ 2 =o at ) v( dt iE ; ay Energy and Relativity. ; 105 equations (8, 26, 27) aré in one aspect mere algebraic variations. Nevertheless, it should arrest attention that the last term in equation (27) is now transterred among positive contributors to energy- -tlux, instead of figuring as a negative quota of resistance. The ty pe prescribed for the first member of equation (1 (0)) is better maintained; separate record is now entered, however, for each branch of a com- posite process * Hquating the alternative mated pairs of terms in the total derivative of equation (25) provides another direct con- nexion with equations (6, 7), and the possible rearrangements of the oo items present fair substitutes for some of relativity’s energetics. Omitting those details as nearly obvious, follow up the more vital comparison which parallels thai of equations (19, 22). On usual assumpuons, electro- magnetic energy, its time-derivative, and the assignment of ac tivity to magnetic energy and to electric energy are Ty: m dv 4 di’ “(My = eee c ae) ga |t [FC a) F : J) a dv ae = + de) mv dv aia 9 ater leet) Tay M,=2inc? + 4m? ; Ty = 8 qa =T,+ Sade Wa AU ae Ge he Meee S229) Ca Hxamine more narrowly the second member and the third of the last equation. The former is co-ordinated instructively with the subdivisions in the bracketed member of equations (18) which Abraham’s electron fits. One stabilized asso- ciation of magnetic activity and power is common ; and so is the type of the (electric) remainder, if ((dm/dt),) displaces (dm/dt). The iast member of equations (29) adheres to the primary conception of equations (1) to (7); the final sub- tractive term harmonizes with their supposition of partial ‘conversion into other energy- forms of work done by (P). The kinetic energy of any ‘“ weight-mass”’ (m,) must be rated mechanical. Perhaps the current doctrine that the * The last two members of equation (28) and the verifications under equation (24) give the foundation for a previous claim (1.), first lines of p. 486, + See, for example, Richardson’s ‘ Electron Theory of Matter’ (L914) pp. 217 sqq. 106 Electronic Energy and Relativity. inertia of the Lorentz electron is wholly electromagnetic may be insecure—at least if equation (1) be made funda- mental. In certain respects, the perfect conversion that Abraham’s. equation of motion predicates (and that our test does not disallow) may be said to introduce the limit-case (m,=0). The consequent perpetual balance of the forces replacing (P, R), not uniquely at the terminal velocity but through ail stages of approach to it, was atits date novel *. Nevertheless. the main issue is not decided there, because we deal with excess only beyond an initial value. A nucleus of mass. for an electron is not yet barred until some inherent contra- diction due to it has been clearly located. The disguise of mechanical energy as electromagnetic 1s. favoured through equations (13, 14). The effective deduc- tion of (im,/4) that appears in the developments for both electrons can be reconciled numerically with a composite rest-masst. Any “ weight-mass” is presumably external to that conversion of energy in the electronic problem, which the scale-factor must seek to cover. A preliminary inclusion of associated mass must be made inoperative in the rectified estimate of electromagnetic activity. The elastic con- formity of one set of equations to varied physical suppositions is strongly marked in all these ae and in other combi- nations Jeft unmentioned that correct algebra permits. Such uncertainties must continue until new evidence decides them. One pertinent inquiry will concern itself with the quantity (y(v)), whose integral powers figure so prominently. Is. this series naturally commensurate with intervals that the physics determines ? University of California. * Abraham, foe. ext. p. 129, eq. VI.; p. 144, eq. (95a). Also, on grounds of formal resemblance, it was sometimes misunderstood to- be nothing else than Alembert’s idea. + The additional “ potential energy’ required by Abraham’s argument Suites in magnitude, within the velocity -range (0, c), with what the particular scheme of ‘equations (27, 28, 29) would set aside as “ pseudo-- conversion.” Compare Abraham, Joc. evt. pp. 189-95. Also the sug— gestion added to equation (12) above. i Oe VI. An Electromagnetic Theory of Radiation. By H. BATEMAN *. 1. QOME interesting suggestions with regard to the nature: of radiation have been made recently by Sir Joseph Thomson { and Leigh Page f. According to the first writer, radiation of energy seems to be generally associated with the formation of closed lines of electric torce which travel away from a source with the velocity of light. In Page’s work radiation of energy is supposed to be caused by a rotation of the field of an electron, and it follows from his expressions for the field vectors in a rotating field that this field may be obtained by superposing on the non-rotating field of an elec- tron a radiant field in which circular lines of electric force travel outwards with the velocity of light. Page’s work is particularly interesting because the amount of energy radiated in one complete revolution of the field is almost exactly 1, hv, where v is the frequency of the rotation and fh is Planck’s constant. In order that v may be identified with the frequency of the emitted light, it seems necessary either to assume the existence of discrete tubes of force or to extend Page’s analysis to the case in which an electron revolves or oscillates with its frequency of rotation. A peculiar feature of the radiant field is that there is a radiation of magnetic charges which travel along straight lines with the velocity of light §. There is, however, no magnetic charge associated with the electron, and so it seems likely that the magnetic charges arise from a distribution of magnetic doublets which in turn arise from moving electric doublets. Whether this is the case or not, the mathematical analysis suitable for the formal development of this theory of radiation is akin to that which has been given in the present author’s speculations regarding the electrical nature * Cominunicated by the Author. + ‘Engineering,’ vol. 101. p. 38] (1916) ; Phil. Mag. June 1920, p. 679. t Proc. Nat. Acad. Sci. March 1920, p. 115. § Aclass of radiant fields in which there is a radiation of magnetic particles in particular directions was found by the present author, Proc. Nat. Acad. Sci. March 1918. It is probable that Page’s radiant field can be built up by superposing a number of fields of this older type. It may be remarked, too, that circular lines of electric force can be obtained in fields of the older type, and in this case it is possible for magnetic charges which are equal but of opposite sign to be emitted in consecutive directions. Circular lines of electric force of opposite senses are then generated in approximately the same region like vortices having different senses of rotation. 108 Dr. H. Bateman on an of the ether, and so it may be worth while to present the analysis In a compact form. 2. Ina former paper * an attempt was made to describe the motion of the lines of force of an electric pole by means of a succession of infinitesimal transformations which trans- form light-particles into light-particles. If for simplicity we consider only transformations which correspond to rigid body-displacements in Minkowski’s four-dimensional space, the path curves or trajectories of the transformations may be written in the form duw=dt | &(1)+¢e(t—7)p—(y—n)ht+(e—f) a], ‘eval dyu=ade | (7) + c(t—7T)q—(#—0)f +(2—e)hy, | (1) dz=dr| (7) +¢(t—t)r—(a@—£)g+(y—m)F |; : edt=dr|le+(«w—E)pt+(y—n)qt(e—f)rj, J where §.7, 6, f, 9: h, p,q, 7 are functions of tr. The infini- tesimal transformation associated with a particular value of t is supposed to be applied to space-time points (a, y, 2, t) which satisfy the relation [a=-€(7) 2+ ly—a(r) P+ eee) =CC es and also to certain space-time points specified by equations of type C= he NST... 7). pa (X,Y, Zap 9 y= POCXS Yo Z, 7), b= Wo Xe VL: ae ene. which may be regarded as solutions of the differential equa- qos (1 ae ye and Z being constants of integration or invariants for the sequence of transformations. The equations (3) may be supposed to give the co-ordinated motions of the different points of an electron of which (€, 7, €, 7) 1s a particular point which we shall call the focus. ‘The lines of force of the electron are supposed to be generated by light-particles fired out from the different’ positions of this focus 8, the direction of projection of the light-particles associated with one line of force varying in a manner indicated by the succession of infinitesimal transfor- mations. In fact, if (1, m, ) are the direction cosines of the line of projection of the light-particles emitted at time 7 we have dl Jp PT mh + ng —Ulp + mg +nr). : Se * Proc. London Math. Soc. (2) vol. xviii. p. 95 (1919). Electromagnetic Theory of Radiation. 109 3. Let us now consider the electromagnetic field specified by the vectors E and H whose components are given by equations of type noe FO (es mi, OG. Tt) 3 : eg Hwee enenye SO 2) ne Or, 1)” Mo=(a—€)at+ (y—n)B4+ (e—C)y +4, By kee (ae ea ean) Y. (6) AU eee (nes tee te | agit tales IS el ra where a, B. y, 6, X, mw, v are functions of 7, and ¢ is the velocity of light. Writing ee Vin a ag lan) CEST SCA anny a 1 ; we see that if oO cat yl —vy’ +m(cv— an! + BEY) —n(cu—rvyé' + af) —I[ lica+ wf’ —vn’) + m(cB + ve’ —NO') +n (ey + An! — pE’)]. (7) we may write E=- = qa les—v_—c(t—7)s|, H=sxE, . (8) where a, w, v, and s denote the vectors with components (a, ise Y); on b, V), (&. ii) > Gy) ce mM, n) respectively. The rate of radiation of energy in the direction s at a very great distance from § is approximately ye re w(3r) oy and this is positive except when s does not vary with 7. Both a and w must be zero for there to be no radiation. The equation (7) is identical with (4) if cat pC’ —vn' =op, ci—BE +yn' =—SH/. These equations generally determine the ratios of e, 6, y, 8, N,v uniquely in terms of f, 9, h, p,q, r- To make the electric charge associated with S a constant quantity, e, we write 4 he : é dard = CM Ui E” —*—€"), LILO Dr. H. Bateman on an and the field is then completely determined when the motion of the electron is known. 4. The part of the field depending on the function @ represents the radiant field added by Page to allow for the effect of rotation. A moving line of electric force in this field is given by the equations @=constant, t=constant. When ¢ is constant the lines of electric force associated with a given value of 7 are cut out on the sphere (2) by the planes through the polar line of a line which meets the sphere in two real points A and B. The directions SA and SB are “associated ’’ directions of projection, the points A and B being collinear with the point T which would be reached by S at time ¢ 1f it continued to move with the velocity which it has at the instant of time 7. Ast and 7 vary the points A and B trace out two associated lines of electric force. These may be regarded as examples of the guiding lines of force of which Sir Joseph Thomson speaks. ‘The other lines of force circle round one of these lines. The lines.of magnetic force being orthogonal to the lines of electric force are circles through pairs of points such as Aand B. It should be noticed that the field vectors in the radiant field may also be expressed in the form 9 Oo a cere ay, oe rae Ae ~where P(#—£) + Q(y—n) + R(e—0) —SG—7) A =|] oO SSeS = — 2 aN 2) 2 ee 8B? SUsqvjit, BVataoey, | Swoon rA=VR—-WQ, w= WP—UR, y=UQ—VP, AB v= Vp The lines of magnetic force are given by @=constant, +=constant, and are traced out by particles which travel ee Ege e ua 63 along the radii for which wow and 7 are constant. There are two such radii for each set of values of ¢, w, and 7. The line joining the positions at time ¢ of the magnetic particles which travel along these radii passes through a fixed point Q(t, 7) given by the equations A=0, B=0, M=0. The polar plane of Q with respect to the sphere (2) divides the . positive magnetic charges on the sphere from the negative mn Hlectromagnetic Theory of Radiation. IGE magnetic charges. The volume density of the magnetic charge may be ‘written in the form SO Oe ee Cane = D(a, ¥, 2) We (M(v . w) + (¢ U ) 6}, and this is equivalent to the expression given by Page when we write ec (, vy w= (1-5 ja, where @ is a vector representing the angular velocity of the ee or its field in the case when v=0. The rate of radiation of energy in a field of type (5) is ed to be iL _9 9 7 : (fe sea = = Shy en Oe e+ OU yan) + See) + tan) where L=m/(c?—v?)? reminds us of a Hamiltonian function used in the theory of relativity *, m being a constant of the ‘nature of mass. If a is zero in the case of an electron this may be in con- sequence of the equations of motion of the electron and it may be necessary to extend the expression for o by adding to. d : | (Se ) a term representing the effect of the external elec- 0& tric field, and possibly also a term 3; to allow for a variation of m with position. The complete coefficient of 2 —€ is then zero on account of the equations of motion, and the complete coefficient of t—7 should be zero on account of the energy equation. This modification of o makes the introducnae of a radiant field depending on @ seem fairly reasonable, because the coefficients in the numerator of 0 may be regardet d as com- ponents of angular momentum when the denominator has been suitably modified. It is possibie, too, that there should also be terms to represent the effect of ‘impulsive couples which act only at isolated intervals of time. These impulsive couples may, perhaps, be produced by magnetic particles emitted from. other sources. 7. It is, perhaps, a little hard to understand why the electron should turn completely round a finite number of times when emitting radiation. It may be that a particular line of force is naturally associated with some particle at a very great distance from S and that the line of force is eventually recaptured by this particle after it has been once torn away. ‘This wo uld account for the return of the line of force to approximately the original direction. A further difficulty in the present theory is that when an * The eee ne expression appropriate in Einstein’s generalized theory of relativity is easily written down, but the expression for rin terms of x, y, z and ¢ is not given by such a simple equation as (2). Angular Momentum and Properties of Ring Electron. 113 electron both rotates and revolves, different lines of force generally revolve at different rates when a=() and there may be only one line of force whose angular velocity corresponds with the frequency of the light. This may, however, be the particular line of force to which we have just referred. ‘To make all the lines of foree revolve with the same angular velocity it seems necessary to make a equal to the acceleration as in Page’s analysis, “put then a non- radiating orbit is an impossibility. 8. According to the present theory the idea that all electromagnetic fields occurring in nature can be built up from the fields of electric poles travelling with velocities less than ¢ seems to be untenable ; it seems almost necessary, in fact, to adopt a more general type of electromagnetic field as fundamental, and it may be that the simple radiant fields described by the present author will turn out to be the most suitable. VII. The Angular Momentum and some Related Properties of the Ring Electron. By H. Stanuey Aven, 1.A., D.Sc., University of Edinburgh *. The Angular Momentum of the Ring Electron. HORTLY before the war the late Prof. S. B. Mclaren was engaged in writing on the magneton f, and it is greatly to be regretted that in consequence of his death on the Western front on 14 August, 1916, the apyelien ions of his work to the theory of complete se Ghiaioia, spectral series, and the asymmetrical emission of electrons in ultra- violet light were never published. Rejecting entirely the idea of magnetic or electric substance, he regarded the magneton as an inner limiting surface of the ether, formed like an anchor ring. The tubes of electric induction which terminate on its surface give it an electric charge, the magnetic tubes linked through its aperture make it a permanent magnet. He found that the angular momentum of any such system, whatever its shape or dimensions, about its axis of symmetry is (87’c)-1N.N,». Here ¢ is the velocity of light, N, is the number of tubes of electric induction terminating on the surface, and N,, is the number of tubes of magnetic induction passing through the aperture. * Communicated by the Author. { Phil. Mag. vol. xxvi. p. 800 (1913); ‘Nature,’ vol. xcil. p. 165 (1913), vol. xevii. p. 547 (1916). Phil. Mag. 8. 6. Vol. 41. No, 241. Jan. 1921. I 114 Dr. H. Stanley Allen on the Angular Momentum No proof of McbLaren’s theorem appears to have been published, and it may be of interest to show that the result can be obtained in a comparatively simple way. Let the position of any point in the ether be referred to cylindrical coordinates, taking the axis of symmetry of the magneton as the axis of zg. Since each Faraday tube is rotating about this axis, the motion of any portion of a tube is at right angles to its length, and the equivalent mass per unit volume of a tube is 47r4N?, where N is the electric polarization or displacement at the point. The angular momentum for unit volume of the tube is consequently AnruN?r?o@, where w is the angular velocity with which the system is rotating about the axis. The motion of the Faraday tubes produces a magnetic field in a direction at right angles to their length and to the direction of motion of magnitude H=47Nro. Hence the angular momentum for unit volume of the tube can be expressed in the form wH?/4arm or Sy, Le Oy) ce 7 Pye TH SL Thus the total angular momentum = — eg where the TT summation extends over the whole space external to the magneton. But XwH?/8a represents the amount of energy associated with the magnetic field, and it is easy to show by considering the energy as distributed in the magnetic tubes that this is equal to $12’, where L is the coefficient of self-induction of the magneton and 7 the current flowing round it. Thus the total angular momentum =Li?/o. But Li=N,,, the number of magnetic tubes linked with the magneton, and i/w=e/27=N,/27, where e is the charge on the magneton. So we obtain finally as the total angular momentum of the magneton 1 9,7 NaNe The difference between this expression and that given by McLaren arises from the fact that he employed rational units. | It is easily shown from a consideration of the dimensions of N,, and N, that the product has the same dimensions as angular momentum both in the electrostatic and in the electromagnetic system. The proof given being perfectly general applies to any magneton of the type considered, whatever may be the shape and some Related Properties of the Ring Electron. 115 of the cross-section, or the size of the cross-section or of the ring. Further, it would seem that the same result would hold not only for a surface distribution of electrification, but also for a volume distribution.as in the case of the ring electron, which is usually looked upon as a circular anchor- ring of negative electricity rotating about its axis with large velocity * 3 Sir Joseph Larmor 7 has pointed out that one or more classical electrons constrained to move round a channel would be like an amperean current. The same method of proof might be applied in such a case, leading to the above expression for the angular momentum. Angular Momentum and Planck’s Constant. In his paper on the constitution of the solar corona, Prof. J. W. Nicholson { first introduced the concept of a natural unit of angular momentum, finding such a unit in the quantity h/2a, where f is Planck's constant. ‘This is the quantity which appears in Bohr’s theory as the angular momentum of a “bound” electron. If we identify this unit of angular momentum with the angular momentum cop) of the magneton, we find Unit angular momentum =//27=N,,N,/27- This identification gives the remarkable result h=N,,N,, or Planck’s constant 1s equal to the product of the number of tubes of magnetic induction and the number of tubes of electric induction associated with the magneton. Assuming Nee, he natural uni Or electric charge, which is equivalent to identifying the magneton with the electron, we find h/e=Nin- Taking e=4:°774 x 107” E.S.U. and h=6°558 x 10~/, this gives NnaHl ie Oma ho UW vrorseet2OselO=* H.M.U., mecumns,¢—2°999 x 10 cm. per sec. * Parson, “A Magneton Theory of the Structure of the Atom,” Smithsonian Misc. Coll. vol. Ixv. me 11 (1915). + Proc. Phys. Soc. vol. xxxi. p. 68 (1919). t Monthly Notices, R.A.S., ane ‘1912 2 I2 116 Dr. H. Stanley Allen on the Angular Momentum The Relation of Lewis and Adams. From their theory of ultimate rational units Lewis and Adams * deduced a relation between Planck’s constant, h, and the electric charge, e, of the form 1d hPc? = 87? (Ame)®, or tn PE SN a ales where e¢ is in electrostatic units. Using the notation employed in an earlier paper?f, in which [ discussed certain numerical relations between elec- {ronie constants, ) x (Are), hcg =2 Tre’, 4 ; where g is a pure number and =7:28077.. x 107°. It is worthy of remark that the numerical factor 877/15, which occurs under the cube root in the expression above, may be written as 487a, where 4 galt te pe F108 B, being the second Bernoulli number = 1/30. The relation given by Lewis and Adams receives very strong support from the e: cperimental determinations of the constants involv ed, as has been shown by R. T. Birge § ina paper on the most probable value of Planck’s constant. He gives the mean value of h as 6°5543+0:0025 x 10777 erg. sec., assuming e=4:°774x 10-" and c=2-9986x 10”. The caleulated value he gives as 6°560 x 107”. By combining MeLaren’s result with the relation of can) ne and Adams we find ms 87° ame N,,ec= v7 = x (Aire)? or j in electrostatic units. Hence 3 87? : Qare N,,.c= a Ce) Nallggre vor ame ey) q * Phys. Rey. vol. iii. p. 92 (1914). + Proce. Phys. Soc. vol. xxvii. p. 425 (1915). In Sommerfeld’s papers on the fine structure of s pectrum lines in the Annalen der Physik for 1916, the constant which I ee med g is denoted by a t Planck’s ‘ Heat Radiation’ (Masius) § 160. . § R. T. Birge, Phys. Rev. vol. xiv. p. 361 (1919). - throug and some Related Properties of the Ring Electron. 117 The left-hand side of this equation gives the magnetic flux gh the aperture of the magneton in maxwells, and we see that the flux in maxwells is equal to the electron charge Gn H.S.U.) multiplied by a numerical factor. Let L' denote the self-inductance of the ring in electro- static units, and n the number of revolutions per second of the electricity in circulation. Then N,,=L'ne, and rma a) ( Or, if L denote the self-inductance in electromagnetic units, 3 S 5 9) Lnje= Ce i) << Gae ON Zulby 15 gq It would be interestin @ to know whether a toroidal surface could be found which would require a factor of the form Ae (5) in the formula for its self-inductance. It is usual to assume that the ring electron is of the form of an anchor-ring of circular cross-section. Webster * has pointed out that, in order to account for the observed mass of the electron, it is necessary to suppose that p, the radius of the section, is very small compared with a, the radius of the circular axis. ‘Ihe self-induction of electric currents in a thin anchor-ring of this type has been investigated by the late Lord Rayleigh +. When terms involving ie square of p/a are neglected, “the formula for the self-inductance is 5) Lom Or. L=4ra [log =i, p where & has the value 2 when the current is limited to the circumference of the anchor-ring, and the value 7/4 when the current is uniformly distributed over the cross-section. In the present problem the term log 8a/p is so large that the difference between the values of # is inappreciable. Substituting this value for L in the previous equation, we find , S- 4qrna 8a 2 Sir Asti 20 log — —A\= a/ = Sage PO loa or a. C p 15 q * Webster, Phys. Rev. vol. ix. p. 484 (1917). + Rayleigh, Roy. Soc. Proc. vol. Ixxxvi. p. 562 (1912), 118 Dr. H. Stanley Allen on the Angular Momentum Now 2mna=v, the velocity of the electricity as it travels. round the ring. Hence 3 (a5 OT e OE =~) x 8m? v 1d T os 8a ae log — +k Q [log — 2 p 2p Since the peripheral velocity, 7, must be less than ¢, the : a velocity of light, log Be —k must be greater than 5 (80\ ee which gives ine greater than 433, showing that p is. excessively minute compared with a. The Mass of the Parson Magneton, and the Radius of the Ring. The electromagnetic mass of the Parson magneton has been investigated by Webster, who found an expression of the form m= —,- log —, - TOU in which the small term & has been omitted. This leads as before to a very small value for the radius of the cross- section of the ring, and it follows that most of the energy and momentum of the field are concentrated very closely around the ring. Lo obtain further quantitative information as to the properties of the ring electron, one further quantity must be known, and we shall assume for this purpose the value of the radius a given by A. H. Compton*. He finds a= (1°85+°-05) x 10-29 em. on the basis of an investigamnd of the scattering of X-rays or y-rays by the ring. Taking m=8'999 x 10-*> om., we ae log . =k) = 2064, which satisfies the condition that log = 8a seat be greater than 433. The value of v/c is 433/2064, so that the peripheral velocity is about one-fifth of the velocity of light. * Phys. Rey. vol. xiv. pp. 20, 247 (1919). It is well to keep in mind the possibility that the radius. of the ring may assume various values, as in the case of the radius of an electron orbit in Bohr’s theory. and some Related Properties of the Ring Electron. 119 By combining Rayleigh’s formula for the self-inductance with Webster’s formula for the mass, we find which, cn substituting numerical values, gives L=4:80x107° E.M.U. Quantitative Results. In the table following are collected the most important numerical constants for the ring electron. ‘The results in the earlier part of the table are based on Millikan’s value for the electron charge, and on the value c=2°999 x 10" cm. per sec. for the velocity of light. The mass is calculated on the assumption that e/m=5°305 x 10" H.8.U. The later results assume the estimate for the radius of the ring a=(1'85+°05)x10- cm. given-by A. H. Compton to be correct, It will be noticed that the strength of the current flowing round the ring is relatively enormous, being nearly one ampere. In consequence, the strength of the magnetic field at the centre of the ring is nearly 3.x 10° gauss. Ata point on the axis at a distance equal to the radius of the ring the strength of the field is about 10° gauss, and at a distance equal to the diameter is about 2°6x 10° gauss. Such intense local fields have been postulated in certain theories of atomic structure. TaBLe.— Properties of the Ring Electron. Property. Numerical Value. Basis for Estimate. PORTER CE. o's oe ove ddiieears vee + 4°774.<10719 B.S.U. Millikan’s experiments, Planck’s constant, h............ 6:°558xX10-7 C.G.8. Lewis and Adams. Angular momentum ......... 1:04410—-27 C.G.8S. h/2z. BRAS GIG UK, 5.202 3.4 oa. owns Wax tore a Aas 4120107 *° H.M.U. DOES, DAE e Aaa ea 8999 x 10778 om. e/m. Magnetic moment, ..,......... 9-232 x 10-7! E.M.U. 3(e/m)(h/27). ACUSTOL TING, Z.0.) 5.002. +. 1:85 x107!" em. A. H. Compton. ANCES GIL IQ Og, Se aa Sa Bea 1:075 10-19 sq. cm. ' za?. Flux density (average) .:... 3°83 x10!" E.M.U. (h/e)+(xa°). Belfinductance )........0.... 4-80 x107© E.M.U. Rayleigh’s formula. Peripheral velocity, v......... 6:29 x10” cm./sec. w/c. PMOOUENEY:, 20 \ Vy > 1}. * Bohr, Phil. Mag. ser. 6, vol. xxx. p. 412 (1915) gives a discussion of what should constitute the quantity in the brackets. Apparently his conclusions lead him to one, viz., the first term in the brackets, but he produces evidence from the separate work of Moseley, Kossel, and Barkla for the inclusion of all the terms as they appear here and in subsequent expressions. Throughout this paper 1 have followed the latter method, but at the conclusion it is shown that a better agreement with the theory can be obtained amongst the results if we assume that the energy required to remove a ‘‘K” electron to a point of zero potential with no kinetic energy =Avg, the atom readjusting itself subsequent to the removal. (See also O. W. Richatdson, ‘ Electron Theory of Matter,’ p. 509 (1916).) be Thin Films of Elements exposed to Réntgen Rays. 23 According to this scheme it must be noted that the so-called ““K” electron is the slowest one, whilst the most rapidly ejected electron is the one that comes from the periphery of the atom, 2. e. the so-called photo-electron *. In a previous paper t I pointed out that the absorption coefficients of these particles in various gases derived by Beatty and Sadler must be regarded as mean values. If the above scheme is correct, not only has the mean to be taken over the various speeds, but the matter is complicated further by our not knowing thé relative numbers possessing the respective speeds. As one might say, we do not know the relative intensities of the respective 8-ray spectral lines. I also produced experimental evidence in support of the above views. Although it has often been stated that. there is a definite “KX” oroup of electrons associated with “ K ”’ X-radiation, and an “l” group associated with ‘ L”’ X-radiation, yet I have been unable to find any experimental determination as to which group will have the greater velocity if the parent atoms are emitting simultaneously both “K” and “L” X-radiations. The nearest approach to any definite statement was inade by Sadler, who pointed out that under certain conditions intense electronic emission from a screen was not necessarily associated with the emission of any (tertiary) X-radiations. that could be detected. ‘‘ For instance, when the secondary exciting beam from silver itself falls upon silver as tertiary radiator, no homogeneous Roéntgen radiation” is produced, and yet a considerable emission of corpuscular radiation occurs ”’ {. Finally, some extremely important deductions have been made from the work quoted above, e.g. it is deduced that the maximum velocity of the Q-rays is never greater than that of the parent cathode ray within the discharge-tube. * It must be pointed out that this statement is not contrary to the experimental results of Barklaand Shearer, who excited first K radiation in a given screen by the incidence of a still harder radiation and then L radiation in a screen of higher atomic weight, by the same radiation (too soft to produce the K radiation in this latter screen). They then studied the maximum velocity of emergence of the 6-rays from the two screens and found them to be equal. These experiments support this conclusion, but an attempt has been made to push the subject farther. Concomitantly with IK emission, a given screen must be emitting L, M, etc. X-radiations, and what one calls the “ K” electron 1s probably that emerging from the atom with the smallest velocity of the groups of electrons associated with the emission of the various X-ray spectral lines. + Trans. Roy. Soc. S. Africa, vol. viii. pt. 1, p. 82 (1919). t bid. p. 354. 124 Mr. L. Simons on the Beta-Ray Emission from In the experiments described below some attempt has been made to find a solution to the following problems :— 1. Is the maximum speed of ejection of an electron from an atom entirely independent of the nature of that atom and dependent only upon the wave-length of the incident radiation ? 2; Apart altogether from the question of the diminution in speed “of emergence of those §-rays which have their origin in the “deeper layers of the material, what is the precise nature of the distribution of speeds amongst the electrons emitted from the parent atoms when these are exposed to X-rays? How is the distribution affected by (a) the nature or the parent atom, (b) the wave-length of the incident radiation ? If the existence of a 6-ray “spectrum” can be proved, what is the interrelation between each G-ray spectral line and the corresponding X-ray spectral line ? Apparatus. The method employed throughout was that due originally to Beatty in his study of the absorption coefficients of the 8-rays. A few modifications were introduced. It was thought that the emission would be simplified by using only very thin are of the various elements. These were As, Se, Zr, Ag, Sn, Sb, Ba, Au, Pb, and Bi. The X-rays which were incident upon these screens in turn were the so-called homogeneous X-radiations, first from silver and secondly from barium *. ‘The primary beam of X-rays was produced by a standard Coolidge tube, and every endeavour was made throughout the whole of the work, a period of about eleven weeks, to keep matters eonsanl The parallel spark, between point and plate, backed up by the tube measured four inches, and the current in the heating spiral was adjusted so that the point and plate were always ore! on the point of sparking. This current was four amperes throughout. The barium secondary radiator was a flat cell of BaOs, the front of the cell being tissue-paper, the back card. When. this was used an extra aluminium sheet 1 mm. thick was inter- posed between it and the ionization chamber and other apparatus in order to eliminate the characteristic L radiation * The fact that the so-called homogeneous K radiation consists of a series of lines of different wave-lengths, whilst rendering-impossible accurate measurements by this method, does not seriously interfere with the main conclusions drawn herein. An attempt is being made to employ really homogeneous radiations and to overcome the necessary mechanical difficulties. Thin Films of Elements exposed to Réntgen Rays. 125 from Ba. When the silver secondary radiator was used, the front of the ionization chamber, being aluminium °75 mm. in thickness. was sufficient to AISNE ithe: L radiation from silver. The Sereens.—These were prepared in the following manner. A brass disk, 12:7 cm. in diameter and 0-17 cm, thick, had a brass rim soldered centrally upon it. The rim was 10°2 cm. in diameter and 0-4 cm. deep. In this manner a flat cell was produced 10-2 cm. in diameter and 0:4 em. deep. Mleven such cells were prepared. These cells were filled with paratiin-wax, the surtace being scraped down flat and level. with the upper edge of the rim and polished with natural graphite. To obtain films of Au, Sn, and Ag electro- deposition from the double cyanide was employed. A brass cylinder, open at both ends, about 10 cm. in diameter and about 7 cm. high, was stood upon the graphited wax plate w (fig. 1). The inside of this cylinder was waxed so as to Bigeed prevent contamination, e is an ebonite block. By an arrangement of an indiarubber ring sealed round the edge of the cylinder, and an annular tinfoil electrode ¢ stuck on the indiarubber ring, good electric contact could be made with the syeieuile surface. The electroplating solution was then poured into the cell, a suitable anode @ dipped in, and deposition started on 8 and judged by the eye. In the case of these three substances, the films were uniform and s 126 Mr. L. Simons on the Beta-Ray Emission from thin that the graphite backing could easily be seen through them. Films of the other substances were obtained in the following manner. It was necessary to obtain an adherent conducting surface in each case. The surface had to be conducting in order that the electrostatic capacity of the ionization chamber, of which the surface really formed the back, shouid remain unaltered when the various screens were substituted for one another. This conducting surface had also to be made of some material of low atomic weight from which there is no electronic emission. ‘To obtain a degree of uniformity, the graphited wax surface was placed horizontally, face uppermost, and an inverted electric hot- plate was brought down for a few moments to within a distance of about 0°5 cm. from the surface. The graphite surface cracks and the paraffin-wax comes up through the fissures of retreat. If the process is watched carefully and is stopped at the right moment, the whole surface is con- ducting, as tested by an electroscope, and it will take a powder in a way that polished ee cannot. It also remains quite plane. Pure As, Se, Bi, and Sb were very finely powdered in a Pott ne lie htly dusted over the prepared plates, the larger grains being wiped off and the plates being vigorously tapped i in order to get rid of any particles not properly adhering to the wax. Microscopie examination showed that each plate was covered with a uniform layer of minute grains. No estimate of the size of the grains was made. Each plate was tested for electrical conductivity. Barium peroxide was similarly dusted ona plate for the Ba screen and red-lead for the Pb screen. An eleventh screen was made containing a flat sheet of carbon (cut from a dynamo brush) instead of ‘the usual parafiin-wax.. Finally, each disk was provided with three screw-legs round the edge outside the rim, and all the parts except the prepar ed screen and the back of the disk were coated with a thin layer of wax and rubbed with graphite. A small brass conical gauge-piece was made 1°45 em. high, which was placed on a sheet of glass and the screen placed over it. The screw-legs were now adjusted until each part of the surface of the screen just touched the apex of the gauge-piece. The screens were then ready to be placed in the ionization chamber, but before doing so each was placed for ashort while under the receiver of an air-pump in which the pressure was maintained at about 2 anni. Thin Films of Elements exposed to Réntgen Rays. 127 The ionization chamber is shown in fig. 2. To the bottom of the brass cylindrical case a sheet of aluminium (a) ‘75 mm. thick was sealed on with Chatterton compound and fixed with a screwed-on brass ring. The heavy top was ground Yl RRQAAN << Se aur ot RRS foe Ss SS ea AGC ee flat for easy removal, the sealing material in this case being a soft mixture of bees’-wax and vaseline. A pure carbon plate (c), about 11 cm. in diameter and 1:5 mm. thick, was mounted flush within a flat brass ring; this ring with its plate rested on three small chairs round the bottom of the case, and when a screen was put in, its legs rested on the brass ring. On the ring there were three small sulphur beads (2), which supported an electrode connected to a Wilson electroscope, the electrode being made of radial carbon filaments. Finally, all the brass parts within the case were painted with a mixture of sugar-water and artificial graphite. Lead diaphragms were arranged so as to keep the X-radiation from falling anywhere but within the prepared surface of the screen. The only gas used throughout the experiments was air, and the pressure within the chamber could be varied from a few millimetres up to atmospheric pressure. Before passing into the chamber the air was roughly dried by bubbling it 128 Mr. L. Simons on the Beta-Ray Emission from through sulphuric acid. A fresh supply of air constantly passed through the chamber during the experiments. Throughout the first set of experiments the silver plate was used as the secondary radiator. First the carbon screen was placed in position, the pressure within the ionization ebhamber adjusted, and the mean ratio obtained of the ionization in it to that in a standardizing electroscope 12 to 14 points on the curves being obtained within the range from zero up toatmospheric pressure. Another screen was substituted and the process repeated. ‘The screens were not used in any definite order. The barium secondary radiator (with its accessory alumininm plate for cutting out the L radiation) was now substituted for the silver one, and the whole process repeated, again using the eleven screens in no definite order. Heperimental results. Fig. 3 (Pl. IL.) gives the points obtained in using the eleven screens successively with silver as radiator, and fig. 4 with barium as radiator. No corrections whatever have been made in putting down the points, the air effect and the normal leak of the instruments being so small that they could be neglected. The ordinates of the carbon curve were stnmacie from these of the other curves. Hach resulting curve would represent the ionization produced in air at various pressures by the @-particles emerging from the respective screens. There is a critical pressure at which the range of the 8-particle is just equal to the thickness of the ionization chamber: viz., 1°45 em. Above this pressure the curves become horizontal straight lines, for the @-particles are now completely absorbed in the air. Below this pressure some of the energy of the particles will be lost by their being absorbed in the carbon front of the chamber. The resulting ten curves, say with silver as secondary radiator, differ widely from one another in their maximum ordinate. In other words, the total ionization by the 8-particles in these experiments shows a considerable variation over the range of substances used. It must be remarked at once that this variation here has but little meaning. It is known that the intensity of the @-ray emission increases with increasing atomic weight of the screen, but apart from this, as the films of substances radiating were so exceedingly thin to a certain extent of variable “thickness, but by no means thick enough to absorb to any extent the @-particles produced from the side of the 1 OR Thain Lilms of Elements eaposed to Réntgen Rays. 129 material in contact with the graphite surface, no importance ean therefore be attached here to the fact that the total ionization observed differs widely for the various screens. Figs. 5 and 6 (Pl HI.).—For the purpose of comparing the curves along their non-linear portions, each was scaled up to the same maximum ordinate*. The result is shown in fie. 5 (Pl. IJII.), which might be called the cathode ionization curves from the ten screens, silver as radiator, and in fig. 6 (PJ. II.), barium as sadeehon. In the paper previously referred to, the writer has shown fully ho ow the first derivatives of the curves shown in figs. 5 and 6 (Pj. III.) give the density of the ionization in a ch ec: containing air at 0° © and 76 cm. pressure at a p 2A 16 ay 3+0 is the pressure at which an ordinate such as AB is drawn, 1-45 em. the thickness of the ionization chamber used, and @ the mean temperature of the experiment, about 24° C. in this ease. This expression las been used in order to convert values of p into corresponding values of the distance from the screen as measured in air at O° C. and 76 cm. pressure. The ordinate AB (=N,) measures the total ionization by B-rays in air (at O° C. and 76 cm. pressure) in the region from the screen up to an imaginary layer 0-1 cm. from the screen. Fig. 7 (Pl. IIT.).—When the curves of figs. 5 and 6 (Pl. III.) were first drawn, there seemed to be little order about their sequence, except that in fig. 5 (PI. IIL.) the curve for the silver screen was lowermost) andl ty flo. 6 (Pl. III.) the barium curve lowermost. In order to show up any sequence, fig. 7 (PI. III.) was drawn. This figure has for abscissee the atomic numbers of the elements of ae screens, whilst the ordinates are taken directly from figs. amo (El NI) as follows. The length CB fig. 5) represents the total ionization by (-rays in air (at 0° C. and 76 em. pressure) included within the region oa an imaginary plane 0-1 cm. from the screen up to the front face of the chamber ; it is therefore a measure of the 6-ray energy crossing this plane, and equals (N,—N,). The line Oe at the 0: Lem. mar k, is drawn right across the 10 curves and its various lengths raised above the respective element distance x 1:45 em. from the screen, where p * The simplest physical interpretation of such an adjustment is to imagine the number of atoms at the surface of each screen so packed that the B-ray emission from each screen gives rise to equal total ionization in the air of the chamber for complete absorption of the particles. é Phil. Mag. S$. 6. Vol. 41. No. 241. Jan. 1921. IX 130 Mr. L. Simons on the Beta-Ray Emission from placed in order of atomic number along the horizontal axis of fig. 7 (PI. IL.). This process is repeated for other ee from the screens. ‘Thus, since the curves figs. 5 and 6 (Pl. ILI.) refer to equal total ionizations, a high ordinate on fig. 7 (Pl. IIL.) at any point indicates a larger proportion of higher velocity electrons passing that plane, and wee versa. The question arises that the marked variations in the shapes of the curves of figs. 5 and 6 (PI. III.) might be due entirely to the ionization in the chamber by the variable tertiary X-radiation from the successive screens, superposed on the ionization by the @-radiations which has a uniform function for each of the screens. Some of the reasons for this view being untenable are set down below. ‘Though this tertiary X-radiation must be present, the ionization produced by it, in comparison with that produced by the §-radiations from the screens, must be negigibly small: Barkla and Shearer neglected this effect in their work already cited. (1) The total ionization in the air in the chamber with carbon ends produced by the incident beam of X-rays is quite small, even at at: eee pressure, in comparison with that produced by the @-radiations from the screens, and it is presumed that the effect of the tertiary X-radiation from the screen would only be a fraction of that due to the secondary X-radiation. (2) The linear portions of the curves in fig. 3 (PI. III.) could not be parallel to each other if the effect of tertiary radiation were appreciabie. ‘The greater the effect of tertiary radiation, the more sloping would these lines become. (3) For a given substance, the thinner the radiating film, the smaller will be the ratio of ionization by tertiary X-rays to that by @-rays in 2 comparatively thin layer of air. The absorption coefficient of the P-rays in the radiating film itself is extremely great, whilst that of the tertiary X-rays is comparatively small. If a film were prepared of such a thickness as just. to absorb the -rays generated at its face in contact with the graphite, increasing the thickness would not increase the tertiary X-ray emission. (4) The apparent independence of the shape of the curves in fig. 7 (Pl. ILL.) on the relative dimensions of the curves in figs. 3 and 4. If the ionization by the tertiary X-rays from the screens were finite, it would, of course, be included in the dimensions of the curves figs. 3 and 4 (Pl. II1.). Thin Films of Elements exposed to Réntgen Rays. 131 (5) Sadler’s remark that the emission of corpuscular radiation is not necessarily associated: with the emission of tertiary X-radiation. This statement will be examined in more detail later. | There are many points of interest in fig. 7 (Pl. IIIL.). First, that ordered curves are obtained for the screens used. Again, if the degree of scattering is the same for all screens, and the phenomenon does not depend upon the actual number of P-particles involved, then, of the screens used, the silver atom emits thie oveutest pr oportion of high-speed electrons when excited by silver X-radiation, and similarly for the barium atom when excited | oy barium X-radiation. Consider for the moment the effect of silver X-rays on the screens successively. There are two important effects that have to be distinguished. ‘The first is that the K, L, M, ete. radiations are excited in those elements below, and only L, M, ete. radiations are excited in those elements above the atomic weight of silver. The second effect is much more obscure. We do not know the relative numbers of atoms radiating each of the Sp yectral lines. With regard to the latter efecr it appears in photographic images of the X-ray spectra and in ionization work generally “that the lines of longer wave-length are the more intense. This can be understood, for the waves of shorter wave-length would soon pass over the atom, and, reacting on an outer ring, be transformed partly into lon er waves, Gnd part of the energy would go into the new corpuscles produced. . The complete mterpretation of the curves in fig. 7 (Pl. III.) would necessitate a knowledge of the relative intensity of the rarious X-ray spectral lines from a thin film of the ele ue There is still another point of very great interest in fig. In 1912, J. C. Chapman, working on 1 homogeneous X-r ie arrived at the formula * where W, is the atomic weight of an element emitting L radiation, W, that of an element emitting K radiation of the same absorbability in aluminium. This formula yields the result that the I frequency from arsenic equals the L fr equency from gold. Moseley expr essed practically the same result ina different form +. He gave * Proc. Roy. Soc. A. vol. Ixxxvi. p. 447 (1912). 4 Phil. Mac. ser. 6, vol. xxvii. p. 712 (1914). K 2 132. Mr. L. Simons on the Beta-Ray Emission from the K, frequency of the elements as abit aac es 2 16,= (GS-5 )N—1)n, and also hale = JN=7 4)Png where N is the atomic number of the element and 1, the fundamental frequency of line spectra (=109720 x velocity of light). These formule give the K, reg of ursenic equals the L, frequency of lead. The resultis ap, proximately the same as Chapman’s. Reference to the curves in fig. 7 (PI. JIL.) will show that this result is applicable not only to X-ray emission hut analogously to B-ray emission. ‘This is clearer in the dotted curves, the distribution in speeds of the electrons about an arsenic screen being similar to that about a lead sereen. There are appearances, too, that the effect is carried on through the range of the elements. An attempt to detect groups of B-particles possessing sub- speeds and to determine the numerical relations between them. (Figs. 8 and 9, Pl. IIT.) The range of the §-particles and the law of absorption that they callew are matters of some obscurity. W hiddington set himself to determine whether the range was proportional to E or H?, where E is the energy of the particle. W. Wilson’s * results give H°? for the rays from radioactive substances. J.J. Thomson f and Bohr t have both deduced complex formule involving EH? as one of their terms. The dificulty arises:from the impossibility of determining the diminution in energy along the complex path of the Bray. Wilson showed that only a “complex distribution of velocities amongst the @B-rays from radioactive substances would olive rise to an exponential law of absorption, whilst Sadler stated that his slow- -moving homogeneous S-rays were absorbed according to an exponential law. In a previous paper § I deduced tentatively a result based upon an exponential law of absorption. The major speed xy * Proc. Roy. Soc. A. Ixxxiv. p. 141 (1910). + ‘Conduction of Electricity throvgh Gases,’ p. 881 (1906). { Phil. Mac. ser. 6, vol. xxv. p. 28 (1913). § Loc. cit. Thin Films of Elements exposed to Réntgen Rays. 133 of the 8-rays from gold (produced by the incidence of silver X-rays) was calculated from the expression Smvy" = 4, (DoD) ee and the minimum speed v, from pMvig” = (hv) p— y (huyt huy + ...). The ratio of vy, to v agreed weil with the experimental result obtained from the logarithmic absorption coefficients of the two sets of rays and by assuming Whiddington’s fourth-power law Av+=constant, where 2 is the logarithmic absorption coeificient. I do not think it safe to assume an exponential or any theoretically derived law * in connexion with these experi- ments. ‘The problem of the range of the sub-groups has therefore been attacked in a totally different manner. Unfortunately, it has to presume the existence of the sub- group, but the result obtained points strongly to the accuracy of this assumption and also to the general agreement with the photo-electric hypothesis of the sub-groups. Figs. 8 and 9 (Pl. III.).—A close examination of figs. 5 and 6 (Pl. III.) shows that the curves cannot be members of the same family, but that the irregularities might be due to the superposition on curves belonging to the same family of other curves varying among themselves. Wigs. 8 and 9 represent the results of an icin to analyse geometrically the curves of figs. 5 and 6 (PI. IIT.). Suppose these Curves. cng. 0 and).6 (rl. Ils). to “be represented by N = Nof(a) ), where No is the ionization due to the total absorption of the 8-ray energy in the air of the chamber, N,, the total ionization in air at 0° C. and 76 cm. pressure between the illuminated screen and the imaginary plane x cm. from the screen. The variation of the density of ionization with distance from the screen is given by N,f'(«), and the variation of a logarithmic absorption coefficient of the @-particles with the distance from the screen by “Jox,N Nya) If the curves of figs. 5 and 6 (PI. III.) could be repre- sented by N,=N,(1—e-™), this operation would yield —) constant. The value of X so obtained gives a measure * The theoretically derived laws deal only with radiators without a boundary; in practice a boundary is necessary. The condition of absorption in the gas near the boundary of the screen will be different from that obtaining in the central region. Unless a guard-ring method were employed, an “empirical formula would be required for each chamber of different dimensions. 134 Mr. L. Simons on the Beta-Ray Emission from of the speed of the particles from Whiddington’s law, Noe = constant But the operation d ; ie 7p Oke Not (@)) performed on the exponential curves would also yield —A, meaning that only if the original curves were exponential, lotting the slope of the logarithms of residues such as BO, fig. 5 (PI. ILI.), would result in the same constant, repre- sented in figs. 8 and 9 (PI. IiI.) by a horizontal straight ines? In this work involving unknown functions, in which the operations have to be performed geometrically, the latter process involving one differentiation is possible, although the physical interpretation of the resulting curves is not so clear as if the former process had been adopted. Figs. 8 and 9 (PI. III.) show the results of operating by the latter method on the curves of figs. 5 and 6 (PI. III.). As these curves are roughly logarithmic, 1t was thought that this. shorter method would show up just as accurately as the ~ method involving two differentiations where the function represented by the one part of the curve changed, if at all, into that represented by the remainder of the same curve. Figs. 8 and 9 show, of course, that none of the curves of figs. 5 and 6 are exponential. Portions of the curves seem to be represented by the same function, definite changes taking place at the points marked. It is not claimed that the resulting lines are true in their smaller detail. A large portion of the end has been omitted in consequence of the impossibility of performing the graphical operation, whilst there can be little accuracy at the beginning because of the difficulty of measuring the small ionization currents. It is claimed that the largest features of the curves are fairly correct. It is presumed that the positions of the minima on these curves give the range ofthe sub-group of electrons ft. * Proc. Roy. Soc. A. vol. ixxxvi. p. 375 (1912). + A little consideration will show that if there were two homogeneous groups of electrons emitted from a given screen and superposed, each following an exponential law of absorption, the curve in this case, drawn in the manner of figs. 8 and 9 (Pl. {II.), would fall gradually to a minimum, marking the range of the slower group, after which it | would run on horizontally. If, on the other hand, the logarithmic absorption coefficient, as it appears In these experiments, increases with distance from the screen, the resulting curve, drawn in the same manner, would still show a minimum or a decided change in direction approxi- wately at the range of the slower group. 135 We can employ the expressions for the photo-electric equation given in the introduction in order to calculate what should be the theoretical value of the energies of the electrons of the K, L, or M type sent out from the various screens when each is subjected, first to silver and second to barium X-rays. In the absence of any definite knowledge of the relative intensities of the a, 8, y, etc. lines of the K, L, M X-rays we must confine ourselves throughout to the lines. Thin Films of Elements exposed to Réntgen Rays. AB IGE le. | | _ Incidence of Silver | Incidence of Barium | | Values of | (K) X-rays. | | | | | | v/ Np. (K) X-rays. | ate Rae | | | | Bl patie Wee | aoe ie | | | | | (S)._- No. K, | L. | M, (©). (0). | (c). | (d). | On -U): )As |....., 38) 783 94) 15 | 747 | 1524) 1618 | 1491 | 2974 | 2868 Se ......| 34 | 835 | 101 | 16 || 681 | 1516 | 1617 || 1431 | 2266 | 2367 Fr | 40 | 1156 | 150 | 28 || 30% 1460 | 1610 || 2054 | 2210 | 2360 a. ie Wes3e 200864) 0 1877 159m | 494 | 2007 | 2347 erie | 50 | 1871 | 254 | 43 | 1336 | 1590 | 215 | 2086 | 2340 Sige | 51 | 1950 | 265 | 46 | 18227) 1587 || 122") 2072 | 2337 | 22 ee | 56 | 2383 | 328 | 60 | (205 M573) ee 1995 1.9398: ena. <.-2| 791! FAT, als UG) 760 | 1477 | 15LO)| 2207 | PDE oc. | 82 | Te WSs! 683 | 1460 | 1433 | 2210 Bie 3. | 83 | 797 | 180 || 656 | 1453 1406 | 2208 lex or | | | Taste I.—Columns 3 and 4 give the values of vu/n for the K, and L, X-ray spectral lines from the various elements 8. These values have been obtained by inter- polation from the observed values quoted by Vegard* ; the values for the M, line are interpolated from the formula fF CNR oe 5: | Vor [M0 = ta? 37N + 40. Column (a) is obtained from the expression ag¥/o)e,—sLUlro)x, + (v/mo)p, + (u/No) ar, ee column (6) by omitting the first of the terms within the square brackets, and column (¢) the first and second terms from the square brackets. Columns (@), (@), and (7) 1n-a similar manner when ,,(v/7o),, 18 substituted for ael(Y[Ro)xc, in the first part of the expression. * L, Vegard, Phil. Mag. ser. 6, vol. xxxv. pp. 293, 801, 3816 (1918). + Loe, et. 136 =©Mr. L. Simons on the Beta-Ray Emission from According to the theory outlined in the introduction, the numbers in these six columns, if multiplied by the constant moh, give respective ely the energies of the K, L, and M electrons emitted by the various screens when they are subjected first to Ag X-rafys and then to Ba X-rays, the K electrons being the least rapid and the M electrons the most rapid ones. This close study by Beatty’s “ pressure variation” method has led me to the conclusion that it is extremely diffienlt to distinguish any variation in. the maximum range in the B-ravs emitted by the various screens all exposed to rays of one type; the practical reason is that the beim of the 6-rays is so attenuated, both by the transformation of its energy into ions and by diffusion near its extreme range, and theoretically from the fact that the numbers in column (c} or in column (7), Table I., are so close together. that the electrons, whose range according to Whiddington’s fourth- power law should be proportional to the squares of these pumbers, are almost “indistinguishable. Or better, the electrons of maximum energy are the peripheral electrons whose potential energy within the parent atom is negligibly small, whatever the “hon may be, in comparison with the imeident quantum. ‘They emerge with practically the whole of the energy of the incident quantum , and their range will be almost independent of the nature of the parent atom, as Barkla and Shearer have already shown. Those eee referring to the ranges which these experiments might be able possibly to distinguish from each other or from dhe mean maximum range are italicised in Table I. These are taken out in Table IJ. One further point worth mention is. that when Silver X-rays (KX) tvpe are incident on silver or on elements just above silver in atomic weight, there is no K emission of X-rays, but groups of electrons possessing a large amount of energy. A similar statement is true for Barium (K) X-rays on barium. This accounts for Sadler’s observation. The distances in the two columns marked “range” are obtained from the positions of the minima of the curves in figs. 8 and 9 (PI. III.) respectively. According to this work, these are approximately the distances traversed in air at 0° C. and 76 cm. pressure by the groups of electrons having ranges witich could be distinguished by this method from those of the fastest groups. From Whiddington’s fourth-power law, which must be regarded as appr oximate, it follows that Rd? should be constant, where EH is the kinetic energy of the electrons of range din air. Apart from the Thin Films of Elements exposed to Réntyen Rays. 137 three results, which are obviously in error, it will be seen that there isa rough agreement with this law for the minor groups. TaBueE II. Incidence of Silver (K) X-rays. Incidence of Barium (K) X-rays. | Hlement’ Kinetic | Range (d) Kinetic Range (d) (S). (energy (E)| in air E/q2. | energy (iH), in air wae of sub- PEM Oats : of sub- at 0° & ae group 760 am. group | 760mm. | | = th, | trom fig 8: — 79h. =n hk. \from fig. 9.) + 7)h. | em cin. | feARSe jai | “228 1550 TIO tae Oo 21-1930 Sener: 6Sleeah 03 Lote lage Oe ano 2120 NA ee ee 304 | 2ib5)7/ Tlie.) | 1054) 438. | 1590 | ee a “ne mee 494 _| 40M P20) 4 SAS... ne | eye EO al 215 128 | *600? Slo es aaa | a 122 “070 | *460? Hebets. Ws = 145 | "333 | 2160 x | ae VBS et (Adee wc. 760...) lie) O20 eae hol @Opesien DOD bol scl OO bi) Oy enone 683 | ‘164 1690) 4: 1433. | OO again LOZ0 ig SP 656 | 164° “| - 1620 1406 | AE | PPA) oe a ST ee eS Ae oe iS he lie igis | GOt | 2090 | S368 | Top | 2368. | | * These numbers are got from the small ranges where the error must be considerable. T Obtained from figs. 5 and 6 (PI. III.). These are the maximum ranges for the incidence of Ag and Ba X-rays respectively. Some further considerations on the potential energy of the electrons. There seem to be two distinct points of view: (1) Tf an electron be removed from a “K” ring an ‘‘L” electron falls into its place, an ‘‘M”’ electron into the place of the ““L” electron, and so forth. During this process the one atom would yield a series of line spectra. This is the view adopted up to the present, and also that all the atoms of a given substance exposed to X-rays of a given type are not necessarily depleted originally of K electrons—a neighbouring atom may be depleted of an L electron or an M electron. (2) If an electron be removed from a K ring, this ring readjusts itself from one stable state to the next, during which there iy an emission of K radiation. It seems to me that according to (1) the potential energy of the K electron should be (hu, +hu,+ etc.),and according to (2), only (hu). 138 Mr. L. Simons on the Beta-Ray Emission from Table III. has been prepared according to this latter scheme, and the results show a very much better agreement than is shown by Table JI. prepared from the former scheme. TAB LTP. _ Incidence of Silver (K) | Incidence of Barium (K) | X-rays. X-rays. | | Values of | Pe | | | ‘Screen ee | Searesey | Kinetic | | eS) a4 energy |p__..| .- || energy | - ‘ | (8) of gee E/d?. | (B) of Bee B/d?. | sub- eS Woe 2 pee lee | em. | em | group | group H oe | L,. +n h. | Mk. |) Noh. nh. | ee = |As...| 783] ... || 850 |0°228| 1780 || 1600 |0595 | 2070" | Se ...| 835] ... |} 798 |0193 | 1820 | 1548 |0-455 | 2300 | Dea) boo | 477 |O-157 | 1200? || 1227. |0:438 | 18607 | Ae Gaal cea | eee eee eee | 750 10140") Sanam | Siieh ype Uae SS) ib oe i 512 | 0128] 1430 | Sb 2 1050)| 265.) 1868 5p! ae 433 0-070 | 1640 | Bai co.| 2363) 328) | 18055 1038 | 2260s a ize ‘einem |Au...| ..°| 717 | 916 |0-175 1 2190 || 1666 (0595 | 2YeoRy Pb Poa 856 (0-164 | 2110 | 1606 |0-560 | 2150 | | Bi pe 836 0164 | 2070 | 1586 | 0-427 | 2430 | | aera oes | | As 1618 060 | 2090 2368 10 | 2870 | Mean of all the values of Bi/d? +n,h=2000. Taste IJI.—Columns 2 and 3 are taken from Table I. Column 4 is obtained from the expression ag(¥/No) x, — s (ufo dx, or L,? the value for K, beiny employed in the latter term for those screens of atomic number lower than that of Silver. Similarly, column 7 is obtained by substituting pa(¥/ No), for the former term, and employing the values for K, in the latter term for those screens below the atomic number of Barium, and L, for those above that of Barium. The values of As at the end are taken from Table I. It will be seen that the values for the constant agree very much better among themselves than is shown in Table IT. Taking m—8'°8 x 10°" om. @—27 x 10* in air aul Ogee and 76 cm. pressure (within an accuracy of 15 per cent. according to Whiddington), np =3°3 x 10% sec.~? (Rydberg’s constant), h=6°6 x 10~?’ erg. sec. (Planck’s constant), we Thin Films of Elements exposed to Réntgen Rays. 189 obtain for the constant * observed in Tables IT. and III. for H/nha: the value 2930. Bohr+, however, calculates from theoretical considerations that Whiddington’s “a” should have the value 1:1 x 10° for air at a pressure of 76 cm. and at 15° C. for slowly moving electrons. Using Bohr’s value corrected to 0° C., we obtain the resu!t 2160 for the theoretical value of the constant tabulated in Tables If. and IIT. It will be seen that the values for the constant in Table III. show better agreement amongst themselves and with the theoretical value than is shown in Table II]. The mean value in Table III. is 2000, whilst the separate values in Table IT. differ so widely that it does not appear legitimate to take a mean value at all. The direct conclusion is that each atom is associated with the emission of one quantum only, and that entirely different atoms of a homogeneous substance emitting various spectral lines concomitantly are associated with the K, L, or M radiations; but further experiment must decide this important conclusion. Finally, it must be clearly stated that the method by which these curves have had to be analysed precludes all possibility of an accurate determination of the ranges of the sub-groups as it turns on the discrimination of those points in the curves of fies. 5 and 6 (PI. III.), where the function represented by the one portion of each curve changes into that represented by the proximate position. To me, the surest evilence for the existence of the sub- groups rests in figs. 8 and 9 (PI. III.), roughly conforming as they do to the requirements of the photo-electric equation. The ten curves of fiy. 8 (Pl. III.) divide themselves into two distinct sections: those of the one section being below, and those of the other section above the silver curve, the positions of the minima, representing the range of the sub- group, approaching nearer and nearer to the screen as we pass up from As to Se to Zr, starting out again at Ba and approaching once more as we pass up through Au to Pb to Bi. In fig. 9 (PI. III.) the division into sections is at the Ba, the sub-group having a smaller and smaller speed as we pass up from As to Sb, starting out again at Au and diminishing to Bi. All this is exactly as is required by the photo-electric equation. Because of the difficulties involved in obtaining experi- mental curves which will “bear differentiation with any co em ih Tacs ners ial bear SINV-A 9 22 7 = ae == , from vf=ad. noha? Nohv> Noh ? Phil. Mag. ser. 6, vol. xxv. p. 28 (1918). 140 Beta-Ray Emission from Thin Films. accuracy, the results of this part of the work must be taken qualitatively rather than quantitatively. I desire to express my thanks to ProfessorO.W. Richardson for his kind interest in this work. ‘ihe veneral conclusions may be summarized as follows :— (1) It is improbable that the maximum speed of ejection of electrons from different substances under the influence of X-rays of definite wave-length is exactly constant indepen- dently Odio mame ch the on bstance, and dependent only upon the frequency of the incident Ke rays. These experl- ments have shown that throughout the whole range of motion the distribution of electrons depends fundamentally upon the substance from which they are emitted, but tie difference, if any, in the maximum velocities is too small to be demon- strable by this method. (2) The experimental results point to the conclusion that there may be speeds of emission of electrons from an atom (S)} when X-rays of frequency uv fall upon it given by either et == (OD (100 [ia V dy eae acne each successive speed being given by the removal of a term from the bracket, commencing with the term, or gm’ = hu—.(hv)x or L or M> the expression in brackets representing the potential energy of the electron from whichever atomic ring it was ejected, Uz, Uz, etc. being the K, L, ete. frequencies of the X-ray spectral lines of the parent atom (S). The experimental data agree rather better with the latter expression. (3) There is a type of electronic emission fundamentally associated with each type of X-ray emission. Assuming that it is not a necessary condition that each of the radiating atoms of one substance should be emitting all possible types of spectral lines, from those of highest energy downwards, then the energy of electronic emission of any one of these atoms is ee anime si to that of its wave-emission; in other words, the “ K”’ electron is the slowest on emergence, the concomitant “ L” electron faster, and so on. ce ae IX. Inertial Frame given by a Hyperbolic Space-time. To the Editors of the Philosophical Magazine. Srrs,— T is interesting to modify de Sitter’s inertial frame to a hyperbolic form. Meterring to Professor Hddington’s » valuable ‘ Report’ on Relativity, re-write his (51.1) thus ds? = — R?}d@’+ sinh’ O(d@ + sin? Od¢”) | + cosh? Odé?.. (51.1) His (51.2) and (51.3) become isinh @= sin Csinw, tan (¢/R)= cosftanw, . . (51.2) ds?= R?(dw + sin? o(df?+ sin? (dé? + sin? Odg?))), (51.3) the only change here being in the sign of R?. His (52.2) changes in the same way, or — dy? pe i Pais dt —=——, (d+ sin” Odd") + ———.._. (52.2) pls ro eae (l—er?)? Ll—er? l—er”’ where «=1/R?. [In the work below I alter the scale of this three-dimensional map, putting r= tanh®; e=1 in place of r=R tanh ©, e=1/R’. | The effects of the modification are first that there is no time-barrier ; to a fixed observer the converse of what occurs in de Sitter’s space will happen—things will seem to move uncannily fast in the distant parts of the universe and distant spiral nebule should show a spurious tendency to approach by a spectrum shift towards the violet. The relative motion of two particles undisturbed by gravitation is excessively simple but, to my mind at least, most extraordinary and unexpected. Let one of the par- ticles be taken as fixed at the origin and let the motion of the other be mapped after the manner of (52.2). It will not move in a straight line but in the ellipse for which ¢/R is the eccentric angle. Thus all such undis. turbed particles retrace their sinusoidal motions for ever and all have the common period 27R (velocity of light=1). Moreover, every observer is the centre of all the elliptic orbits. In the last paragraph ¢ is coordinate time, but there is no 142) Inertial Frame given by a Hyperbolic Space-time. great difference when s the proper time of a fixed observer is used, because tan (s/R) varies (for a given orbit) as tan (é/R). In fact, tan (¢/R) = tan (s/R) . cosh «/ cosh 8, where tanha=a, tanh B=6b; that is, Re and RSG are the true hyperbolic magnitudes of the semi-axes of the ellipse. These results may be proved thus. Introduce five direc- tion-cusines 1,,..., (; to indicate the direction of a radius of the spherical surface in five dimensions specifying the space- time continuum of four dimensions, thus l= COs @ = cosh O COS p> ; ol l,= sin w cos € = cosh @ sin Re l,.= sin wsin cos 9 =i sinh © cos 6, l,= sin wsin fsin 6 cos $=/sinh O sin @ cos ¢, /;= sin @ sin sin @ sin d=? sinh @ sin 6 sin @. Then s being the geodesic between (l,,..., d;) and (1; ..., ls) we have cos (s/R)=L1/ +... 411, For the motion of any undisturbed particle relative to an observer fixed at the origin we may take ¢ =0 and also sinh © sin @=a cosh @ sin (¢/R), sinh © cos 0=0 cosh © cos (¢/R). give z=asin @/R), z=) cos (é/R), cos (s/R)=1,h,' + lsl3' = cosh @ cos (t/R) .,/(1—0”), where J,’ and I,’ are the initial values of J, and J;. The three conditions are the equivalents of the equations eae wal Us Wii These at cnce all three being linear and homogeneous in 1, lo,...d;. It will be seen that the a and 6 are the same as the a and 4 above, but t/R and s/R (s being taken zero whien ¢ is zero) are the complements of s/R and ¢/R previously used. To see that the above results are really extraordinary, consider how the same moving particle will appear to two different observers who are situated ona radius vector of the ellipse but on opposite sides of the curve. P and Q are Newton-Einstein Planetary Orbit. 143 two observers a few metres or kilometres apart and an un- disturbed particle passes between them in a direction perpen- dicular to PQ. They remain fixed at their observation posts watching the particle, for many million years, and at last face one another, and P reports to Q that the particle is now for the first time directly behind him. It seems difficult to believe or to disbelieve that what Q has to report to P at the same instant is that it is directly behind himself also. The relative rest of Q with regard to P is a case of con- strained motion, for if Q has the minutest velocity relative to P, and is undisturbed, he will rotate (in the belief of P) about P in the same period as does the particle. In such case our paradox disappears, but it still remains if we suppose that a constraint bringing about relative rest is so much as possible. Yours, &c., PASTOR OAM TAS: University of Tasmania, June Ist, 1920. eee ~~ _ X. Newton-Hinstein Planetary Orbit. By Sir GeorGE GREENHILL *. (ae modification in the Problem of Two Bodies of the elliptic planetary orbit under ordinary Newtonian gravity to the Sun, due to the additional term introduced by Einstein into the attraction, varying as the square of the angular velocity round the Sun, has engaged the attention of many writers in the recent numbers of the Philosophical Magazine. The result may be stated as a change of Newton’s ellipse into Hinstein’s orbit, from ede COse om SING Oa t Pes SA and so provides an Hlliptic Function application to a Central Orbit. It is proposed here to standardize this problem in the Dynamics of a Particle, and to examine closely the units employed. According to Einstein, a term he denotes by 3in@? must be added to Newton’s term pu? to give the total central acceleration to the Sun ; here @ denotes the angular velocity of the planet, and in a central orbit w=hu’, with w the en’ tpd sn? dp0 in ace * Communicated by the Author. 144 Sir George Greenhill on the inverse radius vector ; so that we may write the central acceleration PS me 24 3me?= O a +3mo?=pu+3mhrut. . (2) In his treatment of the Problem of Two Bodies in Matter and Motion, Maxwell expresses the Newtonian attraction as proportional to w. This follows from the property of the Hodograph ; turned through a right angle, the hodograph of an elliptic orbit is a circle with centre at 8. the pole of the velocity vector HU being at the other focus H. Thus the velocity of P, perpendicular and proportional to HU, can be resolved into two constant components; one perpendicular to SP and proportional to SU, and the other perpendicular to the major axis and proportional to SH. A steamer P for instance, circling past a lightship S in a tideway, keeping the light always abeam, will describe an elliptic orbit over the grotind in true planetary style, the minor axis-being in the direction of the tidal current. Crossing the road in the same way in front of the head- light of an advancing tramear, the path relative to the light § will be described as a planetary orbit. The velocity of the velocity vector being at rizht angles to SPU, the force and acceleration is directed to 8, and varies as w, the angular velocity of SPU. Then in Maxwell’s notation, p/h=hl is a velocity, V, the velocity at the point P where Dp Oa ‘= lab, where p, p’ denote the perpendiculars Sy, HZ from S, H on a tangent, and / the semi-latus rectum. At this point P, SP is parallel to CZ, and HZ 67 ; also if X is the foot of the directrix to the focus 8, and the tangent, at P ents the- major axis in 7, S02 2am HT=2CX; and with T also the periodic time of the ellipse == 3 circumference of the circle of radius a?/b, the radius of cur- vature at the end of the minor axis. Then 8 denoting the mass of the Sun, P of the planet in grammes (g), and G the gravitation constant, = 660 <0 in C.G.S. units; and a denoting the mean (opines: in oe metres (cm), G(S+P)=p=n23= | 7 (4) Newton-Einstein Planetary Orbit. 145 the expression of Kepler’s Law III, and implying that G is a constant for all matter throughout the Universe. In these C.G.S. units, Hinstein’s m must denote a length, in centimetres. It is mysterious then that Hinstein is quoted as calling m the mass of the Sun, as if a mass could be measured in centimetres, by a metre rule, and not in grammes; some mysterious unexplained astronomical units must have been employed, and writers should enlighten us on this point of the theory. The differential equation in Particle Dynamics of the Central Orbit, analytical expression of the normal component of the acceleration, becomes when Hinstein’s term is added, P | os w= ya 0 = +B, MRS OLS t(D) : Bye et crt | and integrating, with h?= yl, 2 2? (‘a) +w=C+ — H2mMw y= 6 gy (6) (Sp) =2mQu—a w= Bu=9), eh) and a, 8, y are the-roots of a cubic in u, where C i i a+ p+y=p By +yat+taB= 7, CON erioe (8) 2m’ In a closed orbit, «, 8, y are positive ; and taken in the sequence so that a>B>u>y, Ae / (4. we i u—ry)? | (9) p?=2m(a—y) = sears | Babe thon ) bringing in an elliptic integral, which is to be reversed in -Abel’s manner ; and then, expressed by the inverse elliptic functions, ~~ - —u ) Pye ant, / get on Ba, go, [a | r (10) ee ea | a—y” equivalent in the direct notation, as in (1), to u=yen?ipd+ snip)... . . (1) Phil. Mag. S. 6. Wok 4 Now 241. Jane 1921; L 146 Sir George Greenhill on the In the degenerate case of pure Newtonian gravity, m=0 ; and then 2=o, 6 and y finite, p=V, £=0; and the orbit reduces, as in (1), to u=y cos’ S646 sin? 30. ||.) ene The apsidal angle @ in (1) is changed from 7 to 2K /p ; and the importance of Hinstein’s theory lies in the applica- tion to Astronomy, where the term m is considered small, in the endeavour to account for the anomalous amount of the advance of the apse in the orbit of Mercury. In that case where m is small, « is very large compared with 8, y; and we take m>0, 2me=1—2m(B+y)>l, a 13 pol Dae ve lle : aia = ae) Gee a $ 2K i es ee > (142 ee eas TigOn a 1 tae) Ao | a Aw 2K Bry in Tele Ae + i 1 A ; ety | 3s mn Ae) Le) Aine ee Le ” ie ga ; T iy making the advance of the apse in one revolution 360° (m/l). The addition of a term varying as uv? or w? to the central attraction would cause the -litferential equation of Newton’s orbit to change from Senger? ; into ps (16) pe 2 (i) =0+ 7 —w, into C+ —C—nw, . AD (J) =B-u.u—y into (L=n)(B—u.u—y), (18) u=y cos? 40+ sin? 46, into } (19) y cos 7490+ B sin? 399, q=/(1—n), the character of the integral and of the equation of the orbit is not altered essentially, except by an expansion or con- traction, in a fan-like manner, of the vectorial angle @. Newton-Hinsten Planetary Orbit. 147 *sThen in G. C. Darwin’s investigation in the Phil. Mag. May 1920, the influence of both additional terms together is considered, by taking physical quantities « and 6, making pe Su sak. (C20) ‘so that his « is our 2m; and then, changing his @ into n, au 1 Ly =3mu? + nut = 2 2u l Hey 2 23) (1a) =2m(u—a.u—B.u—y), - 23 making the orbit, as before, in (11), w= en Spo + Bsns 00, % ww» (24) du - 2 9 9 2 (a) +ur-=2Zmu’ +nu-+ +0, eae (22) dé ‘but with 1 1 — it eS a0 laa prea Pye Tae — . (29) Here with m smail, but n unrestricted, m->0, 2ma=1—n—2m(B+y)<>1l—n, 2 Dy ln — cry) Ea pas pry eee Z ee ee ey) >(1 =) le) Ac Mine! ” I —n) uh (26) iL m RM apne l(1—n)? (27) reducing to the (15) above for n=0. Add a further term to P, 26h?u°, varying as w?, au Cpe Tes 28 + Bm + nut |, Rs eae (28) Qu l lu? ee) OCP able wy. v=. (30) and further progress in the elliptic integral requires some knowledge of the factors of the quartic. S | Then there is the memoir of W. J. Harrison in the Proce. Cambridee Phil. Society (C. P. 8.) Nov. 1919, on the pressure in a radiating current of viscous liquid, and a L2 Wi Ca) an (29>) TNS a + y? = but + 2mu® + nu? + 148 Prof. F. Y. Edgeworth on the further Note on Vibration, O. P. 8. May 1920, where the complete solution requires the elliptic function, as explained above. These are the vibrations in the ideal case where terms varying as 2°. 2° are added to the leading term n?z of simple rectilinear vibration. But these vibrations can be visualized and studied in reality in the various projections of pendulum motion, in the extension to the more general case where the axle is fixed on a whirling arm, as in Watt’s governor, the gyrocompass, or the Gilbert barogyroscope; anda complete solution has been. given in the Report to the Aeronautical Committee on Gyro- scopic Theory, 1914, Chapter VIII; while the verification by differentiation and integration has provided exercises in ‘ Applications of the Elliptic Function,’ 1892. Stroboscopic vision will convert these rectilinear vibra- tions into the central orbits eonsidered above, but with @=mt, so that the shape of the curve is seen, but not the velocity in the orbit. Staple Inv. Nov. 5, 1920. XI. The Genesis of the Law of Error *. Diy keno, Ve EpGrworty, TBA ROFESSOR SAMPSON’S courteous reference to some remarks of mine upon this subject deserved an earlier acknowledgement. I would have sooner expressed apprecia- tion of his criticisms, but that I wished to finish the article which has appeared in the September number of this Magazine.. I hoped to be able to speak more to the purpose about the law of error{ after considering its application to a leading branch of Physical Science. The points at which new light may be expected from Prof. Sampson’s original reflections may be introduced by a. * Communicated by the Author. ‘+ Referring to the article on this subject by Prof. R. A. Sampson in the Philosophical Magazine for October 1919. t The term ‘‘error ” designates according to the context either error: proper, ze. deviations of an observation from the physical quantity which is observed, or generally deviation from an average. The law of error (or the normal law) designates the function (1/en/ w)exp—a?/e considered as approximately representing the frequency-distribution of an aygregate; the first approximation, further approximations being furnished by a series descending in powers of 1//”. where n is the number of independent items in the aggregate. Genesis of the Law of Error. 149 ‘statement of propositions which I should regard as well established, but for his seeming dissent. I. A. (1)* The origin of the law of error is to be sought in the theory of Probabilities. (2) It is obeyed by random agerevates of variable constituents. Games of chance afford the simplest examples. (3) Thus if, a batch of » balls having been taken at random from an immense inedley of black and white balls, the number of white balls in the batch is recorded, and the operation is repeated many times, the series of numbers presented by the record will conform to the law- of-error; more or lessapproximately according as nis larger, and as the proportion of white to black in the medley is nearer to equality. This Se between the bi- nomial series and the law of error was happily employed by Quetelet to illustrate the fulfilment of a law In various kingdoms of nature. In games of chance and organic nature foes not often arise a dif fhieulty which the Reon neienec: may present; namely, that (4) the aggregation of constituents which is treated as fortuitous is non to be determinate. The most familiar instance is the sequence of decimal places in the evaluation of a natural constant, such as wT. Each figure is what it is in virtue of determinate law ; and yet the ensemble presents the arrangement which is an outcome of chance. A more important instance is furnished by the distribution of velocities in a molecular chaos; they obey at once the laws of Dynamics and the laws of Probabilities. The Philosophers } have exercised themselves about this paradox ; but it remainsa mystery §. Onerule emerges that (5) we should not seek for law (other than statistical uni- formity) as the explanation of chance. We must not do as the fatuous gambler who scrutinizes the records of the roulette table in the hope of eliciting a rule to guide his future stakes, In the weighty Seas of De Morgan : “ No primary considera- tions connected with the subject of Probability can or ought to be received it ey, ee upon the results of a complicated mathematical an To use Dr. Venn’s {] phraseology, * The passages headed by bracketed numerals in Section I. are referred to in Section II. H Cp. Venn, * Logic) of Chance,’ p. 112 e¢ seg., ed. 8, and Nixon, Journal of the Royal Statistical Society, vol. Ixxvi. p. 702 (1913). { Renouvier, Venn, von Kries, etc. § Cp. Poincaré, Ke There j is here something mysterious inaccessible to the eae velan.| . » vol. ii.: article on the “ Theory of Eyobubilsty, yf S, aia 4] ‘ Empirical Logie.’ 150 Prof. F. Y. Edgeworth on the uniformities of statistics cannot be explained by other kinds. of uniformity. Of course, other kinds of uniformity, propositions other than those proper to Probabilities, may be employed to: ascertain whether and how far the conditions necessary for the genesis of the law-of-error are fulfilled in any concrete case. The gambler may reasonably inquire whether a roulette table is faked, or the dice loaded. The student of Probabilities may properly expect that a long set of figures- forming the period of a recurring decimal will approximately contorm to the law-of-error ; and that the same will be true- of pairs formed at random from these figures. And yet he may learn from theory or observation that pairs selected in a particular manner from certain periods—e. g. by adding the first place to the (n+1)th, the second place to the (n+ 2)th,,. and so on in a period of 2n decimals—will not fulfil the conditions necessary for the genesis of the law of error *. The principal conditions arethree. (6) First, the number of constituents must be large—how large depends on the degree: in which the other conditions are fulfilled. Secondly, the variation of the constituents must be independent, a condition. which is, perhaps, never perfectly filled in concrete nature, not even in games of chance t. Thirdly, the constituents are- to be aggregated by simple summation. This condition is. sometimes perfectly fulfilled, as in games of chance; but often only approximately, as where the compound is some function other than the sum of the constituents, but such that, when expanded in aseending powers of the constituents, it is equateable approximately to their sum. (7) According as these conditions are more or less pertectly satisfied the law-of-error is more or less perfectly fulfilled. So the path of a projectile is more nearly a parabola the less dense the resisting medium f. | There is a fourth condition which is, indeed, essential. but hardly requires to be stated since it is always fulfilled in concrete nature. (8) The standard deviation and generally the mean powers of deviation for each and all of the constituent variables must not be znjinzte. The definition does not exclude the possibility of an infinite deviation provided that it is sufficiently rare. Thus a group fulfilling the law-of-error is qualified to be a constituent in the genesis * The property is illustrated by the recurring decimal representing 1/19, viz. °052631578, 947368421. 1/1861 furnishes a better instance. _ See “ Law of Error,” Transactions of Cambridge Philosophical Scciety,. p. 129 (1905). +t See Venn, ‘ Logic of Chance,’ ed. 3, p. 77 et seq. t Cp. Phil. Mag. “vol. Xxxyv. p. "499 (1918). Genesis of the Law of Error. 151 of that law, and yet a member of that group may conceivably deviate to infinity. Moreover (9) ‘‘infinite”’ in the detinition a be understood literally, not simply as an equivalent to ‘“immense.’? Thus, considering any frequency-function with infinite mean powers of de sviation, e.g. 1/m(1+47), if we have to do only with values of x which, however large, are me we ean always take n (the number of the constituents) so large that the aggregate, or average, will fulfil the law- of-error. (10) As to the functions which extend to infinity without violating the cundition (8),it may be doubted whether outside the law of error such frequency-groups have any concrete existence. Hvenfor the law-of-error considered as resulting from a finite number of finite deviations the infinite deviation is a limit never attained, probably not even by the velocities in a molecular medley *. B. A right understanding of the theory is promoted, or, at least, evidenced, by a correct appreciation of the dreitars to whom it is due. The foundation of the theory is to be sought in Laplace’s classical treatise f. (11) The first section which he devotes to the subject exhibits the essential features of the Jaw-of-error: namely, that, when the three conditions above specified are present, ‘‘ the peculiarities of the (con- stituent) functions efface themselves in the final result” {. True, he does not advert to the fourth condition. But it was not his wont to dilate upon conditions which might ordinarily be taken for granted. Thus he repeatedly assumes that a function may be expanded in powers of the variable, and powers above the first neglected. In the next section, for instance, he thus obtains a linear equation for the correction of an observation. Of course he knew that there might occur singular points. A caveat was the less necessary in the present case, because in no application of the law which he contemplated could an infinite deviation occur. He could not suppose—nor forsee that anyone would suppose—a really infinite error-of-observation. It is, perhaps, remarkable that Laplace did not extend the law which he demonstrated to account for the prevalence of the law throughout Nature, as shown by Quetelet. (12) It is still more remarkable that Laplace showd not have applied he law to demonstrate that the normal error-function would ‘emerge from the mere superposition of the definite numbers “ That the sum-total energy is finite is not conclusire proot.. + Lhéorie Analytique des Probabilités, Liv. I. cap. iv. { The words are those used by Professor Sampson in denying this conclusion (Congress of Mathematicians 1912, Proceedings, p, 168, pare h), 152 Prof. F. Y. Edgeworth on the of small errors which had arbitrary laws of frequency of their own’ *, ‘* Laplace does not seem to have regarded an error in this light” t, as Glaisher states correctly : ‘* Nowhere does he assume that if one observation only is made its law of facility is e~”’".”~ Laplace evidently did not contemplate this deduction when in the important section dealing with the application of inverse probability to errors of observation he speaks of ‘the complete ignorance in which we are with respect to law of errors for each observation ” §. (13) Laplace’s proof of the law-of-error derives strong support from Poincaré’s theorem that the frequency-function pertaining to the sum of numerous variable elements is such that all its mean powers of deviation are approximately equal to those of the normal error-function ||. Further confirmation is obtained from other proofs. To some of them, in particular the proof by way of partial differential equations, it may be objected that they assume the existence of a final state, an ultimate law of frequency, to which the continued superposition of elements must tend 4. But (14) the assumption is surely not very arbitrary. It is of a piece with the assumption countenanced by Tait ** that. the velocities in a molecular medley tend to a final distribution. It is pleasant to believe that these views are not so much * The aptly-worded predicate of this proposition is borrowed from Prof. Sampson (Phil. Mag. p. 349); but not the proposition itself. + Memoirs of the Royal Astronomical Society, vol. xxxix. p. 106 (1872). Cp. Monthly Notices of the Society, vol, xxxiii. p: 97: he (Laplace) did not himself so apply it (the law).” { Memoirs, Joc. ert. p. 108. § Théorie Analytique, Liv. I. cap. iv. art. 23. | See Phil. Mae. vol. xxxv. p.426(1918). This proof is confirmed ie the solution of Stielttes’ probleme des moments, as presented by Prof. G. H. Hardy in the ‘ Messenger of Mathematics,’ vol. xlvi. (1917) p- 175; regard being had to the rapidity with which the error-function tends to zero (as the variable increases). {| Article on “Probability,” Encyclopedia Britannica, 9th edition, p- 3938. The objection does not apply to the proofs given by Morgan Crofton, Phil. Trans. (1870). ** “ Hiveryone therefore who considers the subject from either of these points of new (ordinary) statistics or the theory of probabilities, must come to the conclusion that continued collisions among our set of elastic spheres will, provided they are all equal, produce a state of things in which the percentage of the whole which have at each moment any distinctive proper ty must (after many collisions) tend towards a definite numerical value.” The proposition is presently extended to sets of spheres “ no one of which is overwhelmingly more numerous than another, nor in a hopeless minority as regards the sum of others.”— “ Kinetic Theory of Gases.” Royal Society of Edinburgh, vol. xxxiil. p. 225. Genesis of the Law of Error. 153 at variance with those of Professor Sampson as seemed betore his conciliatory explanations. There was, however, and, indeed, still is, an appearance of difference which may excuse my previous expressions ot dissent. It may be of more than Sas interest to exhibit the seeming difference. Il. A. Wishing to “discard as tar as possible the language of probabilities © o (Congress*, p. 163), Professor Sampson runs the risk of obscuring the conceptions proper to the seience (1) t. I do not recognize the doctrines of Proba- bilities in the following description of the Laplace-Poisson theory of error: —‘“‘It is a theorem of convergence and must be judged so. It is either true or false. Such phrases as “a@trés peu pres’ are not in the first place admissible. If they are required to help the demonstration out, that means the theorem is false ; for Poisson in particular seems to have held that no conditions were necessary to impose upon the frequencies of the elementary contributin g errors ‘la fonction aura telle forme - Pon voudra’”t. Butit “the view above pre- sented —(6) and (7)—1is correct, the conditions for the genesis of the law-of-errorare,in general, only fulfilled approximately ; the phrase of Laplace “ @ tres peu trés”’ is generally, not to say, universally § app! Opole. On the same view the words. of Poisson are pertectly correct; convergence towards the normal law will set in, svhetever the frequency - -function of the contributory elennenine teh one exception, indeed, formed by the fourth condition (8), an exception which proves the rule as it has no conerete existence (2) and: (10): Discarding the language of Probabilities, Prof. Samp-on writes (Congress, p- 168) : —“The term accidental error has come to carry with it an undefined suggestion of a peculiar quality, but there seems no reason to treat an error otherwise than as a disregarded unknown”... It appears to me that in accounting for the presence of the law of error we cannot discard the peculiar quality of accident or Probability (1) and (2). It is the attempt to get rid of that peculiarity which I described as Professor Sampson’s “ peculiar notion * The reference is to Professor Sampson’s paper on the ‘“ Law of Distri- bution of Krrors”” read before the 5th Congress of Mathematicians (1912) ata joint meeting of the sections on Astronomy and Statistics ; published in the ‘Proceedings of the Congress,’ vol. 1. p. 163. + The numerals in Section II. refer to passages numbered in Section I. i Phil. Mag. p. 349. The references hereinafter thus made are to Prof. Sampson’ s article on the “ Genesis of the Law of Error” in the Philosophical Magazine for October 1918, vol. xxxvi. § Even in games of chance perfect independence is probably only an ideal (6) ; also the number of trials being finite, there can only be an approximation, e.g. (38) to the normal law of frequency. 154 Prof. F. Y. Edgeworth on the of the nature of an error of observation ” (Phil. Mag. p. 347)- Whether we are arguing that the average of arenes obser- vation, or that the errors themselves (as sums of contributory elements), are normally distributed, in both cases, the peculiar character of Probability must be postulated. In the sequel of the passage just quoted (Congress, p. 168) Professor Sampson's treatment of the disregarded unknowns. appears somewhat peculiar. My interpr etation of his formula y=sint for the frequency- -function of an observation was, I think, very natural (Phil. Mag. vol. xxxv. p. 431). But, of course, | accept the explanation which he has given (Phil.Mag. p- 301): ‘‘ All values within the limits + a would be equally likely, and that is what the frequency-graph described would imply **. Yet I fail to see in what respect this conception ie any advantage over Laplace's theory. Laplace also in the very first section of his path-breaking chapter (11) supposes that all values between the limits + @ and —a would be equally likely. Laplace believed that the value of each observation obennred according to determinate laws Tf : but he did not profess to know what these laws were. I am encouraged to surmise that nothing very paradoxical is intended by the passage just quoted illustrating one of the points for which novelty is claimed (Phil. Mag. p. 350), when I consider the passages relating to, another point. Here, too, there occur di ficulties of in terpretation. They are largely due, I dare say, to the obtuseness of the interpreter, Yet the author has candidly taken to himself blame for having thrown out collaterally one misleading statement (Congress, p. 170; Phil. Mag. p. 347). It is not the only puzzling statement in the context. What are we to think of the two statements with respect to the reproduction of form (by the superposition of two functions of the same form) made on the same page ? (Congress, p. 170) :—‘‘ So far as I can determine it, (reproduction) belongs with anv generality only‘ to the functions exp(a’) and exp(a?) ” (sic) ; and a few lines below, ‘it will suffice to show that exp(—/?22) and, of course, also (exp — h?.x?) cos (ku + y) reproduce themselves.” * The explanation continues (loc. cit.): “ If then we suppose that errors are not of mysterious character, suz generis, but are simply the mass of numberless neglected disturbances, each according to regular law and order of its own, it is seen that we ‘obtain the approximation to Gauss’s law which is necessary to begin with by the operation of neglecting the circumstances and order of their origin and scheduling merely 3 in. sequence of magnitude the number of times that each particular value occurs.’ + The determinism of Laplace has often been noticed, especially by Races Essais de Critique générale. Genesis of the Law of Error. 15, Both these statements cannot be true. But they may both have been thrown out collaterally. For, firstly, the property belongs, with perfect generality, to any number of functions, all the members of the family fo.) y={ Cue Comaida —— (2.0) and, secondly, it does not seem to belong in any obvious, or useful, sense of the term to exp( —/h’«?) cos(ka+y). It is true that the superposition of the last written functions results in convergence to the normal law. Of course it does,. if they comply with the fourth condition. And they comply with that condition, since exp(—/’az?) does; and every element of the integral ( wexp(—h?x?) cos (ka+y)da | is less than the corresponding element of the integral COle { wexp(—/x*)dx. —™7 om Accordingly no great addition is made to our knowledge. when the author concludes : “ If we go on piling error upon error, provided each has the fluctuating character indicated above, we shall, as a limit, converge to the pure law of Gauss ” (Pel Mag. ps od0)o Bui why drag in the “ fluc- tuating character”? The proposition remains true when the proviso is omitted. It is true, for instance (for the reason just given), of the function (exp—h?2”)/(1 +2?) No doubt there is some interest in verifying the fact of convergence by actual integration. But this interest is not very great when the deviations, extending to infinity, are such as never could occur in concrete nature. The verifica- tions given by Dr. Burton in the Philosophical Magazine for December 1589 are much more interesting. The author describes his contribution “tas a view by which we can see the law coming into existence, which I submit the other forms of proof, one and all, fail to supply.” Is not the desiderated view supplied by regarding the frequency- function pertaining to an ageregate as of the Form e™ (+R), where Lt is the continuation of Poisson’s ex- pansion in ascending powers of 1/ /n; and observing, say, on the lines of Morgan Crofton’s method, that when a new observation is taken in, n is changed into (n+1)?f Is the * See Camb. Phil. Trans. vol. xiv. pp. 142, 158 ea + Todhunter,“ History of Probabilities,’ Art. 1002 ; ‘amb. Phil. Trans. vol. xx. p. 47 (1905). 156 Prof. F. Y. Edgeworth on the difficulty of apprehending the relations of chance and law (4) and (5) lightened by the introduction of the ‘ holo- morphic function” (Phil. Mag. p. 350), or other novel point ? B. So far, perhaps, 1 may sav with Professor Sampson, ‘the points of difference” between us appear “unsubstantial” (Phil. Mag. p. 347). But with respect to his main criticism of Laplace’s proof, I cannot regard the difference as un- substantial. J cannot retract, but would rather emphasize what I have written on the point (Phil. Mag. vol. xxxv. p. 425 (1918)). Professor Sampson argues: “The con- clusion is, therefore, unwarranted, and there is no proof at all that peculiarities of the functions efface themselves in the final result. I do not believe that the theorem is true ” (Congress, p. 167). 1 can only agree with these statements in the case where the fourth condition is not complied with ; that is, in a case which never occurs in fact. I cannot renee the statement that the attack on the proof given by Poisson and Laplace strikes at all the applications of the law (Phil. Mag. May 1918, p. 422) ; for the objection to the reasoning of Laplace and Poisson can only be admitted on a supposition which would be fatal to every other proof of the law of error*. The negative of that supposition, the axiomatic fourth condition (8), forms the corner-stone of the edifice, whether built according to the design of Laplace and Poisson or some more modern plan. That fundamental support being rejected, the whole edifice would collapse. All the applica- tions of the theory—the Method of Least Squares, both as a good method where the frequency-function for the errors of observation is unknown, and as the best method f when the frequency-funetion for the errors-of-observation is believed by inference from the law-of-error to be normalf{; the representation of statistics by a normal curve (or surface) in the manner of Quetelet, or more exactly by the employment of the second approximation given by Poisson §; the test of Sampling as practised in social investigations by a Bowley Kiar || : the splendid and useful results deduced from normal * The law being defined as above. Note f{ to p. 148. + See as to this distinction, Journal of the Statistical Society, vol. 71, pp. 509,393 (1908), and references there given. Note too (loc. cit. p. 499) that the law-of-error is required to determine the precision of a result obtained by znverse probability from a set of observations for which the error-function is known, but not normal. t Above, p. 149. § As employed with success by Buwley, ‘ Elements of Statistics,’ ed. 2 ; and Journal of the Royal Statistical Society passim. || Referred to in the Journal of the Royal Statistical Society, vol. 76, p. 192 (1912-18). Genesis of the Law of Error. 157 correlation by Pearson and other mathematical statisticians * all these and many other applications of the law to physics and social science, no longer based on reasoniny from Probabilities, would fall in ruins. : With reference to Laplace, mav be noticed a statement which, though not seriously misleading, adds to the perplexity into which the student of Probabilities is sometimes thrown by Professor Sampson’s treitment of the subject. He attributes to Laplace the following argument :—‘ Laplace first offered a demonstration that the error-function possessed a definite form and might be derived from the combination of an unlimited number of small er rors individually following any arbitrary laws of distribution ” (Congress, oaloion Phil. Mag. p. 349). But, as above pointed out (12), it is remarkable, a las been remarked by a leading authority, that Laplace did not make this use of his theory, but applied it only to averages of observations. Was the demonstration attributed to Laplace first employed by Morgan Crofton in the Transactions of the. Royal Society, 1870 + "2 Professor Sampson’s reference to Poincaré (Congress, p. 168) was calculated, on a first hearing at least, to confirm the impression that justice was not being done to the law of error. It seemed as if an advocate referred to the terms of a statute as in favour of his case, omitting mention of a clause ave afforded strong support to the case against him (13). It may be observed that the Poincaré proof 1 is free from the objection which Professor Sampson makes against other proofs, including Morgan Crofton’s, which “ begin by formu- lating the existence of an error-function” (Phil. Mag. p.d48) ft. That objection might be formidable if we uoined the theory te errors proper. We could not @ prior assert that * See Yule, ‘ Outlines of Statistics’; Pearson, ‘ Mathematical Theory of Evolution,’ Royal Society passim, and Biometrika. + The argument is well stated by Glaisher in the ‘Memoirs of the Astronomical Society,’ cited above, dated 1872. t Laplace’s proof is not open to this objection. The observations whose averages he proves to be obedient to the law of error might be like the ‘causes of error” (the contributory elements whose aggregate make up an error of observation) according to Morgan Crofton (Phil. Trans. p. 180 (1870)) such that ‘a function or curve does not assist our conceptions, and we shall do better to consider the points or dots them- selves.” (Consider the version of Laplace’s proot, Camb. Phil. Trans. p. 01 (1905). The expression #(v) employed for the locus of the frequency need not be a continuous differentiable function.) And if the method is emptoyed (though Larlace himself did not so employ it) to deduce the form of the frequency-function pertaining to errors of observation from the fact that an error is approximately the sum of numerous contributory elements, if is not necessary to postulate that the sought function is continuous and differentiable. 158 Geoloyical Society :— errors of observation are distributed according toa continuous differentiable function (Congress, p. 164). But has the objection much weight in general ; with reference, say, to a molecular medley (14), or to an average of statistics prolonged indefinitely under stable conditions ? On the whole it appears from the explanations which Professor Sampson has offered in the Philosophical Magazine that he designed to repair or reconstruct the edifice fonnded by Laplace. But, according to the view here taken, the operation was calculated to weaken the foundation and to damage the structure at other points. Nor were there compensating advantages in the additions proposed as im- provements. On a first view the danger to the structure was more apparent than the intention to make repairs. It was natural, then, that one who was deeply impressed by the use and beauty of the edifice should protest strongly against an attack upon it, though made by an expert whose authority on mathematical subjects commands great respect. XI. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from vol. xl. p. 826. | February 20th, 1920.—Mr. G. W. Lamplugh, F.R.S., President, in the Chair. ts Pee PRESIDENT in his Anniversary Address discussed Some Features of the Pleistocene Glaciation of England, dealing principally with the changes brought about by the ice in the surface-features of our country. With the aid of a sketch-map he showed that over 5000 square miles of English land, or about one- tenth of the whole country, would vanish if the drifts were removed, as the ‘solid’ rocks lie below sea-level in tracts of this extent. A further area of about 10,000 square miles is overspread by drift of sufficient thickness wholly to mask the ‘solid’ land-forms, so that rather more than one-quarter of the country owes its present shape to Glacial and Postglacial deposits. Another 20,000 square miles was glaciated, and more or less modified, but without losing the dominating features of its rocky framework. The remainder of the country was affected only by the intensification of the atmo- spheric agencies, whereby its original features were accentuated. In a general sense, the hill-districts have not been greatly changed, but the lowlands have been in most parts completely altered. The source of the huge mass of material contained in certain of the lowland drift-sheets was next considered, and the opinion was -expressed that a large portion of this was an addition to the land, ‘brought in by the ice from outside our present coast-line. The "7 Lower Carboniferous Chert-Formations of Derbyshire. 159 position and extent of these.drift-sheets could be explained by regarding them as the broad terminal belts of débris concentrated where the ice from the basins thinned off towards its periphery, and where also its motion was checked by the rising slope of the ground. The débris-choked outer margin of the ice may be supposed to have become stagnant after its final forward spurt, and in its waning phase most of its thaw-water probably escaped backward into ie basins, leaving wide stretches of bare boulder- elay unencumbered with water- washed material. Many pecuharities of the drift-features were explicable on the supposition that the ice-movement was not continuous and regular, but proceeded, at the margin of the ice-sheets, by alternations of quick advance with longer intervals of stagnation or relative quiescence, such as haye been observed now in existing glaciers and ice-sheets in many parts of the world. The unequal distribution of the Glacial deposits in the area of scanty drift was then discussed, along with some local peculiarities in the shape assumed by the deposits in several places, and it was shown that the difference between the aspect of the main drift- sheets and the scattered drift could be accounted for by the difference of local .conditions, which led to original irregularity of deposition and to early exposure of certain tracts to exceptionally vigorous erosion. It was also pointed out that the local incidence of giaciation may often have been an important factor, as it is known from existing conditions in Arctic lands that great stretches of moving ice may leave bare land, aside from its path, at lower levels on its flanks. Comment was then made on the curious rarity of peat or other land-detritus in boulder-clay known to have been derived entirely from the land, and this was thought to indicate that the condi- tions for a long period before the actual glaciation had been unfavourable for the growth of timber or peat -producing vege- tation. A brief review was given of the minor changes and new erosion- features produced in the hills as a result of the glaciation. The effect of Postglacial erosion and deposition in modifying the Glacial features was also referred to. February 25th.—Mr. R. D. Oldham, F.R.S., President, in the Chair. The following communication was read :— ‘The Lower Carboniferous Chert-Formations of Derbyshire.’ By Henry Crunden Sargent, F.G.S. The chert-formations occurring in the Carboniferous Limestone and associated rocks of Derbyshire may be classified under two heads :— (1) Those which owe their silica to gaseous or aqueous emanations from igneous rocks. (2) Those which derived their silica from the land by means of chemical denudation. 160 Geoiogical Society. The author considers that in both cases the silica was pre- cipitated direct, and did not, to any considerable extent, pass through an intermediate stage of secretion by organisms with subsequent solution and redeposition. Field-evidence shows that the chert is a contemporaneous deposit, though with possible rare minor exceptions, as noticed in the paper. é Field-evidence also shows that the silica was rapidly deposited, and that it was consolidated before the underlying limestone. Even if siliceous organisms existed on the sea-floor in sufficient numbers to form the chert-beds, which field and microscopic evidence shows to be extremely doubtful, there are no known agencies likely to be present in the sea capable of bringing their skeletons and tests into solution with the rapidity shown to have taken place in the deposition of the chert. Cherts of magmatic origin differ in field-relations and associa- tions, and often in structure, from the cherts which derived their silica from the land. The latter occur mainly in the upper, generally thin-bedded, dark limestone which was laid down near a shore-line. It is believed that, when a sufficient concentration of silica in the sea was attained, rapid flocculation and precipitation would result from contact with the bivalent ions of lime in the presence of carbon dioxide. The author adduces evidence to show that simultaneous de- position of silica and calcium carbonate often took place, and it is believed that, in such cases, segregation ensued, and sometimes resulted in the formation of nodules and lenticular masses of chert. It is suggested that the bedded cherts of terrestrial origin re- sulted from heavier precipitation of silica, comparatively free from calcium carbonate, and spread over the sea-fioor by gentle currents. Metasomatic replacement of limestone and calcareous organisms by silica has taken place at their contact with the chert. Impuri- ties in the silica have tended to limit such replacement. The freer the chert was from impurity the greater was the replacement, but the chert is not a pseudomorph after limestone. Impurities have also had an important effect on the crystallization of the silica. Organisms existing in the sea or on the sea-floor would be en- tangled in the precipitated silica, and their presence in the chert is thus explaimed. The blackness of some chert is shown to be due to the presence of carbonaceous matter. Ferrous iren may possibly have operated sometimes in the same way. The banding frequently seen in chert is believed to be due to a segregation of impurities by diffusion and rhythmic precipitation in the course of dehydration. THE LONDON, KDINBURGH, anno DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. | {i 2 2 < [SIXTH SERIEB.] ee ee Soy FEBRUAR ¥ 1921, XIII. An Asymptotic Formula for the Hypergeometric Func- tion »A,(z). By Dorotray Wrincn, W.Sc., Fellow of Girton College, Cambridge, and Member of the Research Staff, University College, London”. | ae present communication is concerned with the asymptotic forms to which certain generalized hyper- geometric functions tend as their argument which may be real or complex is increased. The expansions obtained include as special cases most of the more familiar asymptotic expansions of the more special functions of hypergeometric type. The investigation originally arose in relation to certain functions which occur in physical problems, more especially those of elasticity, and the need for asymptotic values of these functions has been pointed out yf. The particular instance which gave rise to the investigation is concerned with the lateral vibrations of bars whose cross- section is a function ot the distance from one end. The more important functions which thus arise can be included in a hypergeometric type analogous to the Bessel functions. We consider in this paper the general function oAg[Lt+e, 1+, 1l+y,14+6; <] eo) ‘bl Sop iyete yh iit: anne SRO oni LI (1d +a) (2+). .(ut+) aByd where the parameters a, 8, y, 6 can take all values except ~ * Communicated by the Author. + Vide e. g. Nicholson, Proc. Roy. Soe. 1917. Phil. Mag. 8. 6. Vol. 41. No. 242. Feb. 1921. M. 162 Miss Dorothy Wrinch on an Asymptotic negative integer values. Ihe expansion of this function is developed for all values of <, real or complex, and the procedure is indicated by which we can determine the linear combination of solutions of a characteristic differential equation of the fourth order which takes any of the indi- vidual forms ghey —atB+y+8+3/2) 4 —40l Mat p+yto+3/2) g Mat B+1+0+3) cos [dalt—1 (a+ B+y+6+43/2)7/2], gp Hat B+14+8+82) gin FA atlt — 1 (at B+ +84 3/2)0/2 |, near infinity. Approximation to a Contour Integral. Consider the integral ee £2 al lta fp) z/t Gage \ dt (fee te 2 ee a z being a complex number with principal argument @ and modulus p, round a circle enclosing the crigin. Since both the series t? oAy (t, l+a)= +t alt ; a are regular within the contour, the eae is equal to the residue of the integrand at the point ¢=.0, viz. : p4,(2; l+a,1+f)=1 i Z ' l+a.14-8 + a ot ltea.2+a.14+68.2+6 Thus in order to approximate to ,A,(z; l+a, 1+) as p—>#, we may consider the asymptotic behaviour of the integral I= a | oAr(t, 1+ a)oAr(z/t, 148) — e Formula for the Hypergeometric Hunction »Ay(z). 163 as p— > where OC is a circular contour round the point t=0. Let 6 be the radius of the circle C, then ar 1=| pA; (p/de-%, 14+ 8) A, (de%, L+ajdé. 15 Let y stand for the argument of z, which is such that —17<(y—8@), since pAr(z, 1+ v)~T (1+ v)e%2-° — OO 2 y—n+ 1 VI If —w», . hes : T \ e-Mdz~+h,/7 i — > ~, if a is positive and real, (3 Ca ) °. T a EAS if | Zz rE de~},/7, IL rae 3} << a Vx << 9 . > = Then , : eitte—a\on ae oye 2 p~ Hat), ante) | pret! o. CY apa rae 3 al 2X» 1 7 i ag, a Xp Pte = = p24 tB+2)e— Xr at B+) 22p?%e a 2 || 1 ty. LL Seta) Oe ; ae Jp Bet Bt o—XlatB+3),2pFeXr Next take the case when 7 and therefore also —7 is a critical point; then the integrals p—2¢+P) : €— 203 cos ede, J) —2 and or 3(a+ ) ese o APE COSE Te 0 ee are negligible as ep —>~, after the manner of the asymptotic Hence +7 as critical points theory of the function I.{2). contribute nothing to the asymptotic value of J,. Therefore f ial Ss —il(a 4+ 3-1) —ty,(a+ 3+) 3o20!Xp I, ~T(l+a)FU+8) a 4/mp7 bP TeX TP Ta) o= ; the summation being taken over all X,’s satisfying the conditions —7» if —2r~x with p, and y stands for an argument of <« which makes — 2a < y—@ < 27, then 3 Tl+ePl+6Vil+yP1+6 Sir? oAu(<; I +a, are I+y, 1+6)~ at+B+3 2 (at+B+3) yto+d 4. y+o+ { Z eee. Cy we ae) | e/5 Pi ale 1 1 9 2 20/2 932 10/2 +i eed x ( Cees Oe a) ~ ea a e—2(0/8)2e\x —9)/2 tin(a+B+4), Hither value of y will give the same result, the order of the terms being interchanged. Let us take the smaller in absolute value, viz. ¢ itself. In this expression the lower or upper sign in ety bet air is to be taken in the range in which @ is negative and that in which @ is positive respectively, and the upper or lower sign of et™4+B+2) is to be taken in the range in which y—@ is positive and that in which y—@ is negative - 3610/2 5 2 respectively. When @=0 the term e~~?” etmily ters) is to be omitted. Similarly, when y—@=0, the term 19/2 . 1 pa 208d! et'™(@t+B+2) is to be omitted. Hence ONG ces sry Ss learig th) Vi+ell+6Pi+or1+6 oud z Sar? : [Nit iNo+2Nit2No], where 9 | as Lora Garin) m={ Cee Ay (plore i) : vv —T al s82i8/2 4 (opayher > | xe a ~ dé, Oe ee nll olen merkel aay a {* etm tots) 5,10) — D “—7 a+ 3+ — 9936/24. 2(9/8)2e (x ~ 9/2 e id ( x (p/8 Boome i 2 168 Miss Dorothy Wrinch on an Asymptotic 4 2 y+é+2 N= \ ee net) bc) ane —T a+ B+ a 15° i = 5 S 10/2 _ « ad ix — 0/2 x (p/6 ex %)) ae a 2(p/r)Pe dé, & at T tmi(y +644) +7i(at+B+2),.. i6\— —— 2No= € 5 an aoe) a —7 paths x (p/5 e'lx—8)) a. dé. We are still at liberty to choose 6 as we please, provided only that p/6 and 6 tend to infinity with p, the factors po iain) being adopted according as @ is positive or mi(a+B+4) being pn 2ikelOl2 —a¢pyeybellx—9)/2 negative, and the factors e= adopted ac- cording as $¢—@ is positive or negative. We shall therefore, if possible, choose 6 with a view to ensuring the existence of critical points for the integrals Nj, No, oN,. oNo. Critical values of @ will exist if there exist roots of the two equations dfs 3 Ay ae 16 E é a ¢ é - We therefore choose 5=p?, which gives the only possible case. Consider first the integrals ;N, and .N.. Critical vaiues are given by the equation o=% + 2pr. There will be one, and only one, for each integral, and it will be Hence, using Stokes’s result as before, —HatB+y+041) —if(atB+y+0+) iNi~p é 4 Formula for the Aypergeometric Function Ax). 169 ete Ganon hs ee Ta (ot). 5 tmily+ o+at+B +1) Ze p ae Sy yes $e} x(" 2 ¢ +/ de 2 ee 9 Se ae Oyo to) 2) — (at p+y+o+3/2) = /27 .p e + ig —4pie * +ri(atB+y+6+3/2) x € é according as ¢ is positive or negative. If 6=0 the integral does not occur. Consider next the integrals ,N, and ,N». Critical values are given by the equation g=% +(2u+1)7; o—0=2— Out 1)z. 3 There will be one, and one only, for each integral, viz., O= . BE Fh according as @ is positive or negative, except in the case ow, when there will be critical values 7 and —7 if Oi ae 9 +7; y—O= - : Aer Hence, according as ¢ is positive or teeta, —Matbt+y+dter-(S5a) Flyto+3e oN ~~ p a“ e = a a eo ae elanes -i(5 7) nese -i(§an)* Pie xe hen ine, 5 ies) x ao [2- ole. (a+B+y+o+]) hi aaa aaa 1/? pte 2 ) 3 3/9) sotet(/4-tm/2 ats J 2r[pelio2t™] —L(atB+y+o+3/2) 4pFe? ) if o£0. 170 Miss Dorothy Wrinch on an Asymptotic The values of @ lie between —7 and +7 (+7 not excluded), and the result therefore holds in all cases under consideration, for Stokes’s result holds for all integrals of b the form { edz, when —7@< argaA,(z). 171 Hence oA\a(e 3 Ll+a, 1+, l+y, 1+6) Tl+ell+@6Pl+Prl+6 95/2 q73/2 —}(a 6+8/2 F (a+B+y+ mel [B]. where B is the preceding large bracket, according as ¢ is positive or negative, the last term being omitted if ¢ is zero. Hence in particular, p being real, Wiese fo 048, ley 145) Sei Nate LR Ttea.l+8.l+y.14+68 °°" oils p et b Se earl Fy T+? Mat B-+y+0+3/2) 25/277 3/2 [owt +2 cos 4 4pt - Flat B+y+b+3/2) | | and } aay eae ee Mer eine eae (" cos = 95/277 3/2 X COS | 4ptsin 7 _ il (a+ B+y+6+ 3/2) |). Special forms near infinity. Series of the type >A, occur as solutions of equations of the type —8(S+al(S+b(S4+c)o = zo, which are of frequent occurrence in problems of lateral vibrations of bars whose section is not uniform. If we put @= 2 y, the equation becomes (3—v)(S+a—v)(S+b—v)(S+e—vy = zy. The general solution of the original equation—if we formally exclude the cases when logarithmic solutions are involved— give De tel pace een ar ery eae fee 95/2__3/2 s ene + ex 4 gz* p—glatbter3/2) AT 1 12 ‘ fe AFT etbte432) 4 dt Fmilapo+e+ 97) | | T1—all14b—al4e—q _*totet+8/2- 4 aye¢ —G(—80+b-+0+3/2) WS 2 2 FE 42 +e = Y2 5/23/2 4 e e AN L Tek Borg a —4i2t + 5(—3a+b 3/2 2) eee : 4g Bett 3 a+o+e+ (2) 4. 9 etre seen Pi-b11+e—bV i t+a—b “erbrets2- yt 4 eee : Y3~ 95/2, 32 & 4 | ef $e" ge ea ae ea Ts zi a Piel Fase 4 b= ey 2 Ua 98/2, 3/2 © 4 acre i 172 Miss Dorothy Wrinch on an Asymptotie is therefore, P, Q, R, S being constants, P,Ag1+a, 1+b, 1+¢,1; 2) +27X))Ay(—a, 1+b—a, 1+c—a,1; 2) 2 "RoA,(—b, l+ce—b, l+a—b, 1; 2) +2z7°S,A,(—c, Lta—c, 1+b—c, 1; 2). Let 7; Y2 Y3 Y4 represent these four solutions, which together form the general soiution. Then the results obtained with respect to the asymptotic behaviour of gaa, Is ees ey, iG z) Linear combinations of 4; ¥2 7/3 y4 therefore exist which behave asymptotically near infinity like (a+6+e+3/2) 1 a+b+ce+3/2 1 Raa TACT 424 —————— —424 Zz Ze ee Zz 4 é 5 _at+b+e+3/2 a+b+c+3/2 Zz 4 cos4z4, or z 4 sin 423, Let us consider the linear combination which is asym ptoti- cally equivalent to _atb+e4+3/2 i 2 Ae Rem as Formula for the Hypergeometric Function pA,(z). 173 Writing 95/2, 73 2 atb+te+3/2 ¥ ‘ = = = aren 4 Ys on Peeerktteb) lsc) Ges pea c+3/2 an te 4 io, Le eee il +b—a) V(l+e—a)* i 93/2, 3/2 Caml teneel 2 4 3, oa Sh eo ab) hi 5/2__ 3/2 atb+e+3/2 = ya j i: 4 Pi—c) TU +a—c)T(1+b-c) it is clear that the linear combination | Y.. 1 1 0 | Ye i cosma sinwa | y3, 1 cosm@b — sin arb ne ae: i, COS TC sin 7c 1 will behave asymptotically like Cems simpler form of this expression is, on development, 4y,' sin (rb —c) sin (we—a) sin (ra —b) —4y.' sin me sin wb sin 7(b—c) —4y,' sin 7a sin ae sin 7(c—a) —4y,' sin 7b sin 7a sin 7(a—b) =— e sin (mh —~ ¢} sin a Sin co sin wa sin 7b singe [yD —a)P(—))P(—e) PL D(1+6)0 (+c) + i/2 Whe doe a a6 aa +y3sl(b)PAL—d)P(o—a)P(b—e) Pi 1l—-b4+ a)P(1—e+ 6) +y/0 (co) TA—-e)P(e—b)Pe—a)P (1 —-¢+6)PA—ate). Thus yl'(— a) (—8) F(—0) + yoP(@)T (a= OT (a—d) + ysl (b)P(b-- a) P(b—¢) ty (P(e —a)T (e—b) a+b+co+3/2 4.4 ~Re 4 Buin where A is a function only of a, b, and c. In the same manner we can isolate the solutions which tend to the other individual asymptotic forms, and the value of X is obtained readily in each case. Such determinations are, however, apart from our main object, and this instance will suffice. b) [ele XIV. A Generalized Hypergeometric Function with n para- meters. By Dorotay Wrince, M.Sc., Fellow of Girton College, Cambridge, and Member of the Research Staff, University College, London ™. i N\HE behaviour of the series 2 z 1 a = ? a nm—1 +... 1!Wl+e,) 2! Wd+e,)(2+4,) P= PSll as | <| tends to infinity has been investigated in several particular eases. In the case when n=1 the series is easily to be derived from the Bessel function, and its behaviour on the circle |z|= is well known. In a recent paper by the present author, the case when n=3 was worked out. The asymptotic expansion of the series in the general case has gradually become apparent, and is established in this paper. We may obtain the form of the asymptotic expansion of the series oF n—-1(1 + a, 1 a (24 : i+ An—-15 és) ot ~ a m—1 ? Tobey terial (Ll+a,)(2+4,)...(u+a,) rl when | < | is large after the usual manner. oF ,-1(¢) satisfies the differential equation Ss(S ai a). oe (S ae &n—1)Y = Ye, which becomes S(S + nay) ...(S+ nen i)y=yt", if by analogy with the treatment of Bessel functions we pu 2 (b/n)r, t/n being any one of the nth roots of z. Putting y=ety, the equation becomes (3+t—a)(Sttt+ny—a)...(Itt+ne,1—2)y=ty,. If w be chosen in such a way that the coefficients of the leading power of ¢ on both sides are equal, the asymptotic expansion for oF ,_1{(t/n)"} will be Si(n)etee (1+ : “ oh * Communicated by the Author. A Generalized Hypergeometric Function. 175 where j(n) is independent of ¢ and the summation is taken over the various values of t, which make (Cie. the series y;==1+ a + .. being obtained by solving in descending powers of. t, the equation resulting from the equation (S+t—wx)(S+ti+na,—2)...(S+t+ne,1-v)y =U, by the substitution of the appropriate value for 2. It is evident that the coefficients a, a... are independent of the particular nth root of z¢ chosen. Clearly the relation between the coefficients a,, a .. will involve more than two consecutive ones; we shall therefore not attempt to find any but the first two. The equation may be written in the form Ug Gere g@gt 1 Agt”*. 1A, 3t + 4an) + 3( yy) (gait"} aly gigt”*. Seats on) es ne oa Gp) Gadde’ st jae ihp) +3") =ty1, and the coefficients ,a, can be found by induction and inspection. In order to find only the first few coefficients in the series of descending powers of v, only a selection of the ,a, need be determined. Equating the coefficients of ¢”~1, ¢"-?, t*-8, on either side to each other, we get the equations 14,=0, 1A & + 1 Ag— 9 yy = 0, 11 ky + 1g @ + 43 — (9g a) + 290, a9) +30; a,=0. These equations determine 2, a, and a. As to the value of the ,as coefficients involved, it is evident that n(n—1) 91 =n, 30) = 5) y) : : = The value of ja, is “.—~ =, where & represents the sum 1 Yr of the terms —a, NA, — WW, wee NA 1 —W, taken * at a time. Since ,;a,=0 this gives 176 Miss Dorothy Wrinch on a Generalized The values of the other relevant ,a, coefficients found by induction are as follows : (n—1) (n= 2) 9. n(n—1)(n—2)(3n—S) = 2 2 . 24 tes. 0 SBS s. (n—1) eee —s 3 tos =) o= eae (38-8), = —1)(n—2 oes n(n a ie) eens a, and a, are then found to be 2 2 1Aq/9%, | 12” + (341 — 2a) 1g + oy - 143] /2 2”, respectiy ely. Subsequent coefficients a3, a, ... can be found in the same way, but they will be increasingly cumbrous. They are not, however, of the same practical importance, as a, and ay ll in general, alone be required in applications. The form ae the asymptotic expansion of yF',-1(¢) 1s then s te a7 1 eye | 2 | yD SAUD (14 ae 4 a Vie n~ rT where a1 n— 1 sp = As Sa 2 and ,z, represents an ath root of <, the summation being taken over the nth roots of z Now, in the case of a value of ,z, which has a negative real part, whatever the value of /,(7) the corresponding part of the asymptotic expansion, namely v=l Ny i, ‘ wo ay ia a eo wr{ 1+ > ee | is asymptotically equivalent to zero. Hence, although it is true that pnner Zoe By o()~ || # BL) ere (1) Jn) corresponding to a value of ,z, with a negative real part, can take any value whatever, and there is therefore Fypergeometric Function with n parameters. Were no sense in which any one set of values of /,(m) corre- sponding to such values of ,,z, gives the asymptotic behaviour of of,-1(2) when |z/| is large rather than any other.- It would therefore be misleading to leave (1) as it stands, where certain functions of n are substituted as the values of 7,(n); for the asymptotic equivalence persists, what- ever value may be given to those of the set of f,(2) functions corresponding to nth roots of z with negative real parts. ‘There is, indeed, no sense in which one can talk, for example in the case of n even, of a “sudden jump” * in the value of the function of n multiplying a > l/n 4 \2\"" as arg (z) changes from being the exponential term e In positive to being negative, since the product of e—”|*' and any heaton: of n whatever is asymptotically equivalent to zero, whatever the value of arg(z). We therefore proceed to find the wnique set of functions of 7 1, whose existence is already plain, which makes the asymptotic expansion of F',_1(<) equal to lz ee > de (n)e 2 n@ > Le . ee } R(nz,) = 0 I= = Meno Hea n—l when |z| is large, s, being Sa,+- 5. Sl 74) If C is a closed contour containing the origin of, 1 + a, ee Sob 1+ ay} 5 Z) all acl (L+e).. a1 5 t) J 1+; ari (a) . Les ae) Il+ta,.2+a, Further, if the contour is so chosen that | jt} and | 2/t| tend to infinity as |z| tends to infinity (and it is plain that such a choice is possible), it will be possible to obtain the leading terms of the asymptotic expansion of ,F, by aj proximating to the value, as |z| tends to infinity, of ae integral in which the leading terms of the asymptotic expansion of of n-1(¢) and of eft ( «© of the integral in which the integrand of the integral (1) has been replaced by the leading terms of the asymptotic expansions of the series. We have Le Ne i dt x ( I ae )’ : OR a mn mart yy The only limitations on C, which is a contour in the t-plane, are, that it shall enclose the origin and be such, that on it |¢| and |2/t| are large when |z| is large. We may therefore choose as the contour a circle with its centre at the origin and with radius 6, provided 6 and |z /6 are large when |z| is large. Then z OARan Sy Gi ase TI iG I + ats) 2 l/n a n — an —1 ea Ss { dé s Be uae S n 2 n n NT ay x Coleen =O) Nie S ove va1 2" (a, —-14 v), ; the s-summation being taken over integer values of s, such that i i) = n =O @—@ being the argument of ¢/¢ (requiring sometimes a different value of @ in different parts of the range), which hes between —a and -+f. [The special case when N 2 180 Miss Dorothy Wrinch on a Generalized @~-0=+7 or —7 will not arise.| Let I, and I, represent 6+ 237 Sy a la) i—— - 2 04287 rc) i(p —@) iN T 1 nN | aa —e a : Ss ere [No 5 n 3 i ae “ elt =) ie ~ian(O—8) JO CUMS cs i 6 6+2sr : 7 nglime n ae en so ae a) O°e 208 ye OM TG dh x 2 dé vw—-T7T § =i fe, NG. re ih +v VG respectively. I, is made up of a set of integrals of an oscillatory exponential type and we may approximate to their values by Kelvin’s method. ‘ Critical points ” of the integrands of these integrals occur where the derivative of the exponent is zero, or a 6 Lin, Faas u(G—@) = 0 = lel ) u and are easily seen to exist when, and only when, 6=|2{""*?. Taking this value of 6, we satisfy the condition that on the contour C, |¢| and |2z/t| are large when || is large. The integrals to be evaluated for I, are then of the form 04 2tn api) 1/n+1 linet —t en = je | A é = WT: _(0-4+2tm teehee NG ae Si ha ae wy, |Z 0 n ) “ Critical points” are those values of 6 for which (4 9 vas ae —($—0) = 200 0 + Qtr _ pt 2tr AR 2tir 9) n Suppose 0= @—€ is a critical point of the integrand of ,J,, that is to say @—€ satisfies (3) and is less than or equal to a in absolute value. Then Syptan : i peel. SOS Gans) OS =a | g|ln+] 25 te a i> It | Z| ti @ i "( al (n41 = 2...) ae 5 su w& @—S+7,. nee iy ~ = bail n+1 PS ear eal Ms jf aa a5 Ss n+1?’ since as x tends to infinity re a lis 7 : edz ~4 7m, if See C8) ‘ aa Hypergeometric Function with n parameters. 18L From the equations (3) it is plain ithat every value of ¢ is such that | A Foes is an (n+1)th root of z, and the summation is taken over those values of € alone which are less than or equal to 7/2 in absolute value. Thus the asymptotic equivalent for ;I, is the sum of certain terms of the form ~), ae a (Sp tay +2) Pi neeNe a n+1*r n+1 — I Sati >n ar eats 2 of the terms of the form 2n al / —8$ n+) < pu he aS 12 n+le TUT Nee i (nt r) The question remains as to whether, when all the integrals of the term ;I, are taken into account, any of the (n+ 1)th roots of z with a non-negative real part are excluded (it is of no importance whether those with negative real parts are excluded or not) and as to how many times terms corre- sponding to each of these roots occur. A slight consideration of the equations (3) shows that the first possibility is cut out. As to the second point, it is clear that only values of € less than or equal to 7/2 in absolute value need be considered, In this restricted class of cases, it is plain that the same value of € cannot occur in the case of integrals corresponding to two different values of t. Hence or since nu nt LIV +a,) THU (1+ 4s) 1 il aie S (2+1) ey aa a F or oe a ale) ae ean a (Qin) 2 Vn (Qa)? Vn+1 the summation being taken over values of ,,,,2, satisfying the conditions ~y ae (nies ~ ~ 1 we 7 sure 2 me ale EM eT Y,? (,41%,) »+1 being interpreted to mean the complex number whose argument is a, Sn x are n+17r* The integral I, has no “critical points” in the Kelvin 182 Miss Dorothy Wrinch on a Generalized sense, and therefore by considerations which are sufficiently well known jn applications of this method, the important ‘““oroups of errors” in the divergent development of the functions all arise from I,. We shall, however, show independently that I, is negligible compared with I,. In considering I, we again choose a circle with centre <=0 and radius 6 as the contour, but are free to choose 6 as we please provided only that 6 and |z|/6 are large when |) is large. Then . Ga oe x nde), ve Qn il : slin SE5)) 2 Be = _ { pe : } nN, 2 ob ene sr dt, kee ec ¢ ‘ R(s,)) 27s eee nd!" &— n Dime n aN But let 6 be so chosen that 6 =; (PAOsr iL) Then 1 i ae R(s,,) ? a Fy oe gear 1 2@41) oqay at { Opiate — di|< Ke” 4 ; Cc ) | ct Whatever the value of 7, this is negligible compared with any of the terms a + ice 3) —Sn41 in which R(z,)>0. Hence I, is negligible compared with I, and the asymptotic expansion for 9, results in the form TP (+4) =| (2am) Vn+1 aH SPs « eal) — an 5 2) ~ S eT nn es) ~ Sn t1, arg (n+ 12,) | = is This result is of a similar form to the one assumed for ofn-1 and already proved for F3, and is therefore now established in the general case. The complete form is CO Fypergeometric Function with n parameters. 18 then Il V(1+as) pratt oy, 2 14 ae 3s) ~ Faas } 1)) Cy Sf \ Be Ky x S: +L) pe, 2 (a ), i enh “) T mal (a He ine Gee {arg 412r| = It is worth while to notice the forms which result for of, -1(¢) and for 9F,_1(—z) when z is real and positive. If n—1 oe | e Shc nr @ is real, positive, and large, and s,= > a,+ 5 i ont x xv? eee eT ane LY H(1+ a.) 21 T1(1+ay)(2+a,) r—1 4 ae ae 1/n ee ff S,/t 1 nex Ov M=IL ub E [ise D) (Qn) 2 ./n 2tin = 1/2 1/n 2nt ay Dart Qart ee ce OP tanh COs E © ea) —— — (ss oe ») | | o/n f= ll n Gh Ve As a first approximation we may record the result I MT +2,) 1+ — eo ee Aenea ees —Sp/ n—1 th ete Al c it I1(1 +4,) (27) 2 Jn Be n ni? on 2, arts x ee 42S ener cos’ : cos (na sin a a l , jal n oe or n—\ (1 HT(1+.¢, ie cen Ie Weer aa ergata fim cr ih ! md oP O,.) (27r) Jn 2 van 7 a rain, Qrt Amt ares x coe a en COS ( sin r) bs 184 Miss Dorothy Wrinch on a Generalized and useful results in other cases such as r—l (ur) ee IS ee a da rat Gar. au) aL (Qa)? fr eon 2 x E + 2e ji cos( sin a — =e) I, when 7 is 4, 5, 6, or 7, and in the case of r=8 7 lU1l+a, 8) 1 Ce oy ae Tie Tat aay Cah eae 1 L 1 1 9 ee08 2 ‘ : LIDS: _ Ss xe + 2e eos (u nF it) + 2008(u pals and so on. Similar results are available in the case when z is negative and real. If a is large, real, and positive v—1 1d +2,) a te, i Tee AUNT PAN Wee Gs ie pT q J + we ! ieee ) Vee l i Sa fe (2H) n/n 2t—1 1 Rie Ds (Qt—l)m earl Ey ee | “ Lf? —— . VERS) a SG Sh EAI ce | cos nev” sin —— 7 13 n f 3) OEM OC, PN tame —I 2t—1 ve > — eos | na!/” sin — T————_7( $, + ict n a and ae as a first approximation, we may put OT P II ale +. Ay Saal = nae!” (2t—1)x —~ % u Eaaek 3 i 7 a ek + odd Z g s A 4 (27s Jn an NW i SNe en 2t —1 x COs (na sin T — msn) n n . or nm—-1 ot—1 ( Ny M a te iin n a 1 7 ae or Ow —— (w/2) aS k 4 = Aan = a (2am) 27 a/ nt Aypergeometric Function with n parameters. 185 which gives the corresponding result in the ease when n=2 to the one already pointed out for I,(v), viz. 2 a 4 Jn(v) ~ 7 o8( U5 47); and useful results such as Raina Nas ll Aa DEM rg POY TL 8 Gajey gong PW +a, (2)? /r a 5 ae), Gis. x cos { usin—- —-—"}, Yr for r=1, 2,3, 4,5; and for r=6, 1. @/ey ~ th eee Papa eres a ih let ae 1 ~estri (ufos ee ees (w sin 1/6 —a/6(Sa+5/2)) + cos (u—7/6(Se+ 5/2))]; - ~and so on. | The other (n—1) solutions of the equation S(S + ay)(Stes). (I+, Y= Ty, satisfied by Ups bie eighteens. at ait a 512) 5 are the series —qQ : t ema Cael ea So ee OY ine = eae 1+ Ch ELE z) with r=1, 2,..2—1. Calling these solutions y, yo,...7,_1, since the sum of the parameters for y, is Ye—na,, it is plain that the asymptotic expansion of y, is of the form La—Nna,+ if — 5 mare CROMER Caer k it Sli arse en) or a (0+ ey) ha hae | “ 2 jn Si (n)z Bees 186 © Dr. Houstoun and Miss Dunlop on a Statistical so that we have y= f(n)e mena", — Ane > Sn/ Manzi Hy ae S/n In 3 Yass CaN meee U sean = Syne e nl onzt Les —] where sn=da+4+ a 2 Linear combinations of y, 7;,...y,_,, therefore exist whieh behave, when |z| is large, like 2~*e2"”, or like Wie ane pong = cos (nz! ” sin em = =| : 7 7 XV. A Statistical Survey of the Colour Vision of 1000 Students. By R. A. Houstoun, D.Sc., Lecturer on Physical Optics, and Marcarer A. Duwtor, Thomson Experimental Scholar in the University of Glasgow *. TNDER the title of “A Statistical Survey of Colour- Vision,’ there appeared a paper by Dr. R. A. Houstoun in the Roy. Soc. Proc. A, vol. xciv. p. 576 (1908) tinge paper described the results of a test made on the colour vision of 79 observers by means of an apparatus similar to the Kdridge-Green colour perception spectrometer. Hach observer determined the number of monochromatic patches he saw in a continuous spectrum ; this number varied from o Fig. 1 ” x x x Pad x x x x x x Kx x x x x x x x Oo x x (o) (e324 LO see 3 x ONO RE ORONO IG x eao0qogo0o OC x x * 1e) x*O000000 X09 1@) ooxao0000000 000 00 Ove ie a it Ro 3 10 1S 20 e5 five in the case of the colour-blind to twenty-six in the case of an observer with exceptionally good colour vision. The results were exhibited in the diagram (fig. 1) where the * Communicated by the Authors. Survey of the Colour Vision of 1000 Students. 187 circles represent men and the crosses women. ‘This diagram is a frequency curve of the kind used by statisticians, and the object of the investigation was to find whether this frequency curve was of the type represented by fig. 2 or fig. 3, 2. e. whether the colour-blind were merely the “outhers Fig. 2. Fig. 3. of a homogeneous population” or whether they were an independent or independent homogeneous populations of their own. The second alternative is required by the Young-Helmholtz and Hering theories. The conclusion arrived at was that normal colour vision has quite enough “scatter ” to include colour-blindness as an outlying portion of itself, but that it would require a very much greater number of observers than 79 to positively disprove the existence of separate maxima for the dichromats and mono- chromats. The present investigation is an attempt to get more certainty by using a greater number of observers. At the same time an entirely different experimental arrangement has been employed. It was stated on p. 586 of the former paper that one had to trust to the judgment of the observer as to when the difference of colour appeared and that, while nothing occurred to indicate carelessness on the part of the observer, still it would be better if a method could be employed in which the readings of the observer were checked. - Consequently attempts were made to devise a new apparatus, and after a little trouble the following very simple and satisfactory arrangement was adopted. A low-power microscope was taken, one previously em- ployed in the elementary laboratory course for determining the index of refraction of glass slabs. The focal length of its object-glass was 4:0 cm. and the height of the object- glass above the stage could be determined from a millimetre scale ; when the microscope was focussed its tube moved up and down this scale. The microscope magnified about 25 times. On the stage was placed a slide carrying ten test 188 Dr. Houstoun and Miss Dunlop on a Statistical objects ; the test objects are represented in fig. 4. The test objects consisted of pairs of circular disks, which might either be black or red or green or blue. ‘These disks were Vig. 4. pare. 2@ S® > wat | ® | S | @ © 6 | @ S@ 6 So 88 & tH = \\ BUACK “RED BLUE “GREEN each slightly less than one millimetre in diameter. They were drawn with a pen all on one piece of cardboard, the black ones in Indian ink and the coloured ones in Dove’s water-proof inks. The piece of cardboard measured 42 mm. by 15 mm., was protected by a glass cover, and was mounted on a strip of wood. The red ink, Dove’s carmine, and the green ink, Dove’s deep green, were used at the same strength as they came out of the bottle ; the blue, Dove’s Prussian blue, was too dark when used at full strength, and was con- sequently diluted with water so as to have the same luminosity as the other two colours. The observer’s colour vision was measured by the distance out of focus at which he could Just distinguish the different colours. Before commencing the test the observer was required to read the following instructions which were typewritten on a bie0e of cardboard with the disks entered in their appropriate colours :— (1) First focus the two vertical black spots pd as sharp as possible. Next focus the two horizontal black spots as sharp as possible @@. (2) Screw down until you can just sry whether the object is the two horizontal or the two vertical black spots. Repeat. In this and the following tests the micro- scope is to be as far out of focus as possible consistent with your distinguishing the one case from the other. The distance from focus is to measure the excellence of your vision. © © eee s Survey of the Colour Vision of 1000 Students. 189 (3) Serew down until you can just see which of the above four objects you have, then say whether the red is above, below, to the right or to the left. Repeat. = © 8 © 4 Eu. ee (4) Screw down until you can just see which of the above four objects you have, then say whether blue is above, below, to the right or to the left. Perhaps the nature of the test will be understood best from the figures for a particular case. The vertical black ” spots were ‘focussed sharp at 34°5 and the horizontal black spots at 34:3 mm. The mean, 34:4, was taken as the observer’s focus. He was then given ‘the horizontal black disks very far out of focus, so that there was nothing visible at all in the field, and screwed slowly down until he could just recognize them. This he did at 48. Similarly he could just recognize the vertical black disks at 49°4. The differences of these two numbers from 34: 4,2. e.13°6 and 15, were taken as a measure of his ability to distinguish black from white. The supervisor next placed a red-green pair in the field far out of focus. The observer looking into the microscope saw nothing at all in the field ; he serewed down and recognized the object as “red left”? at 49°7. He was then given another red-green pair and recognized it as “red right” at 52°5. Consequently his ability to distinguish red from green was measured by 15°3 and 18:1. He was next given a green-blue pair and recognized it as ‘blue above ut 49. Finally he recognized “blue below ” at 45°9, so that, iis ability to distinguish blue from green was measured by W456 and 11-5. Very often, when he was first given the black disks out of focus, owing to having not fully under- stood the instructions ine observer screwed down much too far and asked permission to repeat the determination. This was always granted in the case of the first test with the black disks, but not in the case of the other tests. The test with the black disks was arranged principally for giving familiarity with the apparatus before the colour tests were entered upon. As the coloured disks were upon a white background, and as they merged with the background when they were out of focus, the effect of screwing ‘ouit of focus is simply to mix more and more white with ‘the colour in question, until we have finally, in the case of “red above” for example, merely 190 Dr. Houstoun and Miss Dunlop on a Statistical a white field, with a reddish tinge near its centre and immediately below this a greenish tinge. Hach observer took about seven minutes in all. The tests were made between January 1919 and March 1920. Of the observers, one was a member of the University staff, 336 were arts or science men students, 84 were arts or science women stu- dents, 261 were first year men medical students, 79 were first year women medical students, and the remainder 239, 237 men and 2 women, were first year Technical College students, whom we were able to test through the courtesy of Prof. Muir and the staff of the Natural Philosophy Department of the Technical College. All the tests were made in the course of the student’s work in the physical laboratory; each man simply left the optical bench or spectrometer or whatever experiment he was working at, and then returned to it again when the test was completed. Only owing to the large number of demobilised men in the laboratory courses was it possible to carry through the tests with so little trouble; the students were older and, especially the medicals, more capable than usual, and such a favourable opportunity for making such a survey will probably not _ occur again. The numbers are not the full numbers in the laboratory courses. Some students were omitted through absence or through their not reaching the optical laboratory; the observers were not selected in any way, but just taken as they came. One student declined when asked to make the test, as his experiment was at a critical stage at the time, and asit was the last week in the session no opportunity occurred to test him again, but his was a solitary case. No one else declined. The observer could not see the test object except through the microscope. In the tests with the black disks the horizontal spots were sometimes given twice and the vertical spots not at all, or the vertical spots might be given twice and the horizontal not at all, but in the case of the coloured disks it was always a different object each time. After the tests were completed we determined the average distances from focus at which each of the ten test objects had been recognized. They were: black horizontal 13°00 mm., black vertical 12°83, red left 15°24, red above 14°89, red below 15°52, red right 16°13, blue left 15°70, blue above 15:44, blue below 13°59, blue right 13°76. Thus all the objects had not been equally difficult: to allow for this the readings for red right were decreased in the proportion 16°13 to 15:2, those for blue below increased in the proportion 13:59 to 15:6, and blue right in the proportion 13°76 to 15-6. Survey of the Colour Vision of 1000 Students. 191 This does not make the average distance the same in each case, but it reduces the inequality sufficiently to make it negligible. Thus to return to the example already cited by way of illustration, the observer’s values for the black disks 13°6 and 15 were left as they were, his values for the red- green were corrected to 15°3 and 17:06 and his values for the blue-green to 14°6 and 13:2. This gives a mean value of 14:3 for the black, 16°18 for the red-green, and 13:9 for the blue-green. These three numbers were assumed to specify the observer’s ability to distinguish black from white, red from green, and blue from green respectively. Similar TABLE. Men. Women. | Range. DR Ce RE a EEy, | ieee. Red- Blue- | Bias Red- | Blue- green, | green. green. | green, PEO el. be 3 | | I Ga I eam fr 2 | Lins YY, Ses E yan ae 2 He 0) O) (0) Ee (2 ae 1 2 rele 0) NG Gant ae 0 1 1 ) Ge a ae ee 0 2 1 in 0 Olaf aaless. 2 1 3 2 1 Terie) eee ik 5) 4 1 (0) 2 3) Coo nee 8 5) 4. 1 1 il OSG OLO) oacn os 16 Ne} 13 5 2 0 WOR cae Pe 68 23 17/ DA 7 3 HO ee

2226; 13311: | 1:4355 : =H oh) WAR spy cng. eC ele ssl SMa porn tlre: Sir George Greenhill * has recently called attention to the practical importance of the Bessel-Cliftord function, C,(2). Since this function can be expressed in terms of the J functions, 7?.C,(#)=J.2V x), the roots of C,(2) are easily caleulated from those of Jn(z)=0 and the roots of ak C,(a) from those of Jnyi(¢)=0. This readily follows v from the above relation. * “The Bessel-Clifford Function and its applications. * Sir George Greenhill. Phil, Mag. vol. xxxvili. Nov. 1919, pp. 501-6522. F206] XVII. The Determination of Values of Young’s Modulus and Poisson’s Ratio by the Method of Flerures. By H. Carrineton, B.Se., W.Se., Tech.A.M.I. Mech. E.* ‘HE determination of Young’s Modulus and Poisson’s Ratio by the method of tlexures involves the accurate measurement of the principal curvatures of the anticlastic neutral surface of a beam or rod of suitable cross-section when it is bent by couples applied to its ends. The couples should be applied so that the curvature of the longitudinal axis is a principal curvature of the surface. When this is the case, the curvature of the neutral axis of any normal cross-section is the other principal curvature. If the longi- tudinal curvature and the values of the corresponding couples and dimensions of the cross-section are known, Young’s Modulus corresponding with the length of the beam can be calculated. Also Poisson’s Ratio corresponding with longitudinal strain in the direction of the length, and lateral strain in the direction of the breadth, is numerically equal ee ee Lateral Curvature Oiler Longitudinal Curvature’ Values of Poisson’s Ratio for glass were obtained by the method of flexures by M. A. Cornu + (1869), who explored the anticlastic surface of a beam of glass by means of inter- ference fringes produced between the surface of the beam and a glass plate laid upon it. The method was also used by A. Mallock ¢ (1879), who obtained values for a considerable number of materials, including three for white pine and two for box and beech. Mallock placed four short wire pillars in the beams: one pair in the plane of the longitudinal curvature and the other pair in the plane of the lateral curvature. By measuring, by means of a microscope, the distance between the ends of each pair of pillars before and after a beam was bent, he was able to determine the principal curvatures of the surface, and hence the values of Poisson’s Ratio. The writer is not aware of any record of the use of the method subsequent to 1879. The method described in this paper differs essentially from the above in the manner in which the curvatures were measured. The method will be explained by reference to * Communicated by the Author. + Compies Rendus, p. 333 (1869). t Proc. Royal Society, vol. xxix. (1879). Young's Modulus and Poisson’s Ratio. 207 e fic. 1, which refers to the measurement of the longitudinal curvature. A pair of pillars with small mirrors, m, m, pivoted to their upper ends, were fixed to the beam. The lower ends of the pillars were bent in order to make point- contact in the plane of the couples on the upper and lower surfaces of the beam. Forabeam 1 in. wide and } in. thick the distance a in fig. 1 was about 1in, and was symmetrical with ieee respect to the central normal cross-section. The mirrors were adjusted so that the scale was reflected by them along the telescope shown, and the adjustment was made so that the axis of the telescope, axes of the pillars, and the scale were all in the plane of the couples. The full lines show diagrammatically the position of the line of sight before the beam was bent. ‘The chain and dotted lines show their positions when the beam was bent concave upwards. The correct scale-reading from which the curvature should be calculated is 2+y—=z, and it was found that by suitably choosing the lengths a, b, and 0’, the value of y could be made sensibly to vanish, for y=csin 6—c' sin (0+2¢), . ° 0 i a—b'sing—bsin gd +7 (tan ae ) ea Weise rede bead : Aanee cos (0+ 26) 208 Mr. H. Carrington on Determination of Young's Modulus Hence ae tan @—{a—(b' +b) sin }} tan (9+ 26) — y) 1+tan (; -$) tan (9 +29) or a{tan O—tan (0+ 26)} + {(b +0) sin pf tan (6+ 29) Y — a Rae = = . 1+tan z ~ ) tan (@ + 2) Neglecting the square and higher powers of the small quantity in the numerator and writing {+tan (5 -4) tan (0+ 2) =1+ tan tan 0=sec 8, then - ysecO={(l' +0) tan 0—2a sec’6}. *, y=o{(O' +0) sin 6— 2a sec}. 13 iG sae Hence if sin 20 Rs then y=0. This relation is satisfied if, for instance, b=1'5in., b’=2°5 in.., a=1in., and 0@=45°. : The above analysis and the diagram in fig. 1 will also apply to the lateral curvature ; the axes of the pillars, axis of the telescope, and the scale then being in the plane of the central normal cross-section of the beam. Th When using the method, the beam was supported in a hori- zontal position on knife-edges 12 in. apart, and symmetrical with respect to the pillars and the central normal cross- section. Two other knife-edges were placed on the beam at points 8 in. apart, which were also symmetrical with respect to the central cross-section, and the couples were applied by weights suspended from the inner knife-edges. ‘The weights were increased by smali amounts, and after every increase both the scales were read. The readings were then plotted against the weights, and the slopes of the resulting straight lines were proportional to the principal curvatures. Thus, if the values a and / in fig. 1 were the same for both pairs of pillars and scales, then Poisson’s Ratio was given by the ratio of the slopes of the two lines. Also if M was the bending moment and « the corresponding scale-reading for and Poissons Ratio by Method of Mlecures. 209 the longitudinal curvature, then Young’s Modulus (H) was given by S Coa ue b,d* eal : where 6,=breadth and d=depth of the cross-section. The writer used the above method extensively during the war to determine values of Young’s Modulus and Poisson’s Ratio for timber. It was found capable of measuring very small changes of curvature. For instance, the value of Poisson’s Ratio for spruce corresponding with lateral strain in the direction of the grain is about 0°01, so that the lateral curvature is only about 74, of the longitudinal curvature. Tt was, however, possible to measure the lateral curvature in such cases by using a micrometer instead of a scale, and placing it about 150in. from the mirrors, instead of about 50in. as was usually the case. By this means it was found possible to measure changes in curvature corresponding with strains of about | in 5 x 10°. There are a few evident precautions necessary when con- ducting an experiment, such as avoiding longitudinal thrust on the specimen by properly suspending the knife-edges, and preventing bodily rotation of the specimen during test. Also the knife-edges were slightly curved, so that lateral curvature was free to take place. In Table I. are given two sets of values of Young’s Modulus (ff) and Poisson’s Ratio (a) obtained for five TABLE I. | sven, | Wrong | Elastic Type of 4 | Wrought | ats Geeicient, enoenmere Steel, | Ten | Brass. | Copper. | Aluminium. a ee Vee mee E (10° lb./sq. in.). | Flexure. 28°1 27°9 | 14:2 17-6 9 23 E (10° lb./sq. in.). | Tensile 283 | 283 140 | 17-9 9-30 (Marten’s | | | Instrement). H M (10° lb./sq.in.)., Torsion. DED Ny eh MOS | 5:41 6°80 349 ooheene Pee lexune, 0-236) 0245 | 0:333 0305 0-313 RE alos Calculated 0-265 | 0-290 | 0310] 0295] 0-320 from rows 1 and 3. | H=Young’s Modulus. o= Poisson’s Modulus. Af=Moduius of Rigidity. Phat. Mag. 8. 6. Vol. 41. No. 242. Feb. 1921. 1 210 Youngs Modulus and Poisson’s Ratio, different metals. The values of EK in the first row were deduced by the method of flexures, and those in the second row from tensile experiments on the strips, using Marten’s Extensometer. The values of o in the fourth row were obtained by the method of flexures, and those in the fifth E | OTE 1,where J/ is the modulus of rigidity—the values of JZ being deduced from torsion experiments on the strips. It should be noted row were calculated from the equation = that an error of | per cent. in the ratio ai will result in an error of 4 per cent. or 5 per cent. in the calculated value of oc. : The largest difference in the values of o by the two methods occurs for wrought iron. Since this material is fibrous, it is quite possible that the difference may be caused by the assumption of isotropy, which was made when calculating o from values of EH and J/. It is important to notice that, according to the theory of flexures, certain conditions must be fulfilled in order that the method should yield accurate results. These are :—‘* That the greatest diameter of the cross-section, and the third proportional to the diameters in and perpendicular to the plane of flexure, should not be great compared with the radius of curvature of the flexure” *. Since the strips of metal were 0°750 in. wide and 0°130 in. thick the third pro- portional to the thickness and breadth was about 4°3 in. Tbe flexure experiments were continued until the radius of curvature of flexure was about 40 in., and no indications of failure of the method were then noted. In the case of some of the timber specimens, the experiments were continued until the ratio of the radius of curvature of flexure to the third proportional was as low as 6, but even in these cases the method was found quite satisfactory. The dimensions of the cross-sections of the beams used by Mallock + were 1 in. wide and 2 in. thick, so that the third proportional to the thickness and breadth was 4in. The minimum radius of flexure was usually about 200 in., but he notes that the method gave accurate results when the radius of curvature of flexure was much less than this, * Thomson and Tait’s ‘ Natural Philosophy,’ art. 718. + Mallock, Zoe. cat. feeb XVIII. On Talbot's Bands and the Colour-Sequence in the Spectrum. By Nriwau Karan Serut, M.Sce., Assistant Professor of Physics in the Benares Hindu University *. 1. /ntroduction. ‘| Mae theory of the very remarkable system of bands in the spectrum discovered by Talbot which is seen on covering half the aperture of the dispersing system by a retarding plate has been discussed by a number of writers, notably by Airy T, Stokest, Rayleigh §, Schuster ||, Walker , and Wood**. The treatments given make it clear that a spectrum in which the bands are seen is less pure than it would be in the absence of the retarding plate—in other words, that its introduction results in a re-distribution of energy in the different parts of the spectrum as actually formed. But the effect of such re-distribution of energy on the sequence of colours as seen by the eye in a spectrum showing Talbot’s bands, does not so far appear to have been considered t+. Some observations made by the present writer show that the colour-sequence in Talbot’s bands pre- sents some very remarkable features, which become parti- eularly striking and noticeable when the total number of bands in the visible spectrum is not large. It is proposed in this paper to describe the methods of observation used and. the results obtained, and incidentally also to discuss certain other aspects of the theory of formation of Talbot’s bands. 2. Observation of the Colours of Talbot's Bands. The human eye is in reality a very sensitive instrument for detecting changes in colour, and, as was pointed out by the late Lord Rayleigh ff, it is capable under favourable con- ditions of dececting as small a difference of tint as that which exists between the two D-lines of sodium. ‘The eye fails, * Communicated by Prof. C. V. Raman, M.A. fen Drans. 1. p. 1 (184l): { Phil. Trans. ii. p. 227 (1848); also Math. & Phys. Papers, vol. ii. 14. § Scientific Papers, vol, iii. p. 123. || Phil. Mag. January 1904, p. 1. q] Phil. Mag. April 1906, p. 531. ** Phil. Mao, November 1909, p. 758. Ty For a full bibliography of the literature on Talbot's bands, see Appendix to a paper by T. E. Doubt, Phys. Rev. Oct. 17, p. 382. tt Scientific Papers, vol. v. p. 621. Le, Z12 Prof. Nihal Karan Sethi on TValbot’s Bands and however, to perform this important function in an adequate manner when a large number of different colours are pre- sented to it simultaneously without sharp lines of separation, as, for instance, in a prismatic spectrum. But if the colours are presented to the eye separately, and occupying a fairly large area in the field of view, its power in this respect becomes surprisingly great. These facts have to be borne in mind in attempting to discriminate between the sequence of colours in an ordinary spectrum and the sequence as seen in a spectrum showing a considerable number of Talbot’s bands. The following method of observation has been found to be suitable. The eyepiece of the observing telescope is removed and a narrow slit 1s placed in the focal plane of the objective. On putting the eye immediately behind the slit, the whole ot the effective portion of the prism face is seen to be of one uniform colour, and the field of view is also considerably broadened by the diffraction of the light entering the eye through the slit. On moving the telescope by slowly turn- ing the tangent screw, the colour of the different bands and even of the different portions of a band can be easily examined. Having chosen, then, a mica plate of such thickness that it gave about 25 or 30 bands in the whole spectrum, the aperture of the beam was carefuliy adjusted till, with the help of a nicol, the visibility of the bands became about the best. The eyepiece was now replaced by the slit men- tioned above, and it was observed on turning the tangent screw of the telescope that the colour seen changed in a remarkable manner. It remained almost unaltered except in intensity throughout the width of a bright band, but in crossing a minimum, there was practically a sudden jump to another colour quite distinct from the first. This again persisted unchanged in tint until the next minimum was reached, when there was another sudden change, and so on. The gradation of colour and the almost imperceptible passage from one tint to the next observed in an ordinary spectrum was entirely lost. The spectrum showing Talbot’s bands was thus seen to consist not of an infinite variety of tints as found in the ordinary spectrum, but of a limited number only. It is needless to add that this remark applies strictly to that part of the spectrum only for which the adjustment for this purpose is perfect. ~ the Colour-Sequence in the Spectrum. 23 3. Observation of Talbot's Bands with small Retardations. In spite of the convincing nature of the experiment described above, it was felt that the effect would certainly be much more striking if it could be observed without any special aid to the eye and so as to show the different colours side by side at one glance. This would be possible only if the changes were much more sudden, which evidently means that the total number of bands in the whole spectrum must be considerably redueed—to say 5 or 10, and the thickness of the mica plate must be correspondingly diminished. This in turn involves the reduction of the aperture of the beam which beyond a certain limit becomes inconvenient. But the relation between the width of the diffraction pattern and the dispersion of the prism can also be adjusted by reducing the latter instead of increasing the former by making the aperture narrow. “1 ss (L-+eosp')| & gesintlne de Ae a When p'=(2n+1)z, this will give zero illumination what- ever 2; may be. So that in this case we shall have Talbot’s bands with maximum visibility irrespective of the width of the slit. But when the same plate is turned over to the position Pg, ow =: —2h, and +2] i =| = 7 sin*h,z{1+cos (p'—4hyz)jde. . . (5) This integration can be easily performed with the aid of the tables of the Si function given by Glaisher *, for it can be easily put in the form :— I=2| Si a Se — cos p ‘[48i 42,—3S8162, Ly ; Pea —S8i22, + — sin*7, cos 4] Uy SNC Bi COSt) Lamm erie oti. Jose) Gotten y's Uo") (O}) where #=/,2}. When wz, is very small, Siv,=2, and 9 [ 224-2, | —COS p. [ Low, TEE 18.2, ea PARE +- 2x) | =22,(1+ cos p’), a value identical with that obtained from (4). * Phil. Trans. 1870, p. 367. 222 Prof. Nihal Karan Sethi on Talbot’s Bands and When 2; is very large ; a) Sia ay Si ie == =(, 2 vy The equations (4) and (6) reduce in this case respectively to IT=2(1+ cos es and 2 us TT 7 ese. 5 008 p [4-3-1] 9 = so that when the slit is opened very wide we obtain dark and bright bands in position P, of the plate, but uniform illumination in position Py. | The values of A and B in (6) and the corresponding visibility A for various widths of the slit are given below :— TABLE LI. ©). A. B. Visibility = = vis 16 atereatete sve 98 28 “84 is af alx oi aot aletatat ecto late 16 ; 50 66 - eee. 8 1-47 07 05 ef 2-04 pee es 8 ee 2-43 —07 —"03 a § SET 2-68 10 04 8 Le ean 2-8] O17 006 sea tant geen 9-83 “004 001 eet 2:93 001 000 a ane 2:98 ‘001 -000 Fig. 5 is a graphical representation of the visibility for various slit widths. From this it is clear that the visibility of the bands at first diminishes and becomes zero when the width of the slit corresponds to m= nearly, 7.e. when it the Colour-Seguence in the Spectrum. 223 admits about + of the central bright fringe in the diffraction- pattern. Then it reappears with the positions of the dark Fig. Do }. 37 q and bright bands interchanged and increases till the slit admits about 2 of the central portion. The maximum of the visibility in this reappearance is about 16 per cent. of the full visibility. The visibility again vanishes when about 4 of the central fringe of the diffraction- -pattern enters the analysing spectroscope. With further increase of slit width we have another reappearance followed by a disappearance, but these changes can only be seen with difficulty. Actual measurement of the slit width at the different stages showed fair agreement with the results of theory. 7. Summary and Conclusion. 1. The present paper contains some observations and theoretical discussion of the remarkable difference in the colour-sequence as observed in a normal spectrum and in a spectrum showing Talbot’s bands. In the case of a spectrum showing a fairly large number of the bands, these ditferences may be studied by placing the eye behind a very narrow slit placed in the focal plane of the telescope and moving it over the spectrum. With a spectrum showing only : small number of Talbot’s bands in the whole visible region the phenomena may be easily seen without such aid. Talbot’s bands may be satisfactorily observed with such small retar- dations by using a prism combination of small adjustable pepeesion. 2, It was found in all cases that when adjustments were properly made, the colour changed from band to band ina fairly abrupt manner—remaining nearly uniform within the bright regions, but changing suddenly i in passing across the 224 Talbot's Bands and Colour-Sequence in the Spectrum. minima of illumination. ‘The spectrum with Talbot’s bands is thus shown to consist not as usual of an infinite variety of tints, but of a limited number only, depending on the number of bands present. . It is found that the condition for observing this pheno- menon is not the same as that required for observ ving Talbot’s bands at best visibility, 2. e. with perfectly black minima, but is slightly different, the dispersion required in the former case being about # of what is needed in the latter. If the dispersion 1 is less than even this the sequence of colour within a band is actually reversed. 4. The colour at every point in a spectrum showing five Talbot’s bands has been caleulated and the result shown on a graph and plotted on Maxwell's Colour-Triangle. 5. A new method tor observing Talbot’s bands is described, which consists briefly in first forming the laminar diffraction- pattern in white ]1 eht and then analy sing it with a spectro- scope having a wide slit. This method is very convenient when Talbot’s bands with small retardations have to be observed, as it is then unnecessary to unduly restrict the aperture or to use very small dispersive powers. 6. With the foregoing arrangement it is possible also to study the manner in which the. visibility of Talbot’s bands is influenced by admitting more or less of the laminar diffraction-pattern into the dispersing apparatus, and thus to actually trace the successive stages of the process of super- position which results in the interference bands having full visibility when the plate is put on the thinner side of the dispersing prism, and zero visibility when it is on the thicker side. 7. It would be interesting to see if the phenomenon of the colour-sequence in Talbot’s bands could be explained on the simple theory of their formation which has been given by Schuster (Phil. Mag. Jan. 1904). In conclusion, the writer has much pleasure in recording his sense of deep gratitude to Prof. C. V. Raman for his valuable and helpful interest during the progies of the work which was carried out in the laboratory of the Indian Association for the Cultivation of Science. Calcutta, 8th July, 1920. ee | XIX. Notes on Times of Descent under Gravity, suggested by a proposition of Galileo's. By W.B. Morton, M.A., and T. C. Topin, 1. A.* N the series of propositions which form the subject- matter of the third day of his “ Dialogues” Galileo established, by geometrical methods, all the well-known properties of uniformly accelerated rectilinear motion, and, in addition, a number of theorems, many of them of great interest and beauty, which are not included in modern text- books. One of these is Prop. 36, which states that a particle which slides to the lowest point O of a vertical eircle (fig. 1) starting from rest at any point B of the cir- cumference below the level of the centre, will make the Roa - journey in shorter time if it moves along two successive chords BA, AO of the circle than if it goes directly along the one chord BO. This proposition is noteworthy for two reasons. In the first place its geometrical proof, as given by Galileo, is quite simple, but if one attempts it by the straightforward algebraic methods of the present day the work is unexpectedly complicated. And, secondly, the theorem has an historical interest, for it was the first step towards the solution of the problem of the brachistochrone. * Communicated by the Authors. Phil. Mag. 8. 6. Vol. 41. No. 242. Feb. 1921. Q 226 Prof. Morton and Mr. Tobin on Times of Descent It brings out clearly the fact that a path longer in distance may be shorter in time. Accordingly Galileo, in a Scholium to the proposition, states that it appears to follow (“colligi posse videtur”’) that the most rapid passage from one fixed point to another is not by the shortest geometrical path, the straight line, but by an are of a circle. The argument which follows is limited to the proof that the time of descent is continually shortened as more and more chords are taken between the starting point and the bottom of the circle, approaching the are as a limit. In order to establish this Galileo has to make an unproved assumption, which he introduces by the words “ verisimile est” to the effect that the superiority of the two-chord path continues to hold when the particle starts with an initial velocity due to a fall from a level which is not above the centre of the circle. The “ Dialogues’’ were printed in 1638. In 1696 John Bernoulli discovered the cycloid to be the true curve of quickest descent, and published the problem as a challenge to the learned world. He dogs not appear to have been aware of Galileo’s earlier attempt until his attention was called to it by Leibnitz. In a paper which the latter contributed to the ‘Acta Eruditorum’ in 1697*, it was pointed out that Galileo had, through lack of the methods of the differential calculus, gone wrong on two questions, viz., the form of the catenary which he identified with the parabola, and the curve of quickest descent which he supposed to be acircle. To this at a later date Bernoulli added a rather curious comment yt. The catenary, as he had himself shown, can be constructed by the rectification of a paralola, and the cycloid by the rectification of a circle, therefore, in both instances, “ Galilée a deviné quelque chose @approchant.” To return to the proposition, Galileo’s method of proof, slightly simplified, may be stated as follows. _ B is the point of departure on the circle OABC whose lowest point is O. The circle BDA is described with B as its highest point. Then the times BD and BA from rest at B are equal, and the velocity at Ais the same whether the particle slides along BA or GA. Therefore we have to show that the time along DO from rest at B is greater than that along AO from rest at G. The proof turns on the fact that DO is longer than AO. This is proved by Galileo, as a lemma, in a rather clumsy way. It is most readily seen by expressing the angles ODA * Leibnitz, Math. Schriften (Gerhardt’s edition), y. p. 383. + John Bernoulli’s works, i. p. 199. > under Gravity, suggested by a proposition of Galileo's. 227 and OAD in terms of the inclinations, a 8 say, of the chords OA OB to the horizontal. It is easy to show that ODA=a+ BP, OAD=180°— 22. Since Dao wand 0 4 )e. atB<28<180°—28, -ODAOX. We have OBSOA, OB:QD<0G:0OA, OB: DB>O0G: AG. But Ob -VRB=YRB- DB=OV: yD. Obes is ON = Nalp2 OG:AG=OX? ;: XA?, ONG VED SiO IXe OXAY In other words, OY is a larger fractional part of the longer line OD: so OY > OX, and the proposition is proved. In connexion with the proposition and scholium the following questions suggest themselves, and in spite of the well-worn character and the comparative unimportance of the subject, some of the results obtained are perhaps not devoid of interest. (1) The range of validity of the proposition as the upper point rises out of the lower quadrant of the circle and the truth or falsehood of the assumption made in the scholium. This suggests a comparison of the times down one chord and down two chords for the general case when the particle starts with an initial velocity and leads, incidentally, to an analytical proof of the proposition. (2) Extension to the case where the intermediate point A is not on the circle. This includes an examination of the locus of A for a given time of descent down the two lines Q 2 228 Prof. Morton and Mr. Tobin on Times of Descent BAO, and the position A for which the time is a minimum, or what may be called the two-line brachistochrone. (3) Range of validity of the statement in the Scholium that the are is quicker than the chord: involving comparison of are and chord-times in the general case where there 1s initial velocity. (4) It seems obvious that Galileo’s circular are, having a horizontal tangent at the lowest point, will not in general give a shorter time than any other circular are which can be drawn. So we may examine the question of the circular brachistochrone. (5) Comparison of the actual times taken by the different routes : the books give no idea as to how much quicker the cycloid is than, say, the direct line. (1) It is required to compare the times taken bya particle to describe the paths BO and BAO, when its initial velocity , at B is that due to a fall from the level of C. Let the inclinations to the horizontal of the chords OA, OB, OC be «By, then the inclination of BA is 2+8. Tia particle starfs from rest at the level of C its velocity at the levels of BAO will be proportional to (sin? y—sin? 8), (sin?y—sin?«)2, and siny. ‘To get the time along any chord divide the gain of velocity by the acceleration. The con- dition that the path BAO should be quicker than BO is cosec {sin y —(sin* y —sin? 8}3} —cosec a{sin y — (sin’y — sin?a)2; —cosec (4 + B){ (sin? y—sin? «)— (sin? y — sin? B)2}>0. Multiply across by the positive quantity sin « sinf sin (2+) and get rid of the roots by writing sina=sinysin@, sin 8=sinysin @¢. So that cosa=A@, cos8=Ad¢ (mod. sin y). Dropping the factor sin’y we arrive at (sin OAd + sin PAG) {sin (6—O) —(sin d— sin )} —sin @ sin ¢ (cos @—cos f) > 0. From this the factor sin4@sin}¢sin}(d—0) can be removed and the inequality reduces to sin @ {2A — (1+ cos@)t+sin@{2Ad— (1 + cos d)} > 0, or finally cosec 0; 2A8 —(1 + cos @)} + cosecg {2Adg -- (1+ cos f)} >0. under Gravity, suggested by a proposition of Galileo’s. 229 In this the two variables 0 ¢, or « 8, are separated and the discussion now turns on the values of the function S(O y) =cosec 0{2AG—(1+cos 0)?, or, reinstating @, (ey) =cosec a {sin y(2 cos a— 1) — (sin’y—sin?«)2}. When /(47)+/(@ y) is positive the two-chord path is the quicker. Now it is easy to verify that if y<45° the function / is positive for all values of the other angle. So the two chords are quicker than the one provided the level to which the velocity is due is below the centre of the circle, and this is precisely the limitation imposed by Galileo. On the other hand if y>60° then, for all values of a and 8 which are less than y, f is negative and the single-chord path is the quicker. What happens for values of y between 45° and 60° can be seen by examining the graphs of the function f, drawn on meeeeetory—A4A),/ 50°, do°, 60° and a - a A) 4 NH i a A ‘¢ aren twee: meomestace = manner in which the heated platinum wires employed— eleven in number—were mounted, and their mode of in- sertion in the flow-tube. The respective ends of each of the fine platinum wires W were affixed, by means of the smallest amount of silver solder affording a secure junction, to por- tions of considerably thicker copper wires represented by L, L. These copper wires passed tightly through holes in plugs of ebonite P, P, which fitted accurately into rect- angular slots furnished by projections joined to the main tube at right angles, as shown. Precautions were taken that the slots were diametrically disposed with respect to the cross-section of the flow-tube, and after insertion of the plugs the continuity of the inner surface of the tube was. restored by carefully filing and polishing the inner surfaces of the respective inserted ebonite plugs. The resistances of the several platinum wires were made as nen equal as possible in the manner previously explained” ; and by appropriate adjustment of the copper wires to which the platinum wires were affixed, it was possible to secure that the several wires were initially coplanar, and remained equally spaced in the initial diametral plane of the flow-tube on being heated by means of an electric current. ‘The copper wires passing through the ebonite plugs were, after adjustment of the platinum wires as above, secured in position by means of small screws passing through fine brass bushes inserted * Phil. Mag. vol, xl. pp. 641, 642 (1920). Phil. Mag. 8. 6. Vol. 41. No. 242. Feb. 1921. R 242 Dr. J.8. G. Thomas on Thermal Effect of a slow into the-ebonite in the manner shown at la. Alternate bushes were inserted from opposite sides of each ebonite block. The ends of the copper wires were connected to binding screws affixed to ebonite blocks B secured to the anemometer tube as shown, so that each of the platinum wires inserted in the tube was connected with very approxi- mately the same length of lead. The several terminals on the ebonite blocks were connected separately by means of thick stranded copper wire with terminals dipping into mercury cups, so that any or all of the heated platinum wires could be inserted in any external circuit as desired. The insertion of the tube in the main flow tube was effected by means of the spigot unions shown at 8, 8, a device affording a smooth junction between the tubes, with the prevention of eddies. The anemometer tube was wrapped in three layers of felt, affording efficient thermal insulation, and the remainder of the flow-tube wound with asbestos cord. The general method of calibration of the heated wires for purposes of anemometry is given in the papers by the author already referred to*. The fixed ratio arm in the Wheatstone bridge employed was throughout adjusted to 1000 ohms. For purposes of measuring the respective temperatures to which the several wires were raised by the heating current and the air stream employed, the respective values of R, for the individual wires were de- termined from their respective resistances, measured at atmospheric temperature, using the value of the temperature coefficient of the sample of wire from which the several wires were cut, determined by means of a Callendar and Griffiths bridge. The current used for this purpose was 0-005 amp., and allowance was made for the lead resistance. When the platinum wires were heated by the current (1°1 amp.) employed in the series of experiments detailed below, their respective temperatures were deduced from their re- sistances as calculated from the voltage drop occurring across the respective wires. The platinum-scale temperatures so calculated were in all cases corrected to the gas scale. The Weston voltmeter employed enabled readings to be made to 0:001 volt, and it was calibrated before use. Read- ings of current were made either on a calibrated Weston or Siemens & Halske milliammeter. In the present series of experiments the anemometer wires were used in a hori- zontal position, and the flow-tube was likewise placed horizontally. Phil: Mae. focct. Proc .Phys. Soc: loci cr: Current of Air flowing past heated Platinum Wires. 243 Results and Discussion. BhiamMOber OF HOW-GUDOE 2-5. ne~ oeig-7- onteelnd o's aso oelclulasisea soe ut 2:0554 cm. Diameter Of platimUne WAKES .5..2. 2. lac.acaeeicdecsesimssacqocisas 0-101 mm. Temperature coefficient of resistance of platinum wires.. 0°003335 Mean distance between successive Wires ................2.05- 1:3 mm. Number of platinum wires employed .................0.es00s 11 MC GINO] CIGECTU Be staan cenolsnc sels oAaviesis'ecisieeeacidescteceacecmls sms 1-1 amp. Fixed ratio arm in Wheatstone bridge ...................4- 1000 ohms. The respective resistances of the several wires at 0° C.,. and the temperatures to which they are raised by the heating current of 1'1 amp. subsequently employed, are set out here- with in Table I, the wires being numbered consecutively in the direction in which the flow was established in the tube :— & TABLE I. Wire Non, 1 2. 3. 4, 5. 6. ule 8. 9. 10. es POO (em) $2454 02479 0°2507 0°2498 02470 02470 0°2470 0:2498 0°2545 0:2500 0:2535 Initial tem- perature to which wires were raised | ~,,0 ane 5 us (i? Mee Ke 3 Beer by current | 560 674 690 688 695 690" 705 663 672 635° 526 of 1°1 amp. in absence of flow (°C.) The mean value of the resistances at 0° C. of the several wires was found to be 0°2484 ohm. ‘The individual resist- ances, except Nos. 1, 9, and 11, were all within 1 per cent. of this mean value. The percentage variation from the mean value in the case of the Ist, 9th, and 11th wires respectively were 1:2 per cent., 2°4 per cent.,and 2 per cent. ‘The distribution of temperatures amongst the several wires for zero flow may be roughly described as a condition in which the temperature increases considerably from the respective outer wires of the series to the adjacent ones, the remaining wires contained within the four outer wires being all raised to an approximately uniform temperature of 680°C. When the motion due to the slow current of air in the flow-tube was impressed upon the free convection cur- rents arising from the wires heated by a current of 1*1 amp., it was found that, except in the case of the first wire of the series, the resistance of the individual. wires was increased. An indication of the constancy of the air-flow was secured by employing the leading pair of wires, in conjunction with a Wheatstone bridge, as a hot-wire anemometer in the manner already described. A comparison of the respective thermal effects experienced by the several wires due to the R 2 TEMPERATURE CHANGE IN WIRES DUE To FLOW (°C) BOO 244 Dr. J. 8. G. Thomas on Thermal Effect of a slow air-flow is most readily made by determining in each case the change of temperature of the individual wires accom- panying the establishment of the air-flow in the tube. The results given in figs. 2 and 3 indicate how the temperatures ee | { I “i ere | | ulti ae Lebo (8) tS) ie) 15 20 . VELOCITY(CMS. PER SEC, VOLUMES REDUCED TO Oc ano 760mm.) to which the respective wires were raised by the heating current of 1:1 amp. employed depended upon the magnitude of the mean velocity of the air-flow established in the tube. The ordinates in every case represent the increase or de- crease of temperature of the wire above or below the normal temperature of the wire in the absence of flow. The curve representing the behaviour of No. 7 wire is omitted from fig. 2 for the sake of clearness. Its descending portion lies between the corresponding portions of the curves ae i 30 Current of Air flowing past heated Platinum Wires. 245 representing the behaviour of wires Nos. 6 and 8. The TEMPERATURE. CHANGE IN WIRES DUE TO FLow(°C) initial portion of the curve for No. 9 wire is omitted from “100 by) , O l 2 3 4 5 VELOCITY (CMS PER SEC, VOLUMES REDUCED To O°c AND 760MM) fig. 2 for the same reason. Fig. 3 shows on an enlarged scale the form of the various curves contained in fig. 2 for values of the average impressed velocity of the air stream 246 Dr. J.S. G. Thomas on Thermal Eject of a slow flowing in the tube, ranging between 0 and 5 ems. per sec. Briefly, the characteristics of the several curves may be summarised as follows :-— (1) From fig. 2 it is seen that, except in the case of No. 1 wire, the temperature of each wire of the series increases with increase in the mean velocity of the air stream, from its Initial zero value, until a maximum value of the tem- perature increase is attained in the case of each wire, the temperature increase thereafter decreasing continuously, the temperature of each wire ultimately falling continuously below its normal value with increase in the impressed velocity of the air stream. | (2) The temperature of No. 1 wire falls off continuously with increase of the impressed velocity of the*air stream. (3) The maximum rise of temperature of the various wires increases progressively in the direction of flow of the impressed air stream, and, moreover, the maximum rise in the case of each wire occurs ata value of the impressed velocity which is progressively greater the further the wire in question is removed from the first wire of the series. Thus the maximum rise of temperature in the case ef No. 9 wire is greater than that of, say, No. 6 wire, and occurs at a greater value of the impressed velocity of the air stream. (4) The variation in the values of the velocity at which the maximum temperature rise is attained in the case of the successive wires, is seen to be large in the case of the the leading wires of the series (wires Nos. 1-6, fig. 2) compared with the variation in the case of later members of the series. (5) The thermal behaviour of the several wires in the present case enables the wires to be grouped together as follows :— (a) Wires Nos. 1, 2, 3, 4, and 5, for which the initial increase of temperature, where it occurs, progressively increases the later the wire occurs in the sequence in the direction of the air-flow. (6) Wires 6 and 7, for which the initial rate of in- crease of temperature is approximately constant and less than that for class (a). (c) Wires 8, 9, and 10, for which the initial increase of temperature progressively i increases as in group (@). (d) Wires 10 and 11, in which the initial increase of temperature progressively decreases as in group (6). Note.—lIn this classification the first member of each Current of Air flowing past heated Platinum Wires. 247 class after the first may be regarded as affording a transition between successive classes of wires. With regard to (1) above, Table II. gives the maximum rise in temperature, and the maximum temperature attained by each wire in the case of the present series of wires. TABLE II, NWaire mca led Day Loe ied Meet hel A Guanine AN Oho TO. UL, “Maximum in- crease of tem- ORI 2a 2S oZk OMI O eT 2O hon TsO 166s 198 perature. (°C.) | Maximum tem- } perature at-' 560 686 718 740 786 810 825 788 802 801 724 tained. (°C.) It is seen that’ the maximum increments of temperature are by no means small, attaining values equal to about 200°C. in the case of the later members of the sequence of wires. This fact is of some little consequence in connexion with the use of electrical heating coils of fine wire for the purpose of heating a stream of air or other gas. Contrary to what might be ‘anticipated, it 1s easily possible for the temperature of the coil, in the absence of an impressed flow, to be such that actual fusion of one or more turns of the coil occurs when a slow stream of air or other eas is established past the coil. In the case of the wires employ ed in the present series of experiments, the highest temperature is attained by No. 7. The various peculiarities—see (1) to (5) above—of the thermal effects experienced by the respective wires can be readily discussed in the light of the characteristics of the heat transfer by convection, etc., to which attention has been directed in a previous paper”. Owing, however, to the multiplicity of wires employed in the present instance, such a mode of presentation could afford, at best, but a hazy mental picture of the phenomena appropriate to the present ease. ‘The following alternative graphical method of pre- sentation, while possessing no claims to quantitative ac- curacy, has been found of value in the discussion of the heat transference occurring in the present and similar instances :— In fig. 4 the several wires of the system are denoted by A,B, CG D, BE, F,G, H, I,K, L. The full lines extending into the region above the respective wires represent the heated convection currents arising therefrom. The broken lines in the region below the several wires represent the comparatively colder convection currents ascending towards * Proc. Phys. Soc. vol. xxxii. pp. 199-206 (1920). 248 Dr. J.8.G. Thomas on Thermal Effect of a slow the wires. The figure represents the condition of affairs in the absence of any impressed flow of air in the tube. ‘The lengths of the various full lines are drawn so as to indicate Fig, 4. ? i | » rh x fe @ F a: C iB 1A ! s i : i } ; ’ : i i ’ i 5 j 3 ‘ t r) { i j : : : i i j ’ j ; i : i i : : H i ; a i : i E ) ) j 1 : i ’ i i roughly the relative temperatures of the respective wires, and consequently of the respective convection currents arising therefrom (see Table I.). The colder convection currents below the wires are similarly represented by the broken lines, but are somewhat shorter in length, owing to their neces- sarily lower temperature. Fig. 5 represents the altered con- ditions ruling when the heated wires are subjected to the Fig. 5, : \ \ ee \ Bai’ ve ue v a G \ bs % ‘ ‘ ‘, ‘ s \ \ CY ‘ " m \ v . Ne ‘ a 4 Lee 4 \ yn influence of an impressed flow of air moving with small velocity. For the sake of simplicity, consider first of all the two wires A and B. It has been shown elsewhere* that the heat-transfer from the first of the two wires A, say, to B, owing to an increasing impressed air-flow, is characterised by two opposing tendencies, viz. (1) the initial decreasing temperature of the resultant convection current, and (2) the decreasing fall in temperature occurring during transfer >) o from the first to the second wire. The resultant thermal = iPcoes Phys. s0c.docsen, Current of Air flowing past heated Platinum Wires. 249 effect experienced by the second wire is determined by the relative magnitudes of these opposing tendencies. The effect of an impressed flow of air past the wire is, as shown in fig. 5, to deflect the convection currents in such manner to cause an approach to B of the hot ascending from A, and to cause the cold convection current aie towards B to approach meaner tO: oA.) alli) tie: BM is drawn at right angles to BG. It wiil be ‘readily seen that with increase of the velocity of the impressed stream of air, Ax and B@ rotate in a counter clockwise direction, and the triangle ABM goes through a series of values corresponding es to the variation of the thermal effect experienced by the second wire under these circumstances *. If,in like manner, AO be drawn at right angles to B§’, a decrease in AO is to be interpreted as an increase in the cooling effect experienced by the wire A, owing to the impressed. air-flow, and vice ely has’ been pointed out elsewhere}, and ‘the result will be clearly seen, from a comparison of the initial portions of curves (1) and (2) in fig. 3, that for small velocities of the impressed air stream the heat gained by the second wire is considerably larger than that lost by the first. For this, and other reasons, the broken lines in figs. 4 and 5 are shown of shorter length than the full lines in the upper region of the diagram. It is clear that the thermal effects experienced by the respective wires A and B, due to a smal) impressed flow of air, can be roughly represented by the area of the triangle AMB and the reciprocal of AO. The introduction of a third wire (! may now be readily seen by drawing BN perpendicular to Cy’. It is clear that, owing to the impressed flow of air, the cold convection current rising towards C approaches B (7. e. BN = & ED To O'C AND 760mm) S| ee 5 nies 6 coe <>. PER SEC, VOLUMES RED ® at LC) = ~. LOCITY (cm dp E i eof GQ SWIRES EE) 4WikES NOOF WIRESIN AQ 3 WIRES EACH ARM © 2WIRES Fid. 6. OF BRIDGE. =< WIRE of © SWIRES,FLOW | REVERSED. | = i yesu} 50 100 150 200 20 300 350 400 4680 500 550 600 60 WO DEFLECTION ] explained. Analogous results were obtained when the whole series of wires were heated. ‘The sensitivities in these cases were, however, slightly different from those indicated by fie. 6, but the relative sensitivities were of the same order as those Just discussed. In the region of very low impressed 254 Dr. J.S8. G. Thomas on Vhermal Hffect of a slow velocities of the air stream it will be seen from fig. 6 that in the cases where 4 or 5 wires were employed in the re- spective arms of the bridge, the deflexion is initially negative and increases numerically as the magnitude of the impressed ee of the air stream is increased, until a maximum value of the deflexion is attained, the deflexion thereafter decreasing, attaining a zero value and thereafter increasing in the manner already described. A ready explanation of this apparently anomalous result is readily afforded by a consideration of the curves in fig. &d. It will be seen that, initially, the temperatures of wires Nos. 2 to 5 increase more rapidly than those of succeeding members of the series. ‘The temperatures of Nos. 6,7, 8, and 9, it will be noted, increase initially very slowly. It will be seen, there- fore, that whereas for low values of the impressed velocity the rise in temperature of the wire No. 2 is always oreater than that of wire No. 1, and the rise in temperature in the group of wires Nos. 3 and 4 greater than that of the group 1 and 2, by the introduction of wire No. 6 the mean rise cf temper ature of the group containing it may be initially less than that occurring in the corresponding group inserted in the bridge. A similar argument holds in the case of groups of wires in which subsequent wires are included. This reversal of the initial direction of deflexion is shown in fig. 6 in the respective cases where four and five wires are employed in each arm of the bridge, as already explained. That the effect was not attributable to any abnormality of the setting of any of the wires in the flow-tube, was readily proved by reversing the order of the wires in the flow-tube. The calibration curve so obtained, employing five wires in the respective arms of the bridge, i is shown by the broken curve 5R in fig. 6. This, while not identical with the correspond- ing full -line curve—a result due to want of exact equality of the respective wires—reproduces the characteristic feature of reversal of the initial deflexion. Zero deflexion corresponds, of course, io the attainment of the same mean rise of tem- perature, due to the impressed air stream, in the two arms of the bridge. Fig. 6 indicates the attainment of such a state of affairs, in the case of five wires being employed in each arm of the bridge, for a value of the impressed velocity of the air stream equal to ahout 2°7 cms. per sec. The curves given in fig. 3, which are based on results obtained about two months prior to those shown in fig. 6, the wires having been intermittently employed during this period, indicate that zero difference in the mean rise of temperature corre- sponds to a value of the impressed velocity of the air stream equal to about 2°2 cms. per sec. Hxact agreement is not to Current of Air flowing past heated Platinum Wires. 255 be anticipated, as wire No. 11 was not heated in the deter- mination of the curve 5 in fig. 6. Finally, in this connexion it may be remarked that, as seen from fio. 6, the initial sen- sitivity of the arrangement employing three wires in each arm, as already explained, is practically identical with that of the two-wire arrangement. This initial decrease in the anticipated sensitivity is, of course, attributable to the fact that No. 6 wire, with its small initial rate of increase of temperature with increase of the impressed velocity, is in- cluded in the second group of wires constituting sucha device. 30 bs w Cy) Se Se Sd es) nD $ S) VELOCITY (CMS. PER SEC,VOLUMES REDUCED TO OC AND 760 MM @ > | oY D ist AND | HHWIRES ALL REMAINING WIRES | y 9 Q Ist ANDSTHWIRES pe ALSO HEATED V © IstAND 2noWiRES X IstAND @nnWIRES] SUBSEQUENT WIRES | (J Ist AND BH ee NOT HEATED © 20 40. 60. 60 100 20 140 160 160 200 220 20 260 DEFLECTION. The calibration curves given in fig. 7 were obtained by inserting the single wires indicated in the arms of the bridge, intermediate wires functioning as a heating coil. In the ease of the full-line curves in fig. 7, all remaining wires of 256 Dr. J.S8.G. Thomas on Thermal Effect of a slow the series were also heated, and in the case of the broken- line curves, wires subsequent to the pair inserted in the bridge were not heated. The heating current, both in the anemometer wires and in the heating coil, were throughout o adjusted to 1-1 amp., and the galvanometer shunted with 10 ohms. The curves in fig. 7 are therefore, strictly comparable with those given in fig. 6. Considering first the full-line curves, which were obtained with all wires ; of the series heated, it will be seen that, while the ultimate sensitivitv of the anemometer device for comparatively large velocities is ereater, the later the wire employed in conjunction with the first is situated in the sequence of wires, yet this is not so for small values of the impressed velocity. Thus, whereas the respective defiexions corresponding to an impressed velocity of 8 ems. per sec. are 48, 150, and 200 when the 2nd, 5th, and 11th wires are used in the bridge in conjunc- tion with the Ist wire, the deflexions corresponding to an impressed velocity of 4 cms. per sec. are respectively 32, 82, and 57. It is clearly seen that for low values of the im- pressed velocity the arrangement employing the 5th wire in conjunction with the lst is more sensitive than that employing the 11lth in conjunction with the Ist. The explanation of this phenomenon is clear from a consideration of the curves in fig. 3, wherein it will be seen that initially the temperature of No. 5 wire rises more rapidly than that of No. 11 wire, although, as seen from fig. 2, wire No. 11 attains the highest ultimate rise of temperature of the whole sequence of wire. It is clear from fig. 3 that, under the conditions specified, wire No. 5 used in conjunction with wire No. 1 affords the hot-wire anemometer employing one wire in each appropriate arm of the bridge possessing the maximum sensitivity in the region of very low velocities. From fig. 7 it will be likewise seen that, in accordance with anticipations advanced on page 249, the sensitivity of the anemometer device employing wires ‘Nos. l and 2 is greater when subsequent wires are not heated than when such wires are heated. It will also be noticed that while this is also true of the anemometer device emploving wire No. 5, in conjunction with wire No. 1, for values of the impressed velocity greater than about 6 cms. per sec., a rev rsal of the relative sensitivities occurs below this velocity, the greater sensitivity being then shown by the device in w ee subse- quent wires are heated. The case of wires Nos. 1 and 5 being employed differs essentially from the case in which wires Nos. 1 and 2 are employed. As already pointed out, the extreme wires in a sequence are subject, as shown in figs. 4 and 5, to what may be termed an “end effect.” The use Current of Air flowing past heated Platinum Wires. 257 of wires Nos. 1 and 2 alone, results in what is tantamount to the abolition of the end effect, as the end effect in this case is the whole effect, there being only two wires. The effect of the subsequent heated wires upon the temperature rise experienced by No. 2 wire owing to an impressed air-stream moving with slow velocity has already been discussed. The case of wire No. 5 being employed in conjunction with wire No. 1 can be readily discussed as follows :— When wires subsequent to No. 5 in the sequence are not heated, wire No. 5 is one of the end pair of wires, and is subject to what has been termed the “end effeet.”” Its initial temperature and that of No. 4 wire are both less than that of No. 3 wire When wires subsequent to No. 5 in the sequence are heated, wires 4 and 5 no longer experience the end effect. Their temperatures are considerably higher than was previously the case. With the establishment of a slow impressed stream of air, wire No. 5 now experiences a thermal effect, due principally to the approach towards it of the hot convec- tion current arising from No. 4 wire. The convection current being warmer than was previously the case, the rise of temperature of wire No. 5 is materially greater, and the sensitivity of the anemometer device employing wires Nos. land 5 consequently initially ¢ greater when wires subsequent to No. 5 are heated than is the case when these are not heated electrically. Possibly the matter may be made clearer by reference to fig. 3. ‘There it will be seen that initially the temperature rise experienced by wire No. 11 the last of the sequence, is very much less than that ex- perienced by the adjacent wire No. 10. In like manner, it is to be anticipated that when No. 5 wire is the last of the sequence of heated wires, its rise of temperature due to an impressed air-stream will be initially small compared with hat it would be if wire No. 6 were heated, and similarly for subsequent wires of the whole sequence. In fig. 2 it will also be seen that the last wire of the sequence ultimately attains the greatest rise of temperature. Similarly, it is to be anticip: ated that when wire No. 5 is the last heated wire of the sequence, with increase in the velocity of the impressed stream, its temperature rise will ultimately be greater than when culneog ane wires of the sequence are heated. Under these circumstances it 1s to be expected that ultimately the sensitiviby of the anemometer device employing wires Nos. 1 and 5 will be greater if subsequent wires in the series are not heated than would be the case if these latter were heated. In conclusion, it may be remarked that the sequence of wires illustrated in fig. 1 may be used, after the manner Phil. Mag. 8. 6. Vol. 41. No. 242. Feb. 1921. S 258 Lfect of aslow Current of Air flowing past Platinum. employed in platinum thermometry, for purposes of anemo- metry. Thus wires Nos. 1 and 11 might be inserted in a bridge, employing a bridge current of, say, 0°01 amp. Intervening wires would be employed as a heating coil. The deflexion-velocity calibration curves obtained in this manner present the same main features as those discussed in the present paper. With increase in the impressed velocity, the deflexion increases until an upper limit is reached, and thereafter decreases in the manner already described. The sensitivity of the device is, however, owing to the small current employed in the arms of the bridge, very considerably smaller than that of the type of anemo- meter described in the main part of this paper. Thus, using the same electrical apparatus, with its sensitivity equal to that employed throughout this work, in conjunction with what may be termed the thermometric anemometer just described, the heating current being 1:1 amp., the maximum deflexion obtained was 10 scale divisions. The thermo- metric type of hot-wire instrument has been introduced by C. C. Thomas* for the measurement of gas-flow. The electric energy requisite to maintain a constant difference of temperature of 2°C. in a pair of differential platinum thermometers situate one on each side of a heating-coil arranged across the section of the tube, is measured. It is clear that with certain dispositions of the thermometers and heating coil, for very small values of the velocity of the gas stream, the considerations advanced in the present paper become of i importance. Thus, provided the heating coil and thermometers are suitably disposed, for low velocities of the gas stream. the electrical energy necessary to maintain a constant difference of temperature between the two thermo- meters may decrease with increase in the impressed velocity of the gas stream. The work detailed herein was carried out in the Physical Laboratory of the South Metropolitan Gas Company. The author desires to express his sincere gratitude to Dr. Charles Carpenter, C.B.H., for his unfailing and inspiring interest in the research, and for the ready provision of all facilities necessary for the prosecution of the work. Physical Laboratory, South Metropolitan Gas Company, 709 Old Kent Koad, S8.E. 20 Oct., 1920. * Journ. Franklin Inst. 1911, pp.'411-460; Trans. American Soe. Mech. Eng. 1909, p. 655; Proc. American Gas Inst. 1912, p. 339, XXII. On the Period of Vibration of the Gravest Mode of a Thin Rod, in the form of a Truncated Wedge, when in Rotation about its Base*. By Lorna M. Swain, Lecturer in Mathematics, Newnham College, Cambridget. HIS problem was suggested to the author, in connexion with work undertaken at the Royal Aircraft Establish- ment, under Mr. R. V. Southwell, to estimate the effect of centrifugal force on the periods of vibration of airscrew blades. The effect was worked out rigorously for the case of a uniform rod by Mr. Arthur Berry{ and for the case of a rod tapering to a knife-edgeby Mr. H. A. Webb and the author§. As explainedin the latter report, the mathematical work for a rod tapering to an arbitrary depth would appear to be somewhat involved, and it seemed worth while to consider if the problem could not be tackled by some other method, likely to give results sufficiently accurate to be of use. This paper contains an attempt to apply Lord Rayleigh’s method for the calculation of the period of vibration of the gravest mode|| to this case and also gives some estimation of the accuracy likely to be attained. Consider a thin rod in the form of a truncated wedge, symmetrical about an axis (oz), encastred at the base, which contains the origin 0, and rotating with uniform angular velocity w about a perpendicular axis (oy). The breadth, measured perpendicular to the plane yoz, is uniform, * For the case of no rotation, see also a paper by J. Morrow in Phil. Mag. vol. x. p. 124 (1905). + Communicated by the Author. + Advisory Committee for Aeronautics. Reports and Memoranda, No. 488. § Advisory Committee for Aeronautics. Reports and Memoranda, No, 626. || Rayleigh, ‘Theory of Sound,’ vol. i., §§ 88, 89. 8 2 260 Miss Lorna M. Swain on the Period of and the depth, measured parallel to cy, decreases uniformly, remaining finite at the free end. ‘The taper is defined by taking the length to be J and the additional length, which would have to be added to make il taper to a Knife-edge, to be e. The case of a uniform rod is then obtained o making ¢ tend to « and that of a rod tapering to a knife-edge ‘by putting c=0*. The problem is ie find the period of the gravest mode for vibrations in the plane yoz. Let o be the area of any cross-section of the rod, perpen- dicular to oz, at distance z from 0, let I be the moment of inertia of the section about its mean line perpendicular to the plane yoz, let >, I, be the values of o, I respectively for z=0. l+te- l+e-—z\3 Mien gee eee I=1,(———*) | c Le The rotation of the rod causes a tension in it and we proceed to evaluate this tension in the steady motion. Let EH be Young’s modulus of elasticity, F be the shear, M be the bending moment, T be the tension, pasture Lat distance < from a, to act along the axis of f the rod, j p be the density. oe These quantities are measured in gravitational in.-lb.-sec. units. Resolving along the axis of the rod, Integrating this equation and expressing the condition fina 0, when z=l, we find ra [FSO * For this method of treatment the author is indebted to Mr. H. A, Webb. Vibration of the Gravest Mode of a Thin Rod. 261 Now to apply lord Rayleigh’s method*, we require to assume some form for the deflexion curve of the axis of the rod. We shall make two different assumptions and compare the results obtained. Let y be the displacement of the axis at distance < from o. First assume aa : Ua Ns (2? —Alze + O17): where 37 is the displacement of the free end and for small vibrations in the plane yoz is a function of ¢. This is the simplest algebraic curve that satisfies the end conditions for a uniform rod, encastred at one end and free at the other—viz., y= ON dy _ >for eh ie TT a A dy >for z=. 7, (BIS .. =0 | The last two equations become indeterminate for the rod tapering to a knife-edge, as I is itself zero for z=l, and hence we should not expect this assuin ption for the deflexion curve to givea very accurate value for this case. This will in fact emerge in the course of the work. We require to calculate the kinetic and potential energies of the rod. For the small vibrations in the plane yoz, we have kinetic energy due to y and potential energy due both to the bending and to the tension. ~ oan peut ee rode 2? —Alz + 617) dz, =} polit de ree tte? —4x+6)?dx, where lv=z, m9 ile 292 = poly: fie 1+U/c 15 |° * Loc. cit. 262 Miss Lorna M. Swain on the Period of P.H. due to bending L 2 2 as i" Bly (* 2) a, _EI lic—2? aly an (Fo) (12st 2dle + 128) Adz, 0 ae 4 l+c T2EIog ,(° gp NE == ga Tig y (1—.2)*dz, 72EI og 1 . 4 GA sata I PEGE A Es (en Pas >) I 3). P.E. due to the tension L dy 2 See e Wi (pateds as =1{ 9 () dz, 1/(1— x*)\ 4 2 ( e! 1 BAP) Cyl \(2( =) =] =e Spo on | (a —327 + 3.) G (1 — 2?) 3 a Jat L, = pw’ayln’ a a ) ES OTN 15 2 BA aE We have made two assumptions in calculating the kinetic and potential energies. In the first place we have neglected rotatory inertia and in the second we have followed Clebseh in calculating the potential ener ay due to bending ; this is justifiable in most practical cases* Assuming periodic motion of period 27/p, we finally obtain the equation for the period Fale) lie? eni3 Gaal 5 Tims AR : (1+¢/l) a5 3p eae ae Oey el TS © BBE i 1 pawl? Ae Boal where N= gly ) edncoe gHly C * Clebsch, Theorie der Elasticitit fester Korper, p. 258. Rayleigh, ‘Theory of Sound,’ vol. 1.§ 188. Ene. der Math. Wissenschaften, vol. iv. 4 251. ji The neglect of one term in the potential energy due to the tension and the retention of another is legitimate, since it will be found that the ratio of these terms depends on ‘the ea of a radius of gyration of a sec- tion to the length of the rod, which must be small, when the approximate theory for thin rods is used. Vibration of the Gravest Mode of a Thin Rod. —.263 This gives for the uniform rod, w*=12°464+2°35 ... (ia) < 5s) ea se rod, taperine to a knife-edge, w4=39°374+2°39X ... (1b) Secondly, let us assume y= nz, where 7 is the displace- ment of the free end. ? This is likely to give better results for the rod tapering to a knife-edge than the first assumption, as only one of the end conditions now becomes indeterminate. Proceeding in the same way as before, we find Paras enn Be eats 2g Bette K.E.=t poly E | | , P.H. due to bending ly aE ey - (= | ee rie i epee) 4) P.K. due to the tension De cae oa eve [es ioe Ul a pe ao ie Cnn Hence Te 1 eae Bie p= ! ee ee Tina OEE LO Lic Tae ale (re 1) | = ee ek ts D a, This gives for the uniform rod, pw*=20+2,66r... (iia) 9 i ees, LOU. taperino: toa knife-edge, pt=30+2°66r ... (id) To obtain some idea of the closeness of the approximation, (ia), (ib), Gia), Gi 6) have been graphed in figures 2 a, 2 0, and compared with the more accurate results obtained in the two papers already referred to. It should perhaps be pointed out in passing, that for the uniform rod when A=7, «* should be 29-2 and not 321 as given in Mr. Berry’s paper. With this correction, we see that the lower Ray leigh curve for the 264 Miss Lorna M. Swain on the Period of uniform rod is that corresponding to the first assumption and lies almost along the accurate value curve. Its hould of course by the general theory he above it, whereas for the larger values of dit appears to lie just below. This may be Fig. 2 a.— Uniform rod. | i SS ee ee ee Rayleigh (i a), (i6) correspond to the displacement curve y=", (1- 4 +6 7. 2 & (ii a), (il b) ” ” ” 3 YF RE: accounted for by the fact, that the values obtained in Mr. Berry’s paper are only accurate within about 1 per cent. This shows that the values ee from the Rayleigh carve should be accurate within about 4 per cent. For the rod tapering to a knife-edge, the Rayleigh curve corresponding to the second assumption is the lower and gives results correct within about 6 per cent. Vibration of the Gravest Mode of a Thin Rod. 265 Fie. 3. @-39 0-6 C2 aa e-38 05 1 | 0-37 | e-36 35 0-2 ict ae eke i — ‘ eo | Q-34 OL: ak Beh a fave ST ESE A EN SN vee 1¢) ! © (2 3 + L The ENS oh call elton Wa ae OSE Curve marked ois graph of 52 , 292 1 \ 7. ec 3° As We a aceilke (+7) hae 7 PAOQV Esa) 1 | BB ele.) | | gts Curve marked is graph of 4 5999 I —2, } lacamy | \ d | 266 Vibration of the Gravest Mode of a Thin Rod. Thus for any rod, tapsring to a section of finite depth, its period of vibration should be given, with a possible error of about 6 per cent., by the Rayleigh curve corresponding to that assumption, which gives the lower result for the value of ¢/l under consideration. Fig. 4. i 4 as Graph of (14 ‘) el eee | 9) 61,6 - : es l The term in p* independent of w and the coefiicient of X have be engraphed against c// for the two different assumptions as to the deflexion curve (see figures 3, 4), so that it should be easy to see which assumption will give the better result for any particular value of ¢/l or 2. In conclusion the author would like to express her grati- tude to Mr. Arthur Berry for much kind advice and helpful criticism. ro 267 XXII. On the Problems of Temperature Radiation of Gases. (Paper C.) By Mereu Nap Sana, D.Se., Lecturer in Physics and Applied Mathematics, University College of Science, Calcutta * § 1. eae object of the present paper is to discuss and examine the present-day position of the question of temperature radiation of gases. The problem before us is, whether by simply heating a quantity of gas confined within a closed vessel (say, a silica or a oraphite tube), it is possible to make the gas emit its characteristic lne- radiation. The experi- mental results on this subject are somewhat conflicting, and for different elements are widely divergent. While Pringsheim and others+ found that permanent gases like H,, He, Ne, A, No, O,, etc., remain non-luminous even at the highest temperatures which can be commanded in the laboratory, it is known that vapours of many elements easily become luminous at moderate temperatures. Such, for example, are the vapours of I, Br, As, 8, Se, Sb, and other metalloids. If we take the tube-furnace spectra of King ¢ to be cases of pure temperature-radiation, we have to admit that at temperatures varying from 2000° to 3000° K, most of the alkalies, the alkaline earths, thallium, iron, vana- dium, etc., can be made to emit their line-spectra. Gibson § obtained the green line of thallium by simply heating the element contained in a quartz tube. But the conclusion drawn by him—that the intensity of the green line is the same as that of the black body-spectrum at this particular wave-length—is entirely wrong. He placed the quartz tube within a black-body chamber heated to about 1200° C., and observed that the continuous spectrum from the black body was crossed by a black absorption line corresponding to the green emission line. But this black line faded away as soon as the thallium vapour took up the temperature of the enclosure. From this he concluded that the emission of the green line had just become as intense as that of the black body at the same part of the spectrum. But the conclusion is erroneous, for substances in temperature equilibrium within a black body enclosure would ali emit like a black body, and the experiment proves nothing but this simple eee of a black-body enclosure. * Communicated by the Author. + Pringsheim, Verh. d. Deutsch. Phys. Gesellschaft, 1908. Wood, Phys. Zeits. viii. (1907), and other papers. { Kine, Astro. Journal, vols. xxviil., XxXiV., XXXV., XXXVl, § Gibson, Phys. Zeits. xii. pp. 1145-1148 (1911). 268 Dr. Megh Nad Saha on the Problems of According to many physicists, the flame, the are, and the spark represent gradually increasing stages of temperature, viz. 2000° K, 4000° K, and 5000° K. It is well known that spectra produced under these conditions are widely different in their general characteristics, but the hypothesis that these variations can mainly be attributed to the varying values of a single physical variable, viz. the temperature, is not generally accepted. , I wish to point out that the value of the ionization potential, as obtained experimentally by Franck and Hertz, McLennan, and others, or theoretically from the quantum relation eV=A(1,s), has a great bearing on the problem. Asa rule, the higher the Lonization-potential of an element, the greater is the difficulty with which it can be excited to emit its line- spectrum. This will be apparent from the following tables *: TABLE I. Element... Me Ca Sr Ba Na K Rb Cs Re | GOs EE Ole ae sll 432 446 igias Element... Zn Cd Hg H He Ne At Nt ‘potential (24 «© 9 ~——«:10-45«136-17-1 205-256 17-234 16 17-18 The line or lines v= (1, s)— (2, p) form the most important lines of an element, and experiments on the ionization potential have shown that when the vapour of an element is bombarded by electrons, this is the line which is the first to be excited, other lines appearing only when the stimulus is substantially greater. The potential V= z [ (1s) — (2p) ] is therefore called the resonance potential, and may be taken to be a measure of the stimulus which is required so that an element may be just excited to radiation of its funda- mental line. A better name would probably be “ Radiation- Potential.” We may give a number of interesting examples. It is well known that generally it is very difficult to excite helium, the smallest trace of a foreign gas tending to quench the He-lines. According to the present theory, this is due to the fact that helium has the highest ionization and radiation potential of all elements, so that when it is subjected to a * McLennan, Proc. Lond. Phys. Soc.—Guthrie Lecture, Dec. 1918. Franck and Hertz, Verh. d. Deutsch. Ges. vol. xx. (1919). t+ Rentschler, Phys. Rev. vol. xiv. p. 503 (1919). Horton and Davies, Proc. Roy. Soc. Lond., vol. xeviii. p. 124. { Davis & Goucher, loc, cit. Jan. 1919; also Smyth, doc, cit. (1919). Temperature Radiation of Gases. 269 stimulus, this, by preference, passes through the more easily Eble i impurities, leaving the He atoms unaffected. Franck (Zeitschrift fiir Physik, 19 20, vol.i.) describes an interesting experiment on the excitation, by his well-known electron- bombardment method, of helium-lines contained in a tube into which a trace of mercury (less than 1 in 1000 parts) was purposely introduced. Mercury has an I.P. of 10°45 volts and R.P. of 4°9 volts, while the corresponding numbers for helium are 25-6 and 20°5. ‘‘ Wehave then,” in Franck’s own words, “a strong flashing out of the lines of the element with the lowerI.P., at the cost of the lines with the larger I.P.” This also explains qualitatively why, with the range of tempe- ratures available in the laboratory, it is not possible by purely thermal means to excite the permanent gases. In the case of the alkalies and the alkaline earths, the value of the ionization potential is low and gradually decreases as we proceed to elements with higher atomic weight in the same group, and it is found that the difficulty with which the spectrum can be excited lessens in a parallel manner. Thus under all conditions, if we classify the alkaline earths according to the ease with which they can be-excited, the order 1s hBa. Sr, Ca,and Mg. The same can be said at the other elements. In the present paper [ have used the word “Stimulus ” to denote, ina general manner, all physical agencies tending to maka the atoms luminescent. We shall discuss how a high temperature alone can bring about this state. The question can best be approached from the theoretical side. § 2. Modern spectroscopic works have shown that the lines of an element may be grouped under the following headings*:— (1) Lines due to the normal atom.—In fae case ne lines are produced by the quantum-changes of orbit of the outer- most electron, the nucleus and the remaining electrons behaving as a single charge. The characteristic Rydberg 2a* erm : number in the series-formula is N(= Tr): These lines u are produced under a comparatively low stimulus. (2) Lines due to the atom which has lost one electron (the outermost one).—I\n this case the lines are produced by the quantum changes of orbit of the now outermost electron, the nucleus and the remaining charges behaving as a net * For example, compare Fowler’s work on the ‘ Hmission Spectrum of Magnesium,” Phil. 'lrans, vol. 214. 270 Dr. Megh Nad Saha on the Problems oj double ESD Se. The characteristic Rydberg number is AN (= ue a e)'). These lines are produced at a higher ue i generally, but not always”, he in the ultra- violet. (3) The atom may lose two electrons, and now a new series of vibrations may begin with the new outermost electron. The Rydberg number is ON. The lines of any of these groups are as distinct from each other as if they belong to different elements altogether. According to Sommerfeld f, when an atom loses one electron, its spectral properties become similar to those of the element just preceding itin the Periodic Tables. Thus the system of lines of Cat are constituted in the same manner as those of K; similarly, the relation between Sr* and Rb, Bat and Cs. Lines of these different groups may be simultaneously present, but generally one group gains in intensity at the expense of the other. Thus in the flame Cat lines (the H, K) are very faint, the Ca-line (g-line) is very strong. Now ‘““g” begins to lose relatively in intensity as the (H, K) are strenethened with rise of temperature. At the spark- conditions, the most intense lines are the H, K, while the ‘““q” ig almost evanescent. Let us now examine how the transition from the neutral state to the ionized states (from Ca to Ca*) takes place, under the influence of heat alone. In Phil. Mag. Oct. 1920 (“ Ionization in the Solar Chromosphere,” called hencetorth paper A), it has been shown that the problem can be attacked with the aid of the “ New Thermodynamics” of - Planck and Nernstt, and the statistical equilibrium between neutral atoms and ionized atoms can be calculated in terms of temperature and pressure, when the energy of ionization is known. ‘The calculations for alkaline earths will be found in paper A, and those for the alkalies are given in paper B §. According to the mechanical theory, the outermost electron of the neutral atom revolves in a stable orbit when the atom is not subject to any stimulus. Ionization means the transference of this electron to infinity. But the process * For example, in case of alkaline earths. All alkaline earths are distinguished by having large (2,) terms, which causes the principal lines to occur in the visible region. + Sommerfeld, Atombau und Spektralanalyse, Chap. 4, §6 and Appendix. { Planck, Wérmestrahlung. Nernst, Das Neue Warmesatz, etc. Sue Elements in the Sun eden, Mag. Dec. 1920. Temperature Radiation 07 Gases. Zeal is not an abrupt one, for, according to the quantum theory of spectral emission, the electron may have an infinite number of orbits distinguished by different quanta-numbers. The theory of the stable orbits has been formulated by Sommerfeld * in the following manner. Let 7, denote the rotational quantum numbers for an orbit, and n’ the radial quantum number, 2. ¢. 7, 0, @ being the coordinates of the electron ; then, \(SP)aran (35 )ae=nut ey dp =noh, the integration extending over the whole orbit. The energy of the system is now given by the expression N oh eter ta ma, ph oe ee where A=a constant, f(n, n') is a function, the value of which decreases with increasing values of n and n’. The possible orbits can now be thus classified by assigning different sets of values to n and n’. TasueE II. Energy of the System Rotational Radial h quantum-no. quantum-no. N Remarks. N. qi ns : {aru + fm, No; m')\2 il 0 (Al, 8) ) This is the orbit which the electron possesses when it 1 pial — (m, 8) is subject to no stimulus. m= 1, the s-orbits. 9 Las ) a Y (2, P) The p-orbits. 2 m—2 —(m, p) m>2. 3 0 Oe The d-orbits. 3 ies _(m, d) m>s. 4 0 eu) Tne soehiie 4 m—4 —(m,/f) m= 4. b 0 a) The k-orbits. 5 m—5 —(m, k) ees (1,5), (2, p), (3, d),..(m,s),.. are the familiar expressions which, in Paschen’s notation, denote the terms of which a * Sommerfeld, Verh, d. Deutsch. Phys. Ges, vol, xxi. (1919). 2712, Dr. Megh Nad Saha on the Problems of “= P a ~ my . ‘fe ig . e e . series-formula is composed. Thus, for the principal series, the series-formula is p= (1,8) (nas ee When the atom is ina free gaseous condition, and is not subject to any stimulus, it has the energy A—h(1,s). The higher orbits are produced only when a stimulus is applied. The lines are emitted as the electron changes its habitat from one stable orbit to another with less energy. Thus the line (1, s)—(2, p) is emitted when the electron changes its habitat from the p-orbit (2,p) to the s-orbit (1, s), ete. The law according to which these changes take’ place has been thus formulated by Rubinowicz* and Bohr. Let and n denote the rotational quantum numbers of the initial and final orbits. Then m— ny= 110. ore We can thus distinguish among the following different. types of combination :— Group I.—Positive combination, n—nj=1. Combination ......... | (1, s)—(, p) | (2, p)—(m, 2) (3, d)—(m, f) mx, 2 m3 m~ 4 Accepted Appellation. } Principal ae Diffuse Series. ) Bergmann Series. e (2, s)— (i, P) (3, p)- (m, da) m S 3. m Ss 4, ete. Group I1.—Negative combination, n—n=—1. Combination ......... ) (2, p)—(™, s) (3, d)—(m, p) m Ss ie Mm S 2. Accepted Appellation. } Sharp Series. ete. Group I1].—Neutral combination. Combination ......... (1, s)—(m, s) (2, p) —(™, p) (38, d)—(m, d@) Each of the terms (m,s), (m,p), (m, d) may have double or multiple values owing to the different possible orienta- tions of the vibrating electron with regard to the remaining atom f. * Rubinowicz, Phys. Zeits. vol. xix. pp. 441-465 (1918). Sommerfeld, Atombau und Spektralanalyse, pp. 890-408. + For a theoretical treatment of the case, see S. N. Basu, “ On the Deduction of Rydberg’s Law from the Quantum Theory of Spectral Emission,” Phil. Mag. Noy. 1919. Temperature fradiation of Gases. ale, Thus far we have dealt with the electrodynamics of the atomic system, 2. e. the possible stable-orbits of the vibrating electron calculated according to quantum-mechanics. We have now to deal with the statistics of the case, for the higher stable orbits are produced in sufficient proportion only with increasing stimulus, which we obtain only at higher temperatures. The intensity of a line will depend upon the product of the numbers which show the relative proportions of orbits in the initial and final stages at any instant, and also upon the chance of changing from one orbit to another. Thus the intensity of (1, s)—(2, p) will be proportional to A'e, where A is the number of orbits in state (1, s), X’ is the number in state (2, p), and e’ is the chance that the orbit will change from the state (2, p) EOw (ks. S). When the stimulus is sufficiently great, some of the electrons will pass off to infinity, and we shall have partial ionization. This problem can be treated thermodynamically, for here we have to consider a sort of chemical equilibrium between three distinct phases—the neutral atom, the ionized atom, and the electron. But radiation cannot be so treated, for this is a case of internal change of orbits only, not involving any phase-changes. The problem before us can therefore be thus stated :— “* How to find out the statistical distribution of atoms into different possible stable orbits when the mass of the gas is subjected to a given thermal stimulus?” According to the theory of ionization sketched in paper A, if Ca-vapour is enclosed in a vessel, such that the pressure is always main- tained at 10~* atm., we have seen that with increasing temperature the proportion ionized varies as follows :— Percentage of ionization 5... 9 <10—" 2x10—* 9 6 47 Sil 100 Temperature... 2000° 3000° 4000° 5000° 6000° Foy The non-ionized atoms cannot all be with the primitive orbit (1, s), but a good propertion will be found with the other possible stable orbits, for the electron, while detaching itself from the neutral atom, has a chance of taking its habitat in some of these stable orbits , and hence some must be found in these states. The phenomena of line- radiation therefore comes before the ionization becomes complete. Phil. Mag. 8. 6. Vol. 41. No. 242. Feb. 1921. T 274 Dr. Megh Nad Saha on the Problems of ye: At the present time,.the electrodynamical part,—z. e., the manner in which the lines of an element originate from the quantum-vibrations of the constituent electrons of the atomic system—has been worked up in the case of the H-atom alone. For the other atoms, it is only in the process of making *. ‘Bui the thermodynamical part of the problem— that is, how the proportion of different possible stable orbits varies according to temperature, and how the orbits change trom one into the other giving rise to line-radiations— has not even been clearly tormulated. In his interesting development of the quantum theory applied to systems with more than one degree of freedom, Planeckt has laid the foundations ef a new method for dealing with the statistical! aspect of the question. The second aspect—namely radiation following as a result of mutual interchange of stable orbits— has been dealt with by Einsteint. But there are still many difficulties to overcome. When this is done, a new and fruit- ful chapter—that of line radiation of gases—will be added to thermodynamics. We may provisionally proceed along the following lines :— Let us suppose that the orbits having the rotational quanta 1, 2,....m... vary aS a geomeiric progression 21/2) a) ied sill os ip where f is a fraction and a function of temperature and concentration. Similarly, let us suppose that the orbits having the radial quanta 0, 1, 2, 3, ete., vary according to the terms of the geometric progression (1g) go, iG pine Ae where g is a similar fraction. Then at any instant the proportion of orbits with the rotational quantum number n+1 and the radial quantum number n’ is He aaa since at low temperatures and high concentrations almost the whole number of atoms is in the state (1, s) corresponding toma nl=0. We see that fand g are very small quantities under these * Sommerfeld, Atombau und Spektralanalyse. + Planck, Verh. d. D. Phys. Ges. vol. xvii. p. 407 (1915); Ann. d. Phys. vol. 50. p. 385 (1916). { Einstein, Verh. d. D, Phys. Ges. vol. xviii. p.318 (1916) ; Phys. Zeit. 1918, p. 124. Lemperature Radiation of Gases. 219 conditions and gradually increase as the temperature is raised or the concentration is lowered. As an example, we may take that the distribution of the Ca atoms in the orbits at 2000° K follow according to the scheme (we are considering only 7 here) CO oy Ce Oe Ones h sal while at 4000° K, the distribution is Ceea ena ESE nea ip so that the relative intensity of the lines L)-(2p), @-Bd, (3,d—(4)) will be LO es One Wetec. but fully 80 per cent. of the atoms will remain inactive, while at 4000° K, the ratio will be . | Ce eas E ND EAC es but now only 25 per cent. of the non-ionized atoms remain inactive. | The above considerations are not based upon any theoretical argument, but are given here as a sort of construibar vor- stellung of the statistics of line-radiation of gases. We can say that, under all physical conditions, a very small concen- tration of radiant atoms suffices for the production of the series v=(1,s)—(m, p), especially the fundamental line y=(1,s)—(2,p). The (2, p) —(m,d;), (2, s) —(m, Pad) —(m, 7) lines require gradually increasing concentrations of radiant matter in addition to the condition that the stimulus should be sutficiently great. For example, if we take sodium gas at a temperature of 1500° K [an ordinary Bunsen flame tinged with sodium vapour], and gradually decrease the amount of vapour in the flame, the order in which the lines disappear are (Ose) (G0, Oy) (Cg) (rosy (2.9) pet, 2): the (1,s)— (m,p) lines, of which the leading members are the D, and the D,, are the last to disappear, a fact which was recognized by Du Gramont, when he con- rerred the appellation “raies ultimes” upon this class of lines*. This state of affairs persists when the temperature is raised and gradually increasing percentages of atoms are ionized. [fa line is represented by the series formula p= (n, f)—(n', f'), * According to Bunsen and Kirchhoff, 7x 10-12 em. of sodium in the Bunsen flame is quite sufficient to show the D, and the D. line (Pringsheim, Physik der Sonne, p. 121), 2 BiG Dr. Megh Nad the difficulty of detection of the line-will be greater the Jarger are the values of x and n’. In the following table we give the temperature of complete lonization of a few elements, with the temperatures at which luminescence just begins and attains its maximum intensity. But it will be clear from what has been said that the ordinary way of speaking —“ the gas is heated to incandescence or luminescence just begins ’—has no meaning in itself unless we say which particular line is emitted, or whieh orbit is produced. The orbits which are produced are specified in column 3. Under the heading ‘* Remarks,” the manner in which these temperatures have been estimated are briefly described. In this connexion, the following section on absorption should be consulted. The pressure has been taken equal to one atmosphere unless otherwise stated. Saha on the Problems of Ace onnT | | Temperature ae | Lumin- | ote El Orbit | 1. escence is ut ement.| of escence Remarks. ‘ Remarks. lpn ae lvenerated.| Nee aa maximum onization. | begins at | | a | Pe) 2) ait heen ata on ae Temnael 9 2 I 9 fe) is a eae 24,000 2, p) | 50009 K.)ap oP. | 12,0009 Ke | gees | | i ; ; | ANG) FPA LSM DS OE i 6 az | Lemipvor He | 32,000 (2,p) | 11,000° K “A2class: 17,000° K < Bt | | | r T.+ Wie , |Lemp. of | 5 Temp. of | Vics 5 24,000 (3,2) | 7500° KY ao class | | 11,000° K | 4s oF class 10,000- | | Ca ] a Ria cae 40k Ss 000" - (2,p) | 1500° K Bunsen 4000 The oper Be | (Pressure flame are inne oO" Fartm) | | \ The elements N,, O,, A, Ne, etc., resemble H, and He in having large values for the ‘ionization-potential, and therefore they fail to respond to ue temperatures which can be com- manded in the laboratory. The alkali metals (particularly K, Rb, Cs) are more pr ARES in the flame-spectra and less in the arc than the alkaline earths. Mg-lines are rather faint in the flame, but come out very prominently in the open are. Zn, Cd, Hg, Fe, Ti lie between the alkaline earths and the permanent. gases in their spectral properties. All this is in very good qualitative agreement with the hypothesis sketched in the present paper. Temperature Radiation of Gases. at! § 4. Absorption. In this connexion, we may consider the puzzling question of reversal of lines. According to Kirchhoft’s law, we expect that the emission-lines of an element should be reversed when a strong beam of white light is sent through cooler layers of the vapour. But this expectation is not always fulfilled. Wood* has found that if a white light be sent through a column ot sodium vapour, only the lines of the principal pair-series (1, s) —(m, ,), (1,s)—(m, 2), ean be obtained as absorption-lines. None of the lines of the diffuse or the sharp series are reversed. Bevan f has extended the method to the other alkali metals, 7. e. Potassium, Rubidium, and Ceesium, and arrived at identical results. Recently Debbie and Fox (Proce. Roy. Soe: vol. xeviil. p. 147) studied the absorption-spectra of Hg, Zn, and Cd vapour, and found no ee up to A= 3200. But this is due to the fact that the ,s) —(m, lines of these elements lie below 3000 A.U. in tact, Wood found in 1913 (PAys. Zeit. pp. 191-195) that or -dinary Hg vapour absorbs the fundamental line X=2536 (i s)—(, Pr) ' The explanation easily :follows from our theory. The condition for absorption is that in the atoms present, there should be a fairly large number with orbits corresponding to the first term of the pulse of radiation to be absorbed. Thus, in order that a line (2, p)—(m, @) may be absorbed, we must have a sufficient number of atoms with (2, p) orbits. At low temperatures only atoms with (1, s) orbits are present. Hence only the lines corresponding to the combination (1, s)—(m, p) are absorbed. ‘The lines re- presented by the positive combinations (2, p) — (m, @), (3,d@)—(m,f), or the negative combinations (2, p)—(m,s), ean only occur when atoms with (2, p) or (3,d) orbits are present. This can happen only at high temperature or under electrical stimulus. The temperature required for this purpose is very high— much higher than the temperatures used by Wood and Bevan for all elements. In fact , the atoms begin to absorb the lines (2, ») —(m,d) only when they are hot enough to emit the leading terms of the principal series. A line of the Bergmann series will be®in to be absorbed at even a higher temperature, viz. at the temperature of emission of the diffuse series (3, d) — (4, 0). If the views pr ae ore be correct, we may probably obtain the reversal of the lines of the diffuse or the Bergmann * Wood, The Astrophys. Journal, vol. xxix. pp. 97-100. + Bevan, Proc. Roy. Soc. vol. ‘lxxxiii. pp. 4238-428; vol. Ixxxv. pp. 58 76, 278 Problems of Temperature Radiation of Gases. - 1 series of the alkali metals by heating the absorbing columpy of vapour to about 2000° to 3000° K. The most favourable element to start with is cesium, which has the lowest ionization-potential of all elements. In many cases confusion may arise about the proper identification of the (1,s)— (m,p) terms. Thus, what are usually called the ce series of helium and parbelium (viz. the series beginning with the line 20,587 for parhelium and 10,834 for helium) do not really correspond to the combination (1,s)—(m, p), but to the combination (2, s) —(m,p). The (1,s) term for helium is still unknown, and the series (1, s)—(m, p) lie far down in the ultraviolet *. Hence, according to our theory, none of the lines belonging to the combinations (2, s)—(m, p), (2, p) —(m, d) ean be abseries by a layer of helium gas. But if by heating or some ye means we can convert a good proportion to the states (2, s) or (2, ), then and then only can these lines appear as absorption- lines. But ata pressure of 107' atm. helium becomes incandescent, 7. e. emits the lines (1,s)—(m,p), and absorbs the lines (2, p) —(m, d) at Soa not less than 11,000° or 12,000° K,. Vase om in stars of the B-class. But instead of : high temperature we may think of other means. ‘The spark produces mechanically the very same conditions which can be realized at very high temperatures. This is exactly what Pascheny+ has done. He found that the lines of the combination (2, s)—(m,p) for helium and parhelium cannot be absorbed by an ordinary layer of the helium gas. But when a spark is sent through the absorbing layer the lines are strongly absorbed, the absorbed energy being again re-emitted in all directions. The paper thus suggests more problems than it attempts to solve. A critical examination and further development of the hypothesis advanced here requires an overhauling of the whole data on the line-radiation of gases—such as are contained in Kayser’s Handbuch der Spektroskopie and further works. But this programme requires much more time and more extensive study, both practical and theoretical. University College of Science, ‘Calcut tta, e May 25, 1920. * It is quite possible that some of the (1, s)—(m, p) lines for helium and parhelium may be identical with the lines discovered by Lyman in the ultra-violet, and some with the lines discovered by Richardson and Bazzoni in the region of 300 to 400 A.U. by the photo-electric method (vide Richardson and Bazzoni, Phil. Mag. 1918). + Paschen, Ann. d. Physik, ‘vol. xly. p- 625 (1914). variceal XXIII. Possible Magnetic Polarity of Free Electrons. By Artuur H. Comepron, Ph.D.* MM attention has recently been called by Mr. Shimizu to the fact observed by Mr. C. T. R. Wilson that the paths of beta and secondary cathode rays excited by X-rays in air usually terminate in converging helices. The comparatively uniform character of these | curves shows itself clearly in certain of Mr. Wilson’s beautiful stereoscopic photographs which have not been published but which he has very kindly allowed me to examine. For example, one photograph, which shows the complete tracks of 66 secondary cathode particles, reveals 52 tracks of helical form, only two or three tracks showing no general curvature of this type, the remaining 12 rays having ‘paths too irregular to detect with certainty any helical curvature that may ‘exist. These paths may have the form of either a right- or left-handed helix, and the axes of the different helices have a nearly random orientation. A detailed examination of a large number of tracks to prove that the observed curvature of the paths is not a random one, will necessarily involve much time and labour. In default, however, of a complete proof, it is of interest to see whether an explanation can be offered of the apparent tendency of the beta particles to move in a type of spiral orbit. Though the spiral form of the paths would suggest a motion of the particles in a magnetic field, the chance orientation of the axes of the different helices shows that these axes are not determined by any external magnetic field, but are rather characteristic of the individual beta rays. More specifically, the axis of the helix must be parallel with some polarity of the beta particle which is relatively permanent in direction. It is apparent that a simple electric charge can possess no polarity whose orientation will remain constant in spite of numerous collisions with other charges unless it is in rapid rotation. Mr. Shimizu accordingly suggested that Mr. Wil- son’s photographs may be explicable on the assumption that ‘the electron has a definite magnetic polarity which on account of gyroscopic action does not change rapidly in direction. It is clear that a magnetic field whose direction is deter- mined by the electron” passing through it, is capable of producing the type of spiral track that is observed. But a beta particle which acts as a magnetic doublet as well as an electric charge is capable of producing such a magnetic field * Communicated by Professor Sir K. Rutherford, F.R.S 280 Possible Magnetic Polarity of Free Electrons. if the medium through which it passes is susceptible to_ magnetization. For the introduction of such a doublet will induce magnetization in the surrounding medium just as a bar magnet induces magnetization in a neighbouring mass of iron. If the atmosphere acts paramagnetically, the magnetic field at the doublet due to the induced magnetization has the same direction as the magnetization of the doublet. Con- versely, the induced magnetic field due to a diamagnetic atmosphere is opposite-in direction to the doublet’s axis. This induced magnetic field will clearly have the same effect on the motion of the electron as would an exter nally applied field of the same intensity. That is, the beta ray will move in a helical path whose axis is parallel with the magnetic axis of the beta particle. The path may have the form of either a right- or left-handed helix, according as the north or south pole of the beta particle is foremost. This is in accord with observation, which shows paths of both kinds. Furthermore, if the induced magnetic field does not decrease while the velocity of the beta ray diminishes, the tracks of the particles will be converging helices such as appear in the photographs. The intensity of the magnetic field induced at a beta particle due to the effect of its magnetic moment on the part of the medium at a distance greater than r from the doublet may be shown to be Sirus He Orn where yw is the magnetic moment of the doublet and s isthe effective susceptibility of the medium. If the cathode particle is moving at a speed corresponding to a drop through 10,000 volts, the minimum distance of its approach to an electron at rest is according to usual theory about 107? em. Taking this as the value of 7, using for s the value 3 x 1078 of the magnetic susceptibility of air, and for w the moment 10~*° of an electron with an angular momentum h/27, this expression indicates that the intensity of the magnetic field induced at the electron should be of the order of 3000 gauss. This is approximately the field which would be required to produce the observed curvature. The impulsive torque exerted by the beta ray doublet on a magnetic electron at rest when passing it at a distance of 107! em. has, however, a period corresponding to X-ray frequency. It is clearly necessary, therefore, to take into account the inertia of the elementary magnetic doublets of which the medium is composed. Furthermore, the magnetic The Constitution of Atoms. 281 forces acting on an electron due to a beta particle at this close range will greatly exceed the restoring forces due to the other electrons in the atom which for weaker fields may make theatom diamagnetic. In fact, there is no reason for sup- posing that the effective susceptibility of the medium when subjected to such highly intense magnetic pulses of very short duration has any intimate relation to its susceptibility in steady and comparatively weak fields. It seems ratier that the problem must be treated as a statistical one of encounters ofa rapidly moving, electrically charged, magnetic doublet with a random distribution of similar doublets al rest. A preliminary investigation of this problem indicates, how- ever, that the average motion of the beta ray should be rather similar to its motion when passing through a medium of uniform susceptibility. The above numerical discussion is therefore of value as showing that it is not unreasonable to expect a magnetic particle to induce.in the surrounding medium a magnetiz zation of the magnitude re aieired to account for the observed helical paths. If the obvious explanation of these spiral tracks is the correct one, their interpretation yields very valuable results. We have seen that the beta ray seems to act as a tiny gyroscope with a magnetic moment, which is capable of giving rise to torques on other electrons of a duration corresponding to the frequency OLN “rays. The reaction, according to classical dynamics, must result in mutational oscillations of the spinning beta particle. This obviously supplies a mechanism for the production of high frequency radiation by a free electron, which has been suggested by Webster to account for Doppler effects at the tar wet ‘ofan X- -ray tube. The possibility that the electrons are magnetic doublets is also of great impor- tance in connexion with our ideas of the structure of the atom and the nature of chemical combination. Cavendish Laboratory, Cambridge University. August 16, 1920. XXIV. The Constitution of Atoms. By Professor ORME Masson, /.R.S., University of Melbourne *. = it must now be conceded that all material atoms are compounded of positive and negative electrical atoms, it 1s surely time that each of these fundamental and anna sal constituents were known by some distinctive name. This compliment has been paid to the one, but not, * Communicated by Professor Sir E. Rutherford, P.R.S 282 Prof, Orme Masson on the as yet, to the other. For convenience of reference and notation, if for nothing else, it is just as necessary to have a name for positive electrical atoms as for the electrons. Though the hydrogen nucleus has been identified with the positive particle, it “would not be well to adopt a name specially imdlicariveron ite fact ; for hydrogen has no mono- poly in these particles, which are also present in the nucleus of every other atom. Moreover, the electron is just as essential a constituent of hydrogen itself, though not of its nucleus, as is the positive particle. The outstanding characteristic of the electrons is that they mainly determine the electro-chemical characters of the atom; so they are well named. The outstanding charac- teristic of the positive particles is that they mainly determine the mass of the atom. I therefore suggest that they should be called barons (Bapos, weight) *. If this name be adopted, we can conveniently symbolize the baron as b, using e for the electron. We thus have, in what follows, b=one baron (charge +1, mass 1), e=one electron (charge —1, mass negligible). A may stand for the mass of any elementary atom with the atomic number N. * Footnote by Professor Rutherford :— At the time of writing this paper in Australia, Professor Orme Masson was uot aware that the name “ ‘proton ” had already been suggested as a suitable name for the unit of mass nearly 1, in terms of oxygen 16, that appears to enter into the nuclear structure of atoms. The question of a suitable name for this unit was discussed at an informal meeting of anumber of members of Section A of the British Association at Cardift this year. The name ‘‘baron” suggested by Professor Masson was mentioned, but was considered unsuitable on account of the existing variety of meanings. Finally the name “ proton” mice with general approval, particularly as it suggests the original term “ protyle ” given by Prout in his well-known hypothesis that all atoms are built up of hydrogen, The need of a special name for the nuclear unit of mass 1 was drawn attention to by Sir Oliver Lodge at the Sectional meeting, and the writer then suggested the name‘ proton.’ Professor Orme Masson sent the present paper for publication through the writer, and in order to avoid the long delay involved in corre- spondence, his paper is printed in its original form. If the name “ proton ” i generally approved, it is merely necessary to change the symbol “ 6” into “p” in the chemical equations given in the paper. it oot be pointed out that a somewhat similar type of nomenclature for the constituents of atoms has been suggested in the interesting paper of Professor W. D. Harkins, entitled ‘The Nuclei of Atoms and the New Periodic System” (Phys. Review, xv. p. 78, 1920). Constitution of Atoms. 283 The inference from radioactivity work, that most of our elements are not “‘ pure ” but consist of mixtures of isotopes, and that the A value of any pure element, or single isotope, is appreciably integral (O=16 being taken as standard) has been splendidly confirmed by Aston’s recent work. Of the eighteen elements already examined by his mass-spectrum method, only hydrogen gives an atomic weight (1°008), which is not integral to within one in a thousand. The cause of this exception requires further investigation, but in the meantime it may be set aside as related in some way (as suggested by Aston) to the fact that hydrogen is unique in containing no electrons in its nucleus. Tt is unique also in other ways, and especially in the value of the ratio AN, which in H is 1 and in no other atom is less than 2 (nor more than 2°6). If we write A—2N=n, we find from Aston’s work that n=0 in pure elements, He, C, N, O, 8, and also in the lower isotopes of B, Ne, Si, and Ar. There can be no doubt that this also holds for the lower isotopes of Li, Mg, Ca, and, perhaps, some of the other light atoms. It follows, therefore, that the group (/,e) may her ae ded as a secondary unit of positive charge, with mass 2, and that (b,e)~ expresses the composition of the nucleus of any of those atoms. In the higher isotopes of B, Ne, Si and Ar, in the pure elements F and P, and in both the isotopes of Cl, n has small values, ranging from 1 to 4; in As it is 9, in the two Br isotopes it is 9 and 11, and in the six isotopes of Kr it ranges between 6 and 14. The values are higher still in Xe (20 to 27); and in He it apparently ranges between 37 and 44. The numerous isotopes of elements with N values from 81 to 92, contained in the three radioactive series, have all n values from 42 to 54. There is thus a general tendency for nm to increase with N, modified by the fact that it may vary considerably among isotopes with the same N. If the b,e group be still taken as the unit of positive charge, there must be added to it in most cases 7 electrically neutral couplets (fe), each having unit mass. We thus can express the a of all atoms from He to U by the general formula (bse) w(be)n, n having any integral value from 0 to 54 ; and even the unique case of H is included if n be given the special value of —1. If we distinguish the nucleus from the ee electrons | by enclosing the former within square brackets, [ (bse) n(be)n jen becomes the perfectly general formula Ee any electrically neutral atom, while positively or negatively charged ions 284 Prof. Orme Masson on the are similarly indicated with the appropriate decrease in the number of electrons in the shell. We can now express in general terms (which, of course, may be made specific by substitution of the proper numbers for N and n) any action that occurs within the nucleus. Thus an e-ray change depends on the intranuclear action (O52) — hae and is expressed by the sub-chemical equation [ (bee) ar(e)n Jen = [ (Ope) -a(0e)n Jew + (bue)2 where (b,e). is the expelled He nucleus and the other product is the ion of the new element, carrying a double negative charge till, by discharge, it becomes the atom [ (bse) n—2(be)n|en-2. This clearly expresses the character- istics of a-ray action-—that the atomic number is lowered by 2 and the mass by 4, to which may be added the state- ment that there is no change in the number of neutral couplets (x). A @-ray change, on the other hand, depends 4 on another intranuclear action, 2(be) = (Goes thus sub- tracting 2 from n and adding 1 to N, while the mass is unchanged and the main product is the single charged positive ion of an isobare. In full, the equation is Z Te [ (Gee) (Gen Jew = [(bze we 41(be)n—2 len + &, The results obtained by Sir EH. Rutherford by the bom- bardment of light atoms with swift «particles can be expressed similarly. Thus the expulsion of a particle with the mass 3 and a charge of 42 (the nucleus of a lower isotope of helium) can obviously be attributed to the change 2 (be) Ge +(be), while the expulsion of a hydrogen nu- cleus (a single baron) results from the action (b:¢)=) + (Ge). Sir E. Rutherford has shown that the nitrogen atom gives both these particles, apparently by actions which occur independently. The equations for them are TO oe oes. and [ (bo@)7 je7= [ be) 6(be) Jer +0. In the first case the main product is the double charged ative 1 f Aston’s higher isot f bor In the negative ion of Aston’s higher isotope of boron. In second case it is the single charged negative ion of a pre- viously unknown isotope of carbon. ; « Constitution of Atoms. 285 The suggested notation * does not indicate any reason why nitrogen atoms sbould emit both 63e and 6 ions under «-ray bombardment, while oxygen atoms emit b3e but no 6 jicns. Sir Ei. Rutherford seems inclined to infer that barons exist, as such, in the nitrogen-nucleus and not in that of oxygen, just as it has been usual to take the emission of a-rays as evidence that the radioactive atoms contain helium nuclei as such. Of course it may be so, but the inference may not be justified. A similar inference, long ago proved incorrect, is embalmed in the term carbohydrate. The sugar mralee nics do not contain water molecules as such, though they do, under certain conditions, emit water mole- cules and leave a residue of carbon. It need hardly be said that no claim is made that the formule suggested in this paper express the atomic consti- tutions in the full sense of the term. There are really as yet no data to justify such an attempt, so far at least as the nuclear part of the atom is concerned. ‘They do, however, express correctly the nuclear charge (N) and mass (ZN +7) and the shell oa whether of atoms or of ions ; they locate the difference Shetwesn isotopes in the mmmaericel: value of n (and there is no other difference) ; and they serve to correlate the whole system of atoms and their proved pro- cesses of disintegration by means of a comparatively simple notation, which may be employed to illustrate either general rules or pe instances. If the chart of radioactive transformations, as given by oe (Ce Sedans VOLO. 9) WOM be: re- -drawn, making the N scale horizontal and ihe A ale vertical al one unit of the former equal in length to two units of the latter, the n values of all the atoms can be read off on a scale drawn diagonally from N.H. to 8.W. across the centre of the chart. A succession of a-ray changes is thus marked by arrows running along a line of equal n, pointing from 8.E. to N.W.., and such lines of equal n may pass through the symbols of atoms belonging to the same or different series, just as do the vertical isotopic lines or the horizontal isobaric lines. The Universitv of Melbourne, 8th October, 1920. * It may be pointed out here that the (b,e) particle, isotopic with the He nucleus, may be formulated as [b2e)2(de)_1], just as the baron may be writen [(d,e)(e)_, |. These formule are, of course, merely a repeti~ tion of the equations already given for intranuclear changes by which the particles are generated; but the negative value of 2 serves to classify them together and apart from other ions. H is no longer quite unique in this respect. it ho on er) 4] XXV. Hud Discharges as affected by Resistance to Flow. By Wm. JoHN Waker, 8.Sc., Ph.D., Lecturer in Hingineering, College of Technology, Manchester”. PANYPICAL instances of fiuid discharges, greater than i appear to be possible eo theoretical considerations of non-viscous flow, are those of the Venturi meter for low heads, and also, mee Getic conditions, the discharge of steam through nozzles. The latter case is explained b H. W. Callendar’s “‘ supercooling theory” t. The following analysis shows, however, that ‘the discharge of a viscous fluid may actually be oreater, under otherwise similar con- ditions, than that ‘of a non-viscous one. This result appears to be paradoxical in the extreme, and it is with the greatest difidence that the writer advances it here. His excuse lies in the fact that the deduction is the result of the inclusion of a resistance expression which holds in certain cases of fluid motion. The Venturi meter, used principally in water-flow measure- ment, consists essentially of a converging circular tube, the pressures at the entrance and exit of which are measured, he difference between these serving as the measure of dis- charge. The theoretical formula for non-viscous flow is ie 2gh van / 28 of) ay.) ho ri area at entrance where m= ——, area at exit = velocity of fluid in feet per second, and h =pressure difference in feet of water. In practice this is written 2gh a) oe v—=C ce where C is known as the “meter coefficient.” Naturally this is less than unity, having values from °96 to 99. At low velocities, however, values as high as 1°36 tf and as low * Communicated by the Author. t “On the Steady Flow of Steam through a Nozzle or Throttle,” Proc. I. M. E. p. 53 et seq., Jan. 1915. t Coker, Canadian Soc.C.E. March 1902. See also Gibson, Proc. TEC ep. 399. Fluid Discharges as affected by Resistance to Flow. 287 as 0°75 * have been obtained. The curves of fig. 1 showing C against V are typical of such results, each curve tending to either high or low values of C, for low heads. a qative Coefficiené C | Sam 9 SE Ge ESE HEAD (in feet of water) Resistance to fluid motion may be written ¢ Ws Rept { AZ +k i or COR PROCES Va iat MA at Ucn a Rona a where “a” and “6” are constants. Generally, the value of “a” is positive, but cases occur frequently { in which it is found to have a negative value. The presence of this negative coefficient is undoubtedly difficult to explain, but leaving any such explanation aside for the present, it * Herschel, Trans. Am. Soc. C.E. vol. xvii. p. loc. ct. Tt Advisory Committee Aeronautics, 1910-11. { Ihd. See also Villamil, ‘ Motion of Liquids,’ p. 178. 228. See also Gibson, 288 Fluid Discharges as affected by Resistance to Flow. can be shown that curves such as those shown in fig. 1 are an immediate consequence of equation (3). By including the resistance head R in equation (1) we get h= 5 (mi 1) tan tbe", + a foe re The discharge coefficient C is given by : V — aN © ie - Sd Ge > LC (5) ee Solving (4) for v, the equation for C then becomes c=— aa / aN “EIN h where M and N are constants. E M 2 Assuming N=-98 and —— = —°02 when h=256 feet of é water, the higher graph of fig. 1 is the curve obtained. M Assuming, however, ——=+°02, the lower curve is obtained. h Resistance to fluid motion, therefore, of the kind denoted generally by equation (3), may give discharges greater than would be obtained in non-viscous flow. Apart from any practical value, this fact is interesting enough in itself. It implies, of course, that in cases where “a” is negative, regeneration of energy must be taking place. It does not appear that such energy regeneration (or rather recupera- tion) is impossible. ‘The conditions necessary may be present in certain cases, which in the present state of knowledge are not well defined. The coincidence in the form of the curves of fig. 1 with those experimentally obtained is considered too close to be accidental, and it is for this reason that the writer has ventured to submit the results (paradoxical though they appear) of the foregoing analysis. The assumption of a resistance function of the form R« wv” does not give similar results. XXVI. A Quantum Theory of Vision. BG Je LOLS IH iaese eae Theory of Vision herein described originated in views respecting the origin of the latent image which formed the subject of an address ‘to the Photographic ‘Convention of the United Kingdom in 1905 (¢ Nature,’ woleixxn. p. 308). Some couple. of years before the war [ returned to the subject and made experiments on the retinas of oxen and sheep, believing that it might be possible to detect electrons liberated by visible light falling on the retina or on the black pigment. The results were negative. The war interrupted further experiments, but more recently Mr. J. H. J. Poole, using more sensitive apparatus, examined the black pigment of sheep and oxen as well as the fresh retinas of frogs in a state of dark adaptation. These experiments also gave negative results. Further consideration of the whole matter has convinced me that such a surface emission of electrons was hardly probable under the conditions attending the experiment: conditions which involve the unavoidable presence of surface impurities. The failure to detect liberated electrons by no means in- validates the theory herein discussed. Dr. H. Stanley Allen, writing to ‘ Nature’ (Oct. 30, 1919), refers to a theory of colour vision which on January 7th, 1919, he communicated to the Rontgen Society. In this theory he supposes “that photoelectric action takes place in the rods or cones, so that we have a separation of electrons resulting in electrification of the nerve-cells which set up the nervous impulse to the brain.” An essentially similar suggestion was made by Sir Oliver Lodge at the meeting of the British Association in 1919. (1) I assume that the origin of luminous vision and of ‘colour vision is to be sought in the liberation of electrons under light stimulus w ithin a photoelectric substance or substances existing in the retina. The rhodopsin is such a photosensitive substance. In the case of the rods (in which rhodopsin is found) this substance acts as the basis of vision. In the case of the cones the same substance is very probably responsible. A strong argument for this view is to be found in the fact that in the fov ea, where only cones exist, the ‘spectral range of vision is in fair agreement ‘with the spectral * Communicated by the Author. Phil, Mag. S. 6. Vol. 41. No. 242, Feb. 1921. U 290 Prof. J. Joly on a absorption of the visual purple or rhodopsin (see (8)). Further, Kiihne states that the maximum spectral absorption of this substance is at that part of the spectrum whicl: seems brightest to the eye, and which is most active in bleaching the rhodopsin. It seems therefore very probable that all over the retina it is, this substance which forms the inter- mediary between the light and the nerve ; translating the quanta ot light-energy into nerve-stimulus. It is an important feature of the theory herein advocated that the sensitiser in the case of the cones—as the organs of colour vision—should lie outside the cone and should not exist within it as in the case of the rod. That it does not exist within the cones is agreed by all observers. Dr. Edridge Green (‘The Physiology of Vision,’ 1920, p. 43) claims to have seen the unbleached rhodopsin penwmeen but not in the cones of the fovea. He states that when the retina was first examined the fovea was the reddest part of the whole retina. He also calls attention to a confirmatory observation of Kiihne’s in the case of a shark’s retina. Mr. J. Herbert Parsons more recently points out that Hering had recognized such a distribution of rhodopsin as possible (Brit. J. Ophthal., July 1920) *. I assume that in the case of the rods the sensitiser is operative within the nerve. In the case of the cones it is operative from without. (z) The belief is gaining ground that photo-chemical and photo-electric processes are fundamentally alike (Lewis, Physical Chemistry, iii. p. 134). In the case of photo- graphic actions the view that the movement of electrons within the light-sensitive film is responsible for the phe- nomena pbsearedl is supported by facts regarding the various modes in which the plate may be stimulated and by the formation of the latent image. No other theoretical basis affords so general an explanation of the effects of the light. (See ‘ Photo-Electricity,’ by H. Stanley Allen: Longmans Green, 1913.) The range of photographic ‘vision’ may be controlled by the use of a sensitiser, This substance is one which absorbs vigorously in one or more special regions of the spectrum. It sensitises the plate to the same range of wave-lengths as it absorbs. The photographic geeieer are rich in light- absorbing molecules:—chromogens. They are photo- -electric: emitting electrons over the range of frequencies which they absorb. cae Dies Kdridge Gieen believes that vision is due to photo-chemical action progressing in the rhodopsin surrounding the cones. He considers. that the rcds are not percipient. Quantum Theory of Vision. 291 (3) Among the well ascertained facts of photo-electric science the following concern the present theory. (a) The electron is liberated with a velocity which, normally, depends on the frequency of the light only: increasing as the wave-length diminishes in such a way as to render the kinetic energy a linear function of the frequency. (6) The velocity is independent of the intensity (amplitude) of the light. (c) For equal intensities of light of different frequency, the light of highest frequency. liberates most electrons. (d) For lights of the same frequency, the number of liberated electrons increases with the intensity. (e) The electron in most cases absorbs one quantum, the value of this quantum depending on the frequency according to the well-known equation e=hv, where h is Planck’s constant (=6°57 x 10~* erg.sec.) and v is the frequency. In virtue of the absorbed energy, the electron acquires a certain velocity and pursues a certain free path in the medium till diverted by collision. ( 2 The value of the mean free path of electrons taking part in photo-electric emission from platinum has been determined by Robinson (Phil. Mag. 1912, 1913). He concludes that it is of the order 10-’ em. Partzsch and Hallwachs (Ann. d. Phys. xli. p. 247, 1913) concluded that 99 per cent. of the photo-electrons from platinum emerge from a layer thinner than 28x 10-‘ cm. ; from which we may conclude that the maximum range is oe of this value. Patterson (Phil. Mag. 1902, iii. p. 643, iv. p. 652), dealing with the electrical conductivity of dnin cries films, arrives at the conclusion that the mean free path in various metals, including carbon, is of the order 10°° em. He cites a result by Vincent (Ann. d. Chem. et d. Phys. xix. p. 421, 1900) that for silver the mean electronic free path is 6x10-%cm. Vincent’s result also is derived from measurements of resistance. Such of the above determinations as are based on direct photo-electric measurement, using ultra-violet light, require correction for the less value of the quantum associated with visible wave-lengths: that is, if we assume the penetrating power of the electron is dependent on its velocity. On the other hand, judging by the influence of density on the co- efficient of absorption of £8 electrons, a correction for density may be necessary when we venture to so far exter- polate as to apply results on platinum to aqueous solutions of low-density molecules such as the fluid of the retina, Applying t these corrections to the deductions of Robinson and to those of Partzsch and Hallwachs, we find the mean bee 292 Prof. J. Joly on a free path in water for electrons supposed to carry the quantum for yellow light to be 5°5x10-" cm., and the maximum free path to be 154x10-% cm. These are, of course, only approximations and must be regarded as only admissible in the absence of more secure results. The quantities arrived at by Patterson and by Vincent for the mean free path would assign to it a value from five to ten times the above, even if we make no assumption as to the influence of density. A correction on the score of initial velocity is not called for, as the initial velocity affecting Patterson’s deductions is about 7°6 x 10® em.: a value not greatly different from that acquired by electrons carrying quanta associated with visible light. Respecting these figures, we faust of course beac aateametel that the “collisions” refer in general to a deviation of path, not an arrest of motion and loss of energy. (5) The velocity just referred to as acquired by the electron when a quantum of energy is imparted to it by . visible light is very great—of the order 10’ cm. per sec. The course of the electron before its kinetic energy is given up in work of ionization, in thermal agitation, or otherwise, is a brief one: probably less than the billionth of one second. (6) The rods contain the sensitiser in the form of rhodopsin. J assume that this substance emits electrons in the same manner as other light-absorbing and optically unstable substances*. The electrons set free from the sensitiser by the rays absorbed expend their kinetic energy in stimulating the nerve, and, perhaps, establish an electronic current into the ganglion cell with which the nerve makes connexion. I assume that the electronic emission in the rod constitutes an intimate and generally copious source of stimulus—the light being very completely absorbed. The conditions are therefore very favourable to the appreciation of feeble illumination. In this association of the photosensitive substance with the nerve, the quantitative value of the stimulus is developed at the expense of its qualitative value. Colour will not be interpreted to the brain, or only defectively. The liberated electrons are in no degree selectively presented to the nerve. All that escape from the rhodopsin molecules contribute to the total stimulus. In the case of some the energy of the quantum is all but spent ; others reach the nerve possessed of the maximum kinetic energy. The stimuli may overlap * Kiihne dwells on the remarkable instability of rhodopsin towards light and its great stability towards chemical reagents (‘ Photo- henists of the Retina and on the Visual Purple,’ 187 8). Quantum Theory of Vision. 293 in time and space. It is probable that some quanta are expended in conferring purely thermal movements on electrons, and along with the destructive effects of quanta upon the sensitiser the regeneration of this must all the time be progressing within the nerve. A confused flow of stimuli, too impure and too crowded for analysis, is the outstanding character of the sensory contribution of the rod. It corresponds to noise in the case of audition. Nevertheless, the conservation and integration within the rod of the stimuli arising from the interaction of light and sensitiser render it a most sensitive exponent of luminosity reaching the retina. Its sensitivity is, indeed, extraordinary. Henri and des Bancels have shown that the retina is sensitive to an amount of light energy of the value of 5x10°" erg. Now the quantum for green light is 4x 107” erg. We may assume, therefore, that one quantum is sufficient to excite vision (Bayliss, General Physiology, p. 512). That is to say, the liberation of a single electron by green or blue light will excite visual sensation. Again, consider the following relatively commonplace case. A standard candle removed to a distance of 3000 metres projects on each square centimetre luminous energy of the amount 4x 1077 erg per sec. This lummosity will evoke vision ; and such feeble radiants are known to be best appre- ciated when the image falls on parts of the retina rich in reds. How many quanta are involved in this excitation of vision ? The pupil admits, say, the luminosity reaching one-half a square centimetre or 2x107’ erg per second. This corre- sponds to about 7x10* quanta per second ; such quanta as would be associated with yellow light. The number of rods which receive this energy is considerable. The size of the image is indefinite, but it will not bea point image. Suppose it covers one-tenth of a square millimetre, we can roughly estimate the number of rods involved. The total area of the retina is about 1000 square millimetres, and the total number of rods has been estimated as 130 millions, and again as half this number. We shall take the number to be 100 million. There will be about 10,000 rods illuminated. The quanta are in fact distributed over this number of rods : that is, 7 quanta enter each rod per second. We may translate this into 7 electrons liberated in each rod per second. It is evident that these small individual stimuli must be so far conserved us to make their way to the optic nerve. If they did not do so but died out within the nerve, there would be no vision. We have it then, however 294 Prof. J. Joly on a wonderful, however incredible it may seem, that the stimulus arising from one quantum must constitute an appreciable fraction of the threshold stimulus. (7) The cones are structurally different from the rods in that they contain no visible quantity of rhodopsin. Further, they differ in that each cone is connected through a separate ganglion cell to the optic nerve. This prevails in the fovea where only cones exist and where colour vision is at its best. Throughout the retina, on the other hand, the rods are grouped ; several individuals contributing their stimuli to the one ganglion cell. This fact is often referred tu as accounting for the sensitivity of the foveal area. It is very significant. It reveals an effort of Nature to husband and conserve the cone stimnli and to convey them undiluted to the brain. It suggests that these stimuli are more delicate than those coming from the rods, and are of such a character as to bear no intermingling with other stimuli. | It is also noteworthy that in the central fovea the cones lose their characteristic conical form. They attain a re- markable length, at the same time diminishing in diameter till the latter sinks as low as one micron. Taking the diameter of the outer segment of the cone at this latter figure, and the length as 45 microns*, we find the surface amounts to 180 times that of the cross section. To what is this remarkable effort after surface to be ascribed? If we assume the light entering the cone at its inner extremity to be uniformly distributed throughout the cone, it must escape laterally with a luminous intensity reduced 180 times. Plainly there is some advantage on the score of sensitivity in this diffusion of the light. In the central fovea we are told that the cones are so closely packed as to take on a prismatic form where the inner segments approximate one to another. The light must, therefore, at least for the greater part, move as I have indicated. I assume that the outer segment of the cone is bathed in a photosensitive fluid, probably—almost certainly— rhodopsin. In this the light is absorbed: either directly at the surface of the cone or within the thin layer which separates cone from cone. At the meeting of the cone surface with the sensitiser electrons are emitted. They will also be emitted at such distance from the cone surface as the light can penetrate. Some of those freed at the surface enter the nerve with maximum velocity and kinetic energy. Those liberated more deeply enter the nerve with diminished velocity. * Greeff’s drawing, according to Schafer, underestimates the length of the cone. On the drawing it is 38 microns. See Quain’s ‘Anatomy.’ Quantum Theory of Vision. 295 Some do not escape from the sensitiser. Others fail to travel beyond the delicate covering of neurokeratin which is believed to invest the nerve. The fastest electrons carry into the nerve almost the full quantum of energy which is characteristic of the frequency giving rise to them. These are the most effective in exciting a nerve stimulus. If the intensity of the light is considerable there are many such. If feeble, there are only a few; but the speed, trajectory, and energy of these electrons remain characteristic of the frequency. Such stimuli are too few and too brief to confuse one with another by overlapping. They are ap- preciated at their cerebral destination as would be successive notes heard in music. The penetration of the faster electrons into the nerve must be considerable. A maximum free path of 154x107" em. may be assumed. ‘The radius of the cone is about 5 x 107° em. It follows that’ electrons moving in a radial direction may traverse one third of the radius before they become deflected. One half the total cross sectional area of the nerve is traversed by these direct movements. Deviated electrons may be supposed to reach the centre or travel beyond it. As to the nature of the stimuli arising from electronic bombardment and as to the manner and form in which the energy of the electron is transmitted outwards from the retina to the optic nerve, we have much to learn. The trans- Mission is probably electric in character according to many physiologists. It seems, however, to be probable that the velocity of the absorbed slacaam 16 mob dha welmel is quantitatively appreciated by the nerve, the time interval involved is far too small. But we may assume that the disturbance set up by the shock is not so short-lived. It travels relatively slowly from its point (or rather line) of origin. Probably what the nerve appreciates is the energy value of the individual stimuli, and this depends on the quantum of energy associated with the electron ; a. in turn is determined by the frequency of the light, by its “colour.” That each individual electronic eae must possess a certain sensory value appears from the figures cited above respecting threshold vision by the rods. It is necessary to consider a little more fully the specialized nature of the cone functions. In the central foveal area, covering about 0°16 square millimetres, there appear to be some 2x10* cones*. Suppose such an amount of light as would certainly excite * Rather more according to Golding-Bird’s drawing, if this is intended to depict these organs numerically. 296 Prof. J. Joly ona vision in the rods fell upon this area. I assume that the image of a candle flame, which is 1500 metres distant from the eye, is focussed upon the central foveal area and effectively covers this. We now have twelve electrons formed per cone per second. It seems safe to conclude that at this distance the fovea would appreciate the candle-flame. Now it is very evident that enormously greater numbers of electrons might be generated in the sensitiser bathing the cone without risk of overlapping of the stimuli. For not only must we assume that but a small percentage of the received quanta is restored to the cone in the form of what we may call “ characteristic ” electrons (7. e. those possessing. speeds near the maximum speed; in other words, carrying the quantum proper to the wave-length), but we must also bear in mind that the work of each electron occupies at most. but a very small fraction of a second. We have, indeed, arrived at the weak point of the arr angement—its prodigality. Accordingly, we find that Nature, driven to adopt the external disposition of the sensitiser in order to avoid the confusion prevailing in the rods, proceeds now to improve on her design and to develop the cone so as to obtain the maximum number of high-speed electrons. This is effected by increasing the area of the cone wetted by the sensitiser. For the more the activatin light is reduced in intensity per unit area (within limits) the less the penetration of the ray into the sensitiser, and the greater the number of electrons released at the immediate cone surface. Colour vision is in abeyance at very low luminosities. The explanation is that ultimately there are insufficient characteristic electrons to excite the colour sensation. On the other hand, colour vision cannot be excited without the diluting effects of the slower moving electrons appearing. For there must always be many (probably a majority of) slow-moving electrons stimulating the nerve. ‘The sensation arising from these is not characteristic of the frequency which gives rise to them, and a sensation of white light is the result. The colour fails to be ‘“‘saturated.” White sensation is always added to the colour sensation excited by the characteristic electrons. We never experience quite saturated colour sensation. It will be gathered that-the present theory ascribes quantitative sensitivity to the rods; qualitative sensitivity to the cones. The difference being mainly referable to the fact that in the one case the sensitiser is located within the nerve, in the other it is located without the Quantum Theory of Vision. 297 nerve. The sensory stimulus emanating from the rods is- compounded of many sources of stimuli, z. e., such as may originate in electrons possessing every velocity and kinetic enerey up to the maximum proper to the frequency of the- activating light, such as may arise in electronic movements. associated with the regeneration of the sensitiser and in. thermal electronic agitation excited by the quanta taking part in these operations. The stimulus emanating from the: cones, on the other hand, is purified of all stimuli save those: arising from the kinetic energy of electrons, which are activated by the energy of absorbed quanta. The electrons. are, in fact, selectively presented to the nerve: all other sources of stimuli take place outside the cone and are cut off from it by the filament which invests it. In the cone the more intense stimuli tap out to the brain the sensation of. colour which we associate with the intensity of the quanta involved. ‘To this succession of characteristic nerve impulses there is added an underlying accompaniment: the white or luminous sensation made up of all the feebler electrons which impart to the nerve but a fraction of that which is. characteristic of the frequency of the light entering from. the world without. The cone is the more highly specialized. organ of the two and is probably a more recent product of evolution. (8) The higher Inminosity of the colour threshold arises out of the conditions affecting the stimulation of the cones ; the sensitiser being external and hence a part only of the evoked electrons producing visual sensation. In the rods all absorbed radiations are expended on exciting the sensation of luminosity ; the electrons being liberated within the rod. There will be a colourless interval attending foveal stimula- tion of low intensity for the reason that’ the characteristic electrons constitute only a fraction of the stimulus and it requires a certain density of such electrons before colour vision is experienced. In other words, the achromatic effect of very low light intensities or of very brief exposures is due to commencing stimulation by non- characteristic electrons. (9) One consequence of the different disposition of the gensitiser respecting the rod and the cone is that the characteristic quanta stimulating the cone cannot possess the full energy value proper to the originating frequency. ‘That is to say, it reaches the nerve with a quantity of energy such as would be associated with a frequency v’, less thanv. This. is because the electron in the case of the cone must part with some of its energy in penetrating the outer sheath of the cone. We may expect “from this that the entire luminosity curve of 298 Prof. J. Joly ona photopic vision will be shifted towards the red end of the ‘spectrum. ‘This, aecording to the present theory, is the explanation of why the photopic luminosity curve does not quite coincide with the scotopic luminosity curve; the maximum of the first being at D, nearly, and that an the second being at KH (Abney, Colour Vision, p.° 103). Working from these wave- lengths it is easy to show that the electron” must lose about 0-3 x 10-2 erg in penetrating the sheath; 7. e., almost 10 per cent. of its energy. It is, according to ie Quantum Theory, quite unnecessary to seek for any other sensitiser than rhodopsin as the basis of vision. (10) It is probable, according to the present views, that both rods and cones functionate by transmitting electrons from the sensitiser into the optic nerve. The observed current from fundus to cornea attending the light-stimulation of the excised eye finds explanation in “the present theory. _ Bayliss (loc. cit. p. 522) reviewing the researches of Waller, Hinthoven and Jolly, and others, says :—‘ Respecting the results of these researches the main fact is that, in the uninjured eye of the vertebrate, the incidence of light causes an clectrical change in such a direction that the nervous layer of the retina becomes electrically positive to the rod and cone layer.” It will be seen that this points to the liberation of free negative electricity in rods and cones attending the light stimulus ; 7. e., to the presence of free electrons in these terminal organs. (11) Chemical effects have been observed as taking place in the retina when adaptation is changed from dark to light: 2. é., there is a change from alkalinity to acidity. ‘This may be involved in the loss of an electron by the ion HO and the formation of the ion H™. (12) The movement of the cones attending light stimulus, which in the case of most animals certainly occurs, and which occurs in the case of Man, may be a mechanism designed to bring the cones into un-exhausted sensitiser ; the bleaching of the immediate layer touching the cone being fatal to its full and proper activity. The rods carrying the sensitiser within would not profit by such a move- ment, and accordingly do not exhibit it. (13) Mechanical effects, such as pressure, appear to liberate electrons and produce the latent image on a photo- graphic plate. The Juminous sensation attending pressure of the eye-ball may arise in the same way: 2. e., by the mechanical liberation of electrons. It can be referred to what is known as triboluminescence. (14) The degree of spectral analysis attainable according Quantum Theory of Vision. 299 ‘to the foregoing theory of colour vision must be limited. The interpretation of colour is referred to the appreciation by the nerve of the value of the quantum. A complete ‘detailed analysis of the whole gamut of wave-lengths be- tween red and violet on such a basis is, probably, unattainable, even if it was any benefit to the organism. And if attain- able it might result in badly differentiated colour sensations. The evolutionary growth of three highly developed colour sensations corresponding to the central and mean quanta of the spectrum is the result. It is Nature’s compromisegavith her limitations. It is one which is, in part, cerebral in character: the light-sensitive part of the brain accepting as interpretive of the many separate frequencies a commingling of sensations excited by the central and end frequencies. ‘These views do not preclude the possibility of more than three primary colour sensations existing. I assume, how- -ever, that red, green, and violet are alone primary. If, now, rays of the wave-length 5893 A.U., say, are received -on the retina, no sensation special to this wave-length arises, although electrons having velocities quite peculiar to co) it activate the nerve. It is more efficient for the organism to develop special sensitivity towards three representative stimuli, produced by widely differentiated quanta. Thus we feel so much red sensation according to the proximity of X 5893 to 6563, and so much green sensation according to its proximity to X 5461. The combined sensations we call yellow, and yellow becomes a distinct sensation, although it may be really compounded of two other sensations. It is a sort of unconscious memory: the one stimulus ‘ reminding” ‘the colour-visual centre of the stimuli which evoke the representative sensations, red and green. Of course it would be easy to inagine a purely objective explanation of colour vision if rhodopsin exhibited appro- priate absorption bands. But, on the contrary, its absorption “spectrum is remarkably uniform over the range of the visible spectrum. (15) A colour-blind individual is one whose foveal nerves respond feebly to certain quanta. The same abnormality -affects the cones all over his retina. Thus if he is violet- -and egreen-blind, the quanta proper to H or to F produce only a feeble stimulus. But those proper to C and D are fairly normal in the stimulus they excite. His brain has developed no more than the one sensation, the maximum luminosity of which les between Hand D. The abnormality is fundamentally a physical deficiency ; and this leads to ‘mental deficiency, as commonly happens in similar cases, o 300 Prof. J. Joly on a Colour vision curves constructed from the examination of abnormal sight show the curves as overlapping. As I have already pointed out, the entry of spent electrons into the. normal nerve—i. e. electrons possessing less kinetic energy than is proper to the wave-length illuminating the retina— introduces indeterminate stimuli which result in luminous or white sensation; for such, received in the sensory colour- centre, could not be differentiated from the sensation arising when white light falls on the retina, and quanta exciting all thre@ colour sensations stimulate che nerve! cen spent electrons affect the sensations of the colour-blind also,. according to his limitations. They can give no new sensation to one possessed of monochromatic vision. The subject 1s. full of obscurities and difficulties, and I shall not enter upon it. I see nothing in the present theory to accentuate or add to those difficulties. Jt is to be hoped it may contribute to. clearing them up. (16) The spectral limits of the colour-sensation curves find a simple’ physical explanation in the failing absorption of rhodopsin for these wave-lengths. This matter is, of course, Bound up with the limitations imposed by the absorption of quanta (associated with the higher and lower frequencies) by the media through which the light has to pass before it reaches the retina. (17) The light-sensitive substance, rhodopsin, is probably extremely complex. It is said to be related to Anthocyanins. which are rich in chromogens and are held accountable for a wide range of colour in flowering plants. Evidence for the richness in chromogens of rhodopsin is, I think, to be found in its sustained absorpticn of visible spectral rays. “ Fatigue” may evidently be referred to the exhaustion of such chro- mogens as have been deprived of an electron. ‘This appears. to be the natural explanation. (18) When a bright object 1 is looked at for some time and the eyes then closed, ‘“‘ after-images”’ are seen. These are- both positive and negative. The phenomena of positive after-images suggest that. something of the nature of the latent photogr aphic i image is formed in the nerve-substance. Indeed, @ priori, one w ‘ould expect this to happen, the conditions having much in common in the two cases. On this view electrons entering the nerve would in some cases remain attached to atoms within its. substance and an electrostatic field would prevail between them and positive ions in contact with the cone. This system. must break down ultimately—probably is continually breaking down and being rebuilt. The oo attached: Quantum Theory of Vision. 301 within the nerve would be distributed at radial distances trom the surface proportional to the frequency of the light which gave rise tothem. The “red” electrons (i. e. those exciting pre-eminently the red sensation) outermost, then the “oreen ” and lastly the “ violet.”” But in the case of very intense light stimulus the disturbance due to the passage of very many electrons might result in permitting only very few of the red and green, but allowing abundant violet, to collect in this manner ; the latter attaining the outer limits of the field of disturbance. Now in the colour succession of after- images there is found evidence for both these modes of distribution. We must suppose that when the light stimulus is withdrawn the electrostatic field gradually breaks down. We have the red electrons going first ; for they are the most strongly attracted; the green following, and finally the violet. We assume that in the act of reverting to the sensitising molecule the electron creates fresh colour sen- sation. But the energy available must be less than that originally possessed by the electron when entering the nerve. Hinstein’s explanation of the law of Stokes respecting fluorescence (Allen, loc. cit. p. 190) may be invoked. In the present case there should be a diminution of the intensity of the stimulus; in other words a shift of the sensation towards the red end. Now it is agreed by many observers that amidst many variants the after-images appear very generally in the @nger red, green, blue (Parsons, foc. ct: pp. 111, 261). This is for moderate to bright light. For long continued excitation by more intense light, blue takes precedence of all. There is also evidence for a lowering of the spectral sensation. Thus McDougall writes : “ An important feature of the after-images of bright white light is that, after a first short period in which two colours fuse to give yellow, or, as is the case after the brighest lights, all three fuse to give white, the colours that in turn occupy the area of the after-image, alone and unchanging for considerable periods, are red green and blue only. ‘The red is a rich crimson red, decidedly less orange than the red of the solar spectrum, the blue is a rich ultramarine, and the green a pure green having no inclination towards blue or yellow.” He goes on to describe the high saturation of these colours. As regards the repetitional effect generally observed, that is the recurrence of the three colours in the like order, we again find a photographic counterpart in recurrent reversal. This is, according to the electronic theory of photography, due to the break-down of a succession of latent images ; 302 Prot. J. Joly ona accumulations of electrons occurring till a point is reached when these revert to the parent molecules. But in what manner in the case of the nerve could such effects be stored and saved from immediate degradation ? A possible explan- ation suggests itself. The cone during light stimulus contracts; subsequently it again elongates. When con- tracting we may suppose the cone moves towards parts of the sensitiser still unacted upon. [And this is probably the primary object of the movement.| The effect will be to remove the cone from the field of pvsitive ionization and so free the internal, fixed electrons from electrostatic attraction. But when the retina is again darkened the cone moves back into the exhausted sensitiser ; a region rich in free positive ions. Hence as it elongates an electrostatic positive field accumulates till there is break-down and discharge during which the fundamental sensations are successively evoked. If the successive colour cycles attending the movement of the cones overlap there will ensue the irregular sequence often perceived. On the other hand, if the cones move under the influence of a common stimulus and advance, not uniformly, but with pauses of quiescence, then the repetitional colour cycles find complete explanation ; each cycle corresponding to the latent “image” breaking down over one short segment of the cone. That the movement of the cones is general and not due to a stimulus local to each cone is, I suggest, shown by the fact that in the case of the frog, where the cone-movement is very marked, “light on one eye causes reaction on both as also light on the skin so long as the brain is intact”? (Hngelmann, ‘Nahmacher.’ See Parsons, loc. ct. p. 12). Tt is consistent with the view that after-images are of the nature of the latent photographic image that they may persist for very considerable time intervals. The negative after-image is probably explained by fatigue. It appears a little after the positive image, and when the retina is re-exposed to feeble illumination. What was bright now appears as dark, and the colours change to the complementary hues. A discontinous motion of the cones will naturally arise if the nervous actions involved are reflex in character ; as the observations on the frog very surely indicate. In this case electronic stimulus of the cone initiates its retraction and the cessation of the stimulus initiates its extension. Hence when, on the extension of the nerve, the luminous after-image begins to be formed the electronic movements act as the afferent stimulus and extension ceases or con- traction may ensue. Only when the after-image dies out is Quantum Pheory of Vision. 303: the extension of the cone continued. But now a fresh part. of the ‘latent i image ’ > becomes involved and again there is. arrestment: and so on. (19) The momentary electrical response which is noticed in the retina when light is cut off and which is the same in direction as the light response (Bayliss, loc. cit. p. 522) is not difficult to explain on the present theory. It is due to the break-down partially or completely of the latent “ image” in the cones ; that is to the stimulus which arises when “the. anchored electrons are attracted back to the positive field surrounding the cone ; the electrostatic effects of inflowing co) electrons attending light stimulus having ceased to affect. them. (20) Simultaneous contrast effects are, according to my own observations, largely due to imperfect fixation. There- is probably a psy ‘chological factor also involved. The tissue paper increases the effects of adjacent colour-patches because it renders fixation inaccurate. (21) The Purkinje Effect has been explamed by Vow Kries on the assumption that the cones are sensitive to colour, possess & maximum sensitiveness in the yellow, and are responsible for vision at high luminous intensities; the rods being responsible for vision at low intensities. Into. this soater the dark adaptation of the retina enters, for with it scotopic values rise. ‘here is nothing here inconsistent with the Unitary Theory of vision. (22) ‘The dependence of the colour-sensitivity of the cone. upon its surface area is well shown by the increasing colour- blindness of the retina towards the periphery ; the active area of the cone diminishing as retinal colour-blindness Increases. The study of colour vision is hampered by many diffi- culties, chief among which is the elusive and variable nature of the effects under observation. On this account we find disagreement among high authorities as to many phenomena of vision. I shall not here pursue the matter into further details. The foregoing theory is founded on the conception of the quantum. “The nerve is supposed to discriminate between the quanta of three or more representative spectral centres. And should not the quantum be regarded as a vera causa, when we find that a single one of them acting on the retinal nerve suflices to stimulate the sense of vision? What alter- natives have we to a quantum theory at the present time ? One thing seems certain. No interpretation of colour in terms of the frequency seems possible, whether primary or 304 Dr. D. N. Mallik on forced vibrations be appealed to. The late Lord Rayleigh, ain a letter to ‘Nature’ (May 21, 1918) questions the possibility of sound-frequencies of 256 vibrations per second being directly conveyed by the nerves to the brain. “It is rather difficult to believe it,” he adds, “ especially when we ‘remember that frequencies to 10,000 per sec. have to be -dealt with. Even if we could accept this, how deal with ight-processes in action along the nerve repeated 10! times per second ?”’ I have received kind help from many friends. At the ‘time of my pre-war experiments Professor John Mallet Purser gave me much valuable instruction, and since has continually assisted me. Prof. H. H. Dixon, Dr. O’Sullivan, S.F.T.C.D., and the late Sir Henry Thompson also advised me. More recently, I have to acknowledge much assistance in experi- mental work from Prof. Pringle and his Assistant, Dr. Fearon. Dr. Euphan Maxwell has been so good as to place valuable histological specimens at my disposal. I owe much to my discussions with Mr. J. H. J. Poole. Trinity Cotlege, Dublin. Jan. 4, 1921. XXVII. Electric Discharge in Hydrogen. By Dr. D. N. Maui, 2.8.8.4 HE peculiar behaviour of an electric discharge through Hydrogen has been the subject of study for a long time. That it is somewhat erratic, for reasons which are -altogether obscure, has been noticed by several experimenters. In particular, in a paper in the ‘ Philosophical ‘Transactions’ published as long ago as June 1907, Prot. H. A. Wilson and Mr. Martyn described the peculiar behaviour of discharge ‘through hydrogen ina De La Rive tube. They found that the rotation effects were not at all well marked when the contained gas was hydrogen ; although when the tube was filled with air or N,O, the phenomenon was well defined and capable of quantitative measurement. Working with an -induction-coil, as well as storage cells and a tube which had given satisfactory results with several other gases and vapours, ‘I was unable to detect any rotation at all with hydrogen. I was, accordingly, at first disposed to regard this as an ilius- ‘tration of the generally erratic behaviour of hydrogen, which previous experimenters had remarked upon. * Communicated by the Author. Electric Discharge in Hydrogen. 305 On repeating the experiments, however, with tubes of various lengths and with induction coils giving various £.M.F.’s, I found that the rotatory phenomenon is observ- able in hydrogen as in other gases, but the conditions have to be carefully adjusted for the purpose, for reasons which will appear from the theoretical discussion which is given in the present paper. This discussion, however, is necessarily incomplete, in view of our ignorance of the intricate mechanism of the processes which obtain in a discharge- tube. : We know that the equation of continuity in a discharge- tube is [J. J. Thomson, ‘ Conduction of Electricity through Gases,’ art. 137] | OR One. ne aE ay nah =(¢—f) T° where z = =no. of corpuscles generated by collision per unit length of the tube per unit time, B a =no. that disappear through recombination, n=no. of corpuscles per unit length of tube, having average velocity c, 1=length of mean free path. For steady rotation we must have a—B=0, and, moreover, ne) =/(=), where F, f are unknown functions, and X=electric intensity and p the pressure. The condition accordingly depends not merely on the pressure of the gas, but also on X and f. , Now, both experiment and theory have amply shown that X depends not merely on the voltage of the induction-coil, but also on the nature of the gas. We must, therefore, admit that @ also depends on the nature of the gas ; it follows accordingly, that the rotation stage will depend on this quantity, as well as on other factors (pressure, leneth of the tube, and the voltage of the induction-coil). (a e That @ depends on the nature of the gas is & preor evident. The following investigation, further, shows that it ought to be so. Phil. Mag. S. 6. Vol. 41. No, 242. Feb. 1921. x 306 Electric Discharge in Hydrogen. If n=number of electrons in unit volume, then the number of electrons which have speeds lying between c and ctde 8 @2 K ae CN GKG 2 where « is the most probable speed and varies as c. Then the corresponding number of collisions Cn a Aone NGO Hence the total number of collisions corresponding to the range of velecities 0 to ¢ ne Cc WGC: aii, e- > near . Ve é. oC Tp a ae l at y3 Laie If, therefore, only the collisions lying within the range e 7 e e C 0 and c (supposed, small) result in recombination, B x 5. C Mowat Now, we may reasonably suppose ¢ « — for any gas, and Pp e therefore c « is for different gases. Me SE Also c « Xe, i.e. we may take ca —p. if Moreover, if tbe kinetic energy corresponding to ¢ is supposed to arise in accordance with the equation 2 tme= 7 ; where J’ is the distance [supposed, constant] at which the 4 C velocity of corpuscle is zero, then a — Pye In order to satisfy the condition for steady rotation, therefore, we must have (=) =8= le Va FS | pd Collision of « particles with Hydrogen Atoms. 307 xX : Also, since — =const. for any gas, at the rotation stage we have, at this stage, XxX In a previous paper I have shown Una Pe where w=the angular rotation of electric discharge under electric 1 intensity X. We have accordingly po « —, a result which has been experimentally verified. f Moreover, experiment has shown that (at the rotation stage) a varies as the effective voltage of the induction-coil producing the discharge. This voltage, therefore, must vary inversely as the density of the gas used. It stands to reason, therefore, that in the case of hydrogen, the effective voltage required will be much higher than in air, and accordingly, the effect will be much more difficult to observe. XXVIII. On the Collision of « particles with Hydrogen Atoms. To the Editors of the Philosophical Magazine. GENTLEMEN,— N two papers in your Journal (June 1919) I gave an account of some experiments on the collision of « particles with hydrogen atoins, using the scintillation method for determining the number of H atoms set in swift motion. With the microscope used at that time the counting of scintillations was a difficult and trying experience, but during the last year by the use of holoscopic lenses of large aperture and suitable eyepieces, recently constructed in this country, it has been found possible to improve greatly the ease and certainty of counting such weak scintillations. With the old arrangement, using a complex beam of H atoms of different velocities, the observer was very liable to miss the weaker scintillations due to lower velocity H atoms in the presence of the brighter scintillations due to swifter ones. The accurate determination of the relation between the number and angle of ejection of the H atoms for different speeds of the a particles is of great importance in order to throw light on the nature and distribution of the forces in 308 Collision of « particles with Hydrogen Atoms. such close collisions when the nuclei must approach within 3x 1078em. of each other. This problem is being attacked in the Cavendish Laboratory by two methods. The electric balance method, as devised by Mr. McAulay (v. Phil. Mag. Dec. 1919), has the great advantage that it does not involve the eyestrain of counting, and, if the electrical effect is. sufficiently large, is practically independent of the probability variations which make it necessary to count so many particles in the scintillation method. This electrical method should yield approximate quantitative data and has certain advantages over the scintillation method for examining the effect of H atoms near the end of their range where their ionization per unit path is increased due to their reduction of velocity. The relation between the number of H atoms shot out at different angles with the primary beam of 2 rays for different velocities of the particle is under direct exam- ination by Mr. Chadwick and Mr. Bieler using the scintii- lation method. The general results so far obtained by these new methods indicate that the number of H atoms liberated is much larger than the theoretical number to be expected from point nuclei and possibly even greater than my original estimate. They also confirm the observation that the H atoms tend, for the high velocities of the « particle, to be shot forward in the direction of the @ particle but not to the same extent as the preliminary experiments showed. The direct method used by Chadwick and Bieler indicates that even for swift « particles there is present a certain proportion of lower velocity H atoms. ‘The determination of the relation between the number of H atoms and angle of ejection should afford valuable data for calculation in order to throw light on the structure of the nucleiand forces between them. Unfortunately such counting experiments involve a large amount of time, so that there will be some delay before the necessary data are available for adequate comparison of theory with experiment. I would like to take this opportunity of drawing attention to a slip in my original paper (loc. cit. p. 546) which fortunately is of little importance te the main argument. In calculating the emergent range for H atoms scattered at an angle @ with the direction of the « particle, the correcting factor for absorption under the experimental conditions was taken as cos 9. As Mr. C. Darwin pointed out to me some time ago, this is only a rough approximation and the average value for the correcting factor is a compiex function of “6 which can readily be calculated. EK. RuTHERFORD. a ee LONDON, KDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. M IR lo i (SIXTH SERIES.) Se ARCH 1921, XXIX. The Intensity of Reflexion of X-Rays by Rock-Salt. By W. Lawrence Braae, M.A., Langworthy Professor of Physics, The University of Wahiehesten R. W. James, M.A., Senior Lecturer in Physics, The University of Manchester ; and ©. H. Bosanquet, Balliol College, Oxford *. Introduction. lly ape comparisons of the intensity of reflexion oe of X-rays by crystal-faces were first made by W. H. Bragg {, who measured by the ionization method the energy of the X-rays reflected by various faces of rock-salt. He showed that if X-rays of definite wave-length are reflected at a glancing-angle 0 by a face with one or more even crystallographic indices (e. g. (100) (110)), the in- tensity of the reflected beam can be eae ey approximately by the formula A sin? @ eet —Bsin2@ — e 5 where A and B are constants and Ig measures the intensity of the reflected beam of rays. In the case of even orders of reflexion from faces with odd indices, the intensities of reflexion may be expressed by the same formula. When the observed intensities for various faces and orders are plotted * Communicated by the Authors. + W. H. Bragg, Phil. Mag. vol. xxvii. p. 881 (May 1914). Phil. Mag. 8. 6. Vol. 41. No. 243. March 1921. Y 310 Prof. W.L. Bragg and Messrs. James and Bosanquet: against sin 8, the points lie on a smooth curve, showing that the intensity is a function of @ and does not otherwise depend on the indices of the face or the order of the reflexion. Theoretical expressions for the intensity of the reflected beam have been deduced by Darwin * and Compton +. Any theoretical formula for the intensity of reflexion or diffrac- tion by a crystal must contain a factor whose value depends upon the number and arrangement of the electrons in the atoms. In the case of reflexion, the intensity falls off more quickly with increase of glancing-angle than the theoretical formula would indicate, and this falling off must be ac- counted for by the factor referred to above. In his paper, Compton makes use of the relative measurements of intensity made by W. H. Bragg, and assuming the theoretical formula for the intensity of reflexion to be correct, he obtains the relative value of the factor at different angles, and thence deduces certain possible arrangements for the electrons in the atom. A very interesting paper by Debye and Scherrer ft deals with the same question from a slightly different standpoint. The relative intensities of the diffraction haloes obtained by Debye’s powder method are measured photometrically. The authors come to the same conclusion as had been drawn from the reflexion measurements §, that the falling- off in the intensity with increasing angle of scattering must be partly ascribed to the variation in the factor we are considering. The experiments described in the present paper have been made with the object of extending the measurements of intensity over a larger range of glancing-angles. Further, a direct comparison has been made between the energy of an incident homogeneous beam, and its reflexion by the- crystal. The results so obtained have been compared with those given by the theoretical formula ; and it will be shown that there is strong evidence that the formula is accurate. From the observations it is possible to calculate not only the relative values of the factor, which depends on the arrange- ment of the electrons in the atom, but also its absolute value over arange of angles, so that a direct comparison may be made between the observed value and the value calculated for various models of the atomic structure. * C. G. Darwin, Phil. Mag. vol. xxvii. pp. 315 & 675 (Feb. and April 1914). + A. H. Compton, Phys. Rey. ix. p. 1 (Jan. 1917). + Phys. Zettschr. xix. pp. 474-488 (1918). § Cp. W.H. Bragg, Trans. Roy. Soc. A, ccxv. pp. 253-274 (1915). The Intensity of Refleaion of X-Rays by Rock-Salt. 311 Comparison of the Intensity of Reflexion by diferent Faces. 2. The method employed is fundamentally the same as that described by W. H. Bragg. Homogeneous rays are emitted from the focal spot on the tar get, and are limited to a narrow beam by a slit termed the bulb-shit. The beam falls on the crystal, and the reflected beam is received by an ionization-chamber through a second slit. If the chamber- slit and the bulb-slit are equidistant from the axis of the instrument with which the crystal face coincides, the chamber-slit can be set so that it receives all rays of any particular wave-length refiected by the crystal face, although the reflexion may take place at various points on the face owing to irregularity of the crystal structure. This focussing effect has been described by W. H. Bragg and one of the authors * | The chamber may therefore be placed so as to receive all reflected homogeneous rays of any required wave-length— for instance, those corresponding to the K,-line of Rhodium. If this is done, and the crystal is slowly rotated about the axis of the spectrometer, very little effect is observed until the crystal planes come into the position where the equation for reflexion is satisfied. The ionization then rapidly rises to a maximum, and falls away again as the crystal passes beyond this position. The more perfect the crystal, the narrower the range over which reflexion takes place. No crystals are perfect ; in all cases the rays are reflected by a number of facets making a small angle with each other. As the crystal turns, these come, one aiter the other, into the . correct position fon reflexion, the reflected lcm lexan each falling on the chamber-slit in virtue of the focussing effect described above. ; The intensity of the reflexion cannot be measured by the effect observed when the crystal is set at the position which gives the most intense reflected beam, for the strength of this beam is dependent on the degree of perfection of the crystal face. It is measured by sweeping the crystal with uniform angular velocity through the entire range over which it reflects, and by observing the total ionization produced in the chamber during this process. In this way, every part of the pencil of homogeneous rays will fall at some time on a portion of crystal which reflects it, and will contribute its share to the whole effect. Experiments made with different erystals show that the intensity, measured in this way, is not dependent on the degree of perfection of the crystal, * ¢X-Rays and Crystal Structure,’ p. 31. Y 2 312 Prof. W. L. Bragg and Messrs. James and Bosanquet: that it is the same for any one face and order, however the crystal may be distorted, and is, in fact, a definite physical quantity on which theoretical calculations may be based. 3. The X-ray spectrometer is of the type devised by W. 4H. Bragg, which has been described in former papers. The ionization-chamber is filled with methyl bromide, and a potential of 320 volts is applied between the outer walls of the chamber, and the inner electrode which is connected to the electrometer. This potential is sufficient to prevent appreciable recombination of ions with the strongest ioniza- tion produced. A Lutz-Hdelmann string electrometer is used to observe the charge communicated to the electrode, its sensitivity being adjusted to about 100 divisions to a volt. A null method is employed to measure the charge. The inner rod of a small cylindrical condenser is connected to the electrometer, the outer cylinder being raised to any desired potential by a potential divider and ‘battery . When the crystal is swept thr toe the reflecting angle, a charge is communicated to the electrometer. This charge is neutral- ized by adjusting the potential of the outer cylinder of the condenser until the string in the electrometer returns to its zero. The potential applied to the condenser is then pro- portional to the total charge which has passed through the jonization-chamber. 4. It is necessary that the incident beam of rays should remain constant in intensity. A Coolidge bulb is used, in which the anticathode consists of a button of rhodium embedded in a tungsten block. A large induction-coil and Sanax break supply a current of 1:5 milliamperes at a potential of about 50,000 volts. It is possible to keep the intensity of the rays constant to within 2 or 3 per cent. and with the Coolidge tube it is also possible to repeat the conditions of the experiment on successive occasions in a satisfactory manner. Such variations in intensity as do occur are probably due to the irregular action of the break. 5. Superimposed .on the homogeneous rays, there is a general radiation of all wave-lengths which is also reflected by the crystal. In making a measurement of intensity it is. necessary to allow for this general radiation. When com- paring the intensity for two faces or orders, a preliminary survey is made in each case to enable a measuberment to be: made of the effect of the general radiation. The chamber is set at a series of angles over a range including the angle at which it receives the homogeneous beam. At each position of the chamber, the crystal is swept through the corre- sponding reflecting angle and the total ionization measured. i Lhe Intensity of Retexion of X-Rays by Rock-Salt. 313 A series of readings plotted in this way is shown in fig. 1. The readings are at first approximately constant, being due to the general radiation. As the position at which the chamber is set approaches that at which homogeneous rays Riese a ft) NS lonization in Abritrary Units > 24 25° 26 AT Chamber Angle are received, the jonization rises rapidly, remains constant again as long as the whole pencil of homogeneous rays enters the chamber, and then falls to a value approximately equal to its former steady value when the homogeneous rays are no longer received. . 6. When comparing two crystal faces, this survey is made in each case. One of the faces is then mounted in the spectrometer, the chamber set so that it receives the homo- geneous beam, and a series of readings taken by sweeping the crystal backwards and forwards. The crystal faces are interchanged, the chamber reset, and a series of readings taken for the other face. This process is repeated several times, and the means of the intensities for the faces are compared. The preliminary survey indicates what fraction of the total intensity observed must be subtracted, for each face, in order to allow for the general radiation ; and when this has been done, the ratio of the corrected readings gives the ratio of the intensity of reflexion by the two faces, A series of readings obtained in this way is given below. It is a comparison of the reflexion by the (311) face of NaCl, mounted so as to face left on the spectrometer, of the same face turned through 180° so that it faces right, and of the third-order reflexion from the face (100) mounted so as to face right. he difference between the values for (311) L and (311) R is due to inaccurate grinding of the crystal él4 Prof. W. L. Bragg and Messrs. James and Bosanquet : surface, the effect of which will be discussed later. It can be shown that, although they differ greatly, their mean represents accurately the strength of reflexion if the face were cut true. In taking the readings, the crystal was turned 5 minutes of are for every beat of a metronome, beating 100 to the minute. Comparison of (311) L, (311) R, and (300) R. Sweep of | Chamber Potentio- Mean of Face. . crystal. angle. meter Readings. readings. oy ORY Onna scale. (311) L & 50-11 20 20 50 2 (71, 73, 73, 72) (2:2 (311)R 1005-12 35 21 00 2 (57, 55, 58, 57, 56, 56) 565 (3811) L 8 50-11 20 20 50 2 (73, 74, 74,75) 740 (3800) R 17 30-20 00 38 25 3 (77, 78, 78, 78) 778 (311) L 8 50-11 20 20 50 2 (72, 71, 70, 72, 71, 70) 71:0 (3800) R 17 80-20 00 38 25 3 (78, 78, 79, 80, 80) 79:0 A survey of the three reflexions showed that the homo- geneous radiation was responsible for 76°9 per cent. of the total effect in the case of the (300) R reflexion, 33:0 per cent. for the (311) R, and 32-2 percent. for the (311) L reflexion. Since the intensity is very much greater for the (300) face than for the (311) face, different scales on the potentiometer were used. A reading of 72°2 on the second scale represents 72:2 per cent. of a tutal voltage of 15°72, the corresponding voltage for the third scale being 22°79. Taking this into account and allowing for the general radiation, one gets a ratio Mean intensity, face (311) _ 3°22 Intensity, face (300) R 13°45 In another experiment, (300) R and (3800) L were com- pared, and in this way the relative mean intensities of (311) and (300) measured. In order to have a uniform system of indicating both the order of reflexion and the face at which it is taking place, the convention of multiplying the indices of the face by the order has been adopted. Thus, by the reflexion from (622) is meant the second order of reflexion from the face (311). The crystal is not turned continuously during each reading; its setting is altered five minutes of are at each beat of a metronome by means of a series of spokes on the tangent screw. It would be preferable to turn the crystal with a uniform angular velocity, but it is unlikely that any ap- preciable crror was caused by the method used. In order = (02395, The Intensity of Reflexion of X-Rays by Rock-Salt. 315 to make certain that this was the case, the crystal was turned slightly between each reading, in order to ensure that the halting-places did not occur at exactly the same angles. 7. The faces used in this experiment were prepared by grinding, and were of sufficient area to intercept the whole of the incident beam of rays. In general, faces were prepared 3 or 4 centimetres in length and breadth. The perfection of the crystal structure may be judged from the range of angles at which reflexion takes place. In most eases the greater part of the effect was observed to take place within less than a degree of are as the crystal was turned, the faces being prepared from large blocks of rock- salt which were very little distorted. The face shou!d be cut so that it is as nearly parallel to the planes of the crystal structure as possible. If this is not the case, there will be a difference in the intensities of reflexion when the crystal is mounted facing right and left on the spectrometer table. This effect is described and explained in the paper by W. H. Bragg referred to above (Phil. Mag. loc. cit. p. 888). When the crystal face is not parallel to the planes of the structure, the incident and reflected beams do not make equal angles with the face of the crystal. If the glancing-angle of incidence is less than that of reflexion, the rays suffer less absorption in the crystal than when the reverse is the case. The smaller the angle of incidence, the greater is this effect. As an example of this effect, the following table gives a comparison of the Ist-, 2nd-, and 3rd-order reflexions from a natural face (110) of a ruby crystal (Al,O3). The crystal had the form of a six-sided prism bounded by the faces (110), but had so developed that this prism tapered towards one end. The angle between the face of the prism and the erystal planes could be measured by comparing the angles at which reflexion took place on the right-hand and left-hand sides. In this case it was 1° 49’. It will be observed what a large effect is produced by the small deviation from truth in the orientation of the face. - Comparison of (110), (220), (330) A103. COMA Miak. camiiaranscknc isch eies 110 220 330 Right-hand side ............ 100 61°0 41°5 Meft-hand'side i....2.) sn 0: 52 38°9 39°5 316 Prof. W. L. Bragg and Messrs. James and Bosanquet : The corresponding figures for the face (100) of NaCl are given below :— (100). (200). (300). Right-hand side ......... 100 213 5:08 * Left-hand side ............ 116°6 21°8 The error in the orientation of the (100) face was in this case too small to be measured with accuracy; it was less than 30 minutes of are. Since the effect of inaccurate grinding of the face is so much less for the second order than for the first order, the intensity of reflexion from (200) NaCl was taken as standard, and all other intensities compared with it. On account of the difficulty of grinding the face accurately, there was generally a difference in the intensities on the two sides. It can be proved, however, that the mean of the intensities on the two sides can be taken as the right value without making an appreciable error so long as the difference in the value does not exceed about 25 per cent. The intensities were in all cases measured on both sides and the mean taken. In most cases the difference between them was small; for the higher orders it did not exeeed D) per cent. 8. It is necessary to use faces which have been ground. The strength of the reflexion is very different, especially for small glancing-angles, when a cleavage surface is compared with one which has been ground. As an example, a com- parison is given below of “the reflexion from a very perfect cleavage face of a rock-salt crystal (A) with a similar cleavage face on a crystal (D), which was afterwards ground until a layer 1 millimetre thick had been removed. The intensity of reflexion from crystal (D) was measured with two orientations of the face. In the first, the crystal was set so that the edge, on which the knife was pressed in cleaving the crystal, was horizontal. In the second position, the edge was vertical and therefore at right angles to the horizontal beam of X-rays. Face. _ Intensity. A (100). (Cleavage face) ......... ..... 50°8 D (100) before grinding :— Kirst position...) 9.0.52 cipeastest 25-4 Second position 242s... -.5-¢ se 12°9 DT T00)yatter srinding) 27). 0 soc eo. 100 Hence D (100), after grinding, reflected eight times as well as in the second position before grinding. * The difference between right-hand and left-hand sides was less -than the error of determination. The Intensity of Reflexion of X-Rays by Rock-Salt. 317 Comparison of Ist, 2nd, and 3rd orders :— (100). (200). ~—-(300). Crystal D (after grinding) ......... - 100 184 Aol Crystal A (cleavage face) ............ 50°8 18:1 4-7 These comparisons were made for reflexion on the right- hand side only, and must be regarded as approximate. In this case, as:in the case of the error due to inaccurate grinding, the effect is much greater for the first-order reflexion. On account of this effect, intensity measure- ments of the (100) reflexion are doubtful, and this provides an additional reason for using the reflexion (200) as standard. The difference in intensity for the two positions of the D cleavage face indicates that the reason for the imperfect reflexion may be due to a distortion of a freshly-cleaved surface. A cleaved surface has a rippled appearance, the ripples being parallel to the line on which the knife-edge was pressed in cleaving the crystal. The range of angles, however, over which the crystal reflects is no greater for a cleavage surface than for a ground surface, so that it would not appear that the imperfect reflexion is due to small variations in orientation of the face. Measurements made at various stages in the grinding down of a crystal face indicate that the effect is deep seated, and a depth of a millimetre at least was removed from the face D (100) before it was used in obtaining the results given in this paper. 9. Certain precautions must be taken in order to ensure an accurate result. The crystal must be swept through a range of angles sufficiently great to ensure that all the facets add their share to the total effect. A range of three degrees is generally used. To check whether this was sufhcient, a larger range was used in certain cases. This increased the amount of general radiation, but when this was sub- tracted, the intensity due to the homogeneous rays was found to be the same as when the smaller range was used. The crystal planes must be parallel to the axis of the instrument. The crystal is mounted on a table which can be rocked about a horizontal axis parallel to the faces, and the intensity of a high-order spectrum is measured for various tilts of the face. The crystal is fixed at that setting which gives a maximum effect. Unless this precaution is taken, the beam on reflexion may be thrown upwards or downwards, and not be completely received by the ionization- chamber. 318 Prof.W.L. Bragg and Messrs. James and Bosanquet : A simple calculation shows what the width of the slit of the ionization-chamber must be in order that all the homo- geneous rays may enter it. It is advisable to limit this as much as possible, as the amount of general radiation is directly proportional to its width. The breadth of the homogeneous beam may be found by a survey with a very fine chamber-slit set at a series of angles. For the higher orders, the chamber-slit must be wider in order to include the two components of the K, doublet. 10. The comparisons which have been made are tabulated below, and the results are plotted in the form of a graph inate. 2. Plane. Intensity. VA Intensity. Cosec @. = x 10°. a Ss x 103, 100 100 10 9:21 612 24:80 200 19-90 4-46 4-60 122 11:05 300 4:87 2-21 3:07 29:8 5-49 400 0:79 0:89 2:30 4:85 2:20 500 0-116 0:34 1-84 0-71 0:84 110 50-4 710 6:50 310 17-60 920 6-10 2-47 3-25 37:3 6-12 330 0-71 0:84 217 4:35 2-08 111 9:00 3-00 10-62 551 745 992 33:1 575 531 202 14-25 333 0:58 0:76 3.54 3:55 1:89 444 2-82 1:68 2:65 17-2 416 55D 0:137 0:37 212 0:84 0:92 311 1-17 1-09 5:56 7-22 2:70 622 2-69 1-64 2-78 16-40 4-06 331 0:81 0-90 4-93 4-95 2:23 All. 061 0°78 3°54 3°74 1:93 711 0°302 0:55 9:58 1:87 1:37 The figures in the column headed “ Intensity” were initially expressed in terms of the (100) reflexion, which was put equal to 100. Since it»was discovered later that the (200) reflexion was a more reliable standard of intensity, all the other intensities have been determined relatively to it. In order to facilitate comparison with figures given by other authors, its value has been fixed at 19°90, since the ratio 100:19°90 was the most reliable value for the ratio Intensity The Intensity of Reflexion of X-Rays by Rock-Salt. 319 (100) : (200). The values for (100) and (111) are, how- ever, difficult to measure accurately, since the nature of the crystal face has so great an effect on the intensity, and the figures given here must be regarded as approximate. Fig, 2. % s ae oe et 1SBEeY ay Pace Cosec 8 For the other faces, it is believed that the average error does not exceed 2 per cent., except in the case of the very small intensities of high order, The smallest intensity measured is that denoted by (500), and is little more than one erat of the (100) reflexion. Wt Lf a, 9° EL xi (o>) = Ss re) Cp fay) > fo Oo 320 Prof. W. L. Bragg and Messrs. James and Bosanquet : The figures obtained by W. H. Bragg are given below for the sake of comparison :— Plane. Intensity. TOO. eye ees 100 DOO Fees Ike J0Q eee 6°25 LO: Fas eee 41-0 ; pend A ce 705 PARA SORE Ne 24:4 444 od... 4-20 11. In fig. 2 the square-root of the relative intensity has been plotted against the cosecant of the glancing-angle. By plotting the intensities in this way, the approximate relation found by W. H. Bragg—that the intensity varies inversely as sin? @—is made evident. All the points lie on two smooth curves, showing that they form two groups within each of which the intensity is a function of the glancing-angle alone. For instance, the reflexions from the faces (511) and (333) occur at the same angle, and the corresponding intensities 0°74 and 072 are denna within the error of observation. The points which he on the lower curve are those for which all the indices are odd—the faces (111), (311), (831), (333), (511), (711), (555). These reflexions are from planes which contain alternately sodium and chlorine atoms. The wave-train reflected from the planes containing sodium atoms is 180° out of phase with that reflected by the plane containing chlorine atoms. ‘The other reflexions are either trom planes which contain both sodium and chlorine atoms and are identical in their nature, or are reflexions of an even order from planes containing sodium and chlorine atoms alternately. -In both cases the sodium and chlorine atoms reflect wave-trains which are in phase with each other. Since the square-root of the intensity has been plotted, this may be taken as being proportional to the amplitude of the reflected wave-trains. The upper curve, therefore, represents the sum of the amplitudes due to sodium and chlorine atoms, the lower, the difference of these two amplitudes. Comparison of Incident and Reflected Beams. 12. The rays from the bulb consist of heterogeneous radiation of all. wave-lengths over a certain range, super- imposed on the homogeneous radiation whose intensity of The Intensity of Reflexion of X-Rays by Rock-Salt. 321 reflexion has been measured. In order to compare directly the energy in the incident and reflected homogeneous beams, it is necessary to obtain a homogeneous beam by reflexion from a crystal face, and observe the total amount of radiation reflected by a second crystal, turning with constant angular velocity, on which this homogeneous beam is allowed to fall. Fig. 3 shows the arrangement of the apparatus to effect this. The rays from the anticathode were reflected by the Fig. 3. e erystal C, so as to pass after reflexion through the collimator slits of the spectrometer. The incident beam was not limited = by slits, but by being forced to pass on reflexion around the lead wedge W%, the edge of which was pressed against the crystal face. The position of the bulb and the orientation of the crystal face were adjusted until the reflected beam passed truly through the axis of the spectrometer. This beam fell on the second erystal C., which was rotated with uniform velocity , and the total amount of radiation * Cp. Seeman, Phys. Zeit. xv. p. 795 (1914). 322 Prof. W. L. Bragg and Messrs. James and Bosanquet : reflected was measured in the ordinary way. In this case, since the rays are homogeneous, no allowance for general radiation need be made. The amount of energy reflected is proportional to the intensity of the incident beam, and inverscly proportional to the angular velocity of rotation. The quantity is therefore a constant characteristic of any one face and order, where EK = Total amount of energy reflected when the crystal is rotated with angular velocity w radians per second. I = Total amount of energy passing into ionization- chamber when the incident beam enters it for one second. ee Ko _. as This constant a will be defined as the Reflecting Power ” of the crystal face for the wave-length i. The chamber was placed so as to receive the whole of the incident beam (2.e. that reflected from C,), and the effect measured when the rays entered the chamber for a known time. It was then turned so as to receive the beam reflected from C, when the second crystal was mounted on the spectro- meter table and turned with a known angular velocity. This was done for the face (100) of NaCl. The reflexions from other faces are so much weaker that it was not con- venient to compare them directly with the incident beam. As a check, an absolute measurement was made of the reflecting power for (222), which was found to be in agree- ment with that calculated by a comparisen with (100). A series of measurements gave for the constant H® 9-000612 for NaCl (100)*. Since the reflecting powers of the other faces have been determined in terms of that from the face (100), their absolute reflecting powers may now be calculated. These values are given in the fifth column of Table I. . (The reflecting power of a face can only be defined satis- factorily in this way. It may be of interest, however, to give approximately the proportion of homogeneous radiation reflected when the crystal face is set at the most favourable * More recent determinations of this value have shown that the ficure 0:000612 is too high. The value varies somewhat with the nature of the erystal face, and a better mean value is ‘00055. The Intensity of Reflexion of X-Rays by Rock-Salt. 323 angle, although this depends on the state of perfection of the face as explained above. A direct comparison showed that when a narrow beam of X-rays falls on the face (100) set so as to reflect it, the intensity of the reflected beam is about one twenty-fifth of the incident beam. | Theoretical Formula for the Intensity of Reflexion. 13. Formule for the intensity of reflexion have been deduced by Darwin and Compton (loc. ct.). The formula given by Compton is directly applicable, for he calculates the total amount of energy reflected when the crystal is turned at a uniform rate through the reflecting - angle. That given by Darwin may be extended to this case, and is in agreement with Compton’s formula. These formule are based on the amount of radiation scattered by a free electron when set in oscillation by rays of given intensity. It has been shown by J. J. Thomson * that the amount of energy 5 radiated per second by a single electron is given by Ss P where P is the energy of the incident radiation falling on 1 sq. cm. per second, e and m are the charge and mass of the electron respectively, and ¢ is the velocity of light. This expression is confirmed by Barkla’s work{ on the total amount of radiation scattered by elements of low atomic weight, from which he deduced that the number of electrons in the atoms of these elements is approximately equal to one-half the atomic weight. If the incident radia- tion is plane polarized, the relation between the amplitude of the electric vector of the incident radiation, and that of the radiation scattered in any direction perpendicular to the direction of the electric vector, is given by BCs, AW me? sR: where R is the distance from the electron. To simplify matters, we will consider that the radiation reflected from the crystal face is plane polarized in such a manner that the electric vector is perpendicular to the plane of incidence, and allow for the “polarization factor’’ at a later stage of the calculation. e* Sir ~ 3 mc? * J.J. Thomson, ‘ Conduction of Electricity through Gases,’ p. 321. 1 C. G. Barkla, Phil. Mag. vii. p. 543 (1904), and xxi. p. 648 (1911). 324 Prof. W. L. Bragg and Messrs. James and Bosanquet : The following is a brief summary of the calculation, treated in a slightly different manner from that in Darwin’s and Compton’s papers, where it is worked out more com- pletely. | If an atom contains Z electrons, and the waves scattered by these electrons are in phase, the amplitude of the scat- tered wave will be owes e” A'=5-4Z 9. - . 4) If the spatial distribution of the electrons is such that the scattered waves are not in phase, the factor Z must be replaced by a function F, which depends on the angle of scattering and the positions of the electrons. F tends to its maximum value Z at small angles of scattering. Let rays from a source § fall at a glancing-angle @ ona plane containing n atoms per unit area, and be reflected. The amplitude at any point P is equal to one-half the total effect due to the scattering by the atoms lying in the first Fresnel zone around the corresponding point of incidence I. The area of the zone is equa! to ao versie sin 6 0 7447,” where 7;=SI, 7%=I1P. The number of atoms it contains is therefore NTN Ty" sin 6° 7,479” and the amplitude at P is equal to 2 12nmr nro : none eine yee e” 2° asin 0° 7, +72 sinO@ 7,47, 1 me? If 7, is great compared with 7,, so that the incident rays may be considered as a parallel beam, we get the relation Amplitude of reflected beam ny Ny Th ica Amplitude of incident beam ~ Dc sind.) tee (2) Considering now a thin slip of crystal consisting of p planes at a distance d apart, the reflexion will be most intense when mr = 2d sin @. C—O ee The Intensity of Reflexion of X-Rays by Rock-Salt. 325 At a glancing-angle (@+e) the amplitude of the reflected beam will be | 2d cos 0 sin(pm.e. — ae, et neP 2d cos @ p-7.€.— + 2d cos 0 x 3 Ii we put Oi — DiGi". eye aay A: and the energy of the reflected beam is proportional to Amplitude If, now, the crystal is rotated with constant angular velocity w, the total amount of radiation reflected is pro- portional to TACO ey, (De. { ae dt -—-*7 leeiisime cy) de a IN 2 42 jee =(D") py e a ee ST Gh rn dd Since sina ai: 2 ib=T, Mes this becomes (CO) apo ep Wr? hen pr Q9deos@.w ‘sin?@° mc!’ 2wd cos 0° If N = number of atoms per unit volume, t = thickness of crystal slip, we have m= Nid, ce. The energy in both the reflected and incident beams is proportional to the square of the amplitude of the electric vector. If the energy of the incident beam falling on the Phil. Mag. S. 6. Vol. 41. No. 243. March 1921. = Z 826 Prof. W. L. Bragg and Messrs. James and Bosanquet : crystal per second is I, and it is all intercepted by the slip of crystal, the total energy reflected will be given by Mes 6 Coane We sint@ me 2ad cos. 6” or Eo N2A2¢ Bes [I = 2sin? @ cos 6) me In this calculation it has been assumed that the absorption of the radiation is inappreciable. As a corollary, we can calculate the reflecting power of a homogeneous fragment of crystal of volume V. The volume of the slip irradiated by a narrow pencil of rays is equal joe, 2 ese S ie | to --—, .t, where S is the area of cross-section of the pencil. sin @ From the above formula Ko N?r? NE ca t Il ~ 2sinOcos@° °m?c*’ sin @’ Now, I=SIy when I is equal to the intensity of the beam irradiating the crystal, defined as the amount of energy falling on one square centimetre per second, whence Ko Nr? 5 e == “e e ae ELy e We e . . - ee sinc m4 (4) b) This result shows that the ‘“ Reflecting power” of a homo- geneous fragment of the crystal is proportional to its volume, if the fragment be so small that absorption in it is inappreciable. We will now assume that the crystal consists of a number of such homogeneous crystalline particles, set approximately parallel to each other, but not exactly so. When the rays are reflected from the face of a crystal, the reflexion by particles below the surface is diminished by absorption. It will be assumed that the linear coefficient of absorption pw isa constant. (This assumption will be discussed more fully below.) Rays reflected by a particle-at a depth z beneath the erystal surface suffer absorption by passing through a distance = 6 of the crystal. They are therefore reduced in intensity in the ratio ‘ _ 22 [ase asim), The Intensity of Reflexion of X-Rays by Rock-Salt. 327 By equation (3) the reflecting power of a thickness ¢ of the crystal is given by the formula Bom Ne eek, 2 sim 6). COS Oe inc The total reflecting power of the crystal face is therefore equal to : 4 N7\3 Cray aa wale : NEN an e ne dz 2 sin? 6 cos 6 mc A N?A3 she) MSG. eizrsems (cos. 8 inte. 2a N?2X3 ‘ et Zusin 20) me (9) Compton gives his formula in the form ae E, if Nervegey? i a3 HW 281m 20 AG” which agrees with this, since his factors. N*¢é7* have the 4 ; e same meanine as N?K? defined as above. Compton 5 mct : derives the equation by a more complete mathematical treatment, and has discussed very fully the effect of imper- - tection of the crystal and of the length of the wave-train. He arrives at the same formula, whatever assumptions are made. This expression must now be multiplied by a “ polariza- , 1+ cos? 26 TN tion factor’ and a “ Debye factor” e~Bsim?, &) The complete expression for the reflecting power R is tuerefore j meee 2) 3 4 29 Be NIA . 3 E ie tute O ,-Bsin2@ e (6) i 2m sin 2@ mc 2 ; The Debye Factor for Rock-salt. 14. Debye * gives the formula for the factor which expresses the diminution of the intensity of reflexion with rise of temperature 1n the form * P, Debye, Ann. der Phys, (4) xl. p. 49 (1914). | Li 2 328 Prof. W. L. Bragg and Messrs. James and Bosanquet : where the constant B is a function of the temperature, the wave-length A, the atomic weight, and the characteristic temperature @ of the crystal. W. H. Bragg * made a series of measurements of the effect of temperature in reducing the intensity of reflexion by rock-salt. Intensities were compared at 288° K and 643° K. The results were, within the errors of experiment, consistent with the ratios given by Debye’s formula. The latter gives different values for B according to the assump- tion or otherwise of the existence of the “ Nullpunktsenergie.” Compton (loc. cit. p. 47) gives, as the two values for B in NaCl, 4°6 and 3°6 respectively. The mean value for B at 288° C., calculated from W. H. Brage’s results, is equal to 4°12 ,and this will be assumed in the galeule rome which follow. “To assume that the effect of temperature on both curves of fig. 2 is the same, is equivalent to supposing that the average amplitude of vibration of sodium and chlorine atoms is the same. This is very probably not the case, and the authors intend to measure the effect of cooling down the crystal in order to obtain an empirical law over a wider range expressing the temperature effect. However, the factor e~ Bs does not affect very greatly any but the smallest intensities measured, and will therefore not make much difference to the conclusions to be drawn from the curves. The Linear Coefficient of Absorption “ p.” 15. The coefficient of absorption by rock-salt of the homogeneous radiation was measured in the usual way by interposing plates of rock-salt of various thickness in the path of the direct beam (fig. 3) and measuring the diminu- tion in energy of the beam. Experiments were made with plates from 0°05 cm. to 0°15 cm. thick. The linear coefficient of absorption « was found to be 10°7. In the theoretical formula it has been assumed that p is constant. Now, W. H. Bragg has shown that in the case of the diamond, when the crystal is set so as to reflect the radiation, the absorption-coefficient is abnormally large. It was theretore interesting to try whether such an effect is observable in the case of rock-salt. A slip of crystal 0:92 mm. in thickness with faces parallel to (106) was set on the spectrometer table at right angles to the homogeneous beam from Cj; in fig. 3 and the. absorption measured. It was then turned through an angle of about 66° until the (100) * W.H. Bragg, Phil. Mag. Joe. cet. p. 897. The Intensity of Reflexion of X-Rays by Rock-Salt. 329 planes at right angles to the crystal face reflected the radiation, the reflexion being observed in the usual way. On redetermining the absorption- coefficient it was found to have increased by about 15 per cent. This effect is diseussed in Darwin’s paper referred to above, and will reduce the intensity of the reflected beam. Its effect will be smaller for higher orders of reflexion, since the increase in absorption is due to multiple reflexion within the crystal interfering with the primary beam, and reflexion is so much weaker in the higher orders. The effect will not be taken into account in the caleu- lations, since it is not obvious what allowance should be made for it. It is to be remembered, however, that the reflexions from ee and (110) must be diminished by the increase in the absorption-coeflicient. It may possibly be the case that the effect of grinding a cleavage face, which increases so greatly the intensity of reflexion, is due to the fact that grinding breaks the crystal up into a number of small homogeneous crystals oriented in slightly different directions, so that absorption at the reflecting angle plays a less important part in diminishing the intensity of reflexion. The Comparison of the Theoretical and Observed Results. 16. The formula for the reflecting- power of a face states that Ho) Nee ea lala Cost 20, \ me 2 posi 20 “ines. 2 Since all the quantities have been measured except I’, we can calculate the absolute value of F fora Tange of laos of 8. If the effect of the electrons in the chlorine atoms be represented by Iq, and of those in the sodium atom by F'y,, then for reflecting-powers corresponding to points on the upper curve of fig. 2 we have i oy ats Fya 3 for those corresponding to points in the lower curve F=Fq—Fya. From the formula Ko me ets “sin20 = +P sin k = int, ae ae 5) . e - 3 Woe eee 1 -+ cos? 26 where N is the number of molecules of NaCl in unit volume of the-crystal. Value of F 330 Prof. W. L. Bragg and Messrs. James and Bosanquet : Taking 2 a= $30) se UES p= 1077, | ies : es -10 i FA ix QLOr 2, Nasa ¢ =3x 10”, N=0°615 x 1078, B= 4:12, this reduces to \ eo 9) — 9143 He) / sim 200) saieen c y 1 9¢ SE e rh cos. co The dotted curves in fig. 2 represent one-half the sum of the ordinates, and one-half the difference between the ordinates, of the upper and lower curves. From these Fig. 4. Sin @ Values of F for Sodium. The small circles indicate the observed values. dotted curves the absolute value of Fq for chlorine, and Fy, for sodium, can be caleulated directly. They are tabulated below, and the values are plotted against sin 6 in figs. 4 and 5. The Intensity of Reflexion of X-Rays by Rock-Salt. 331 Fig. 5. 20 nS ——— Value of F Values of F for Chlorine. The small circles indicate the observed values. The values of F for Chiorine and Sodium are:— Glancing-angle @, sin 0, Fo. F Na: ©) /) (5 44) (0-100) (11°67) (6:90) 7 30 01305 10°11 6°88 10 0 01736 87 6°26 12 30 02164 T72 4:98 15 0 0:2588 6°88 4:18 17 30 0:3007 614 347 20 0 0°3420 5°56 2:95 22 30 0:3827 5:00 2°41 25 0 0:42.26 » 4°50 L-91 27 30 04617 4-01 1-49 30 0 0:5000 343 0-83 The angle between the scattered and incident beams is twice the glancing- angle @, 332 Prof. W. L. Bragg and Messrs. James and Bosanquet : It will be seen at once that the values of F are of the right order of magnitude. F' should tend to a value 18 for chlorine, and 10 for sodium, as sin @ approaches zero, assuming the atoms in the crystal to be ionized. The greatest value of Fo, is 11°67, and of Fy, is 6°90, when sin @=0°10. 17. It now remains to take various models of the atom and see hcew the form of the function F calculated for these models agrees with that actually observed. It is not intended here to lay much stress on the agree- ment between the calculated and observed forms of F for all values of 0. The object of the comparison is to demonstrate that any probable arrangement of electrons gives a close agreement between theory and experiment at small glancing- angles, and therefore to prove that the formula for the intensity of reflexion is very probably the true one. The first atom model is one in which the electrons are supposed to be distributed uniformly throughout a sphere whose radius is 1702x1075 cm. in the case of chlorine, 0°67 x 1078 cm. in the case of sodium. In the second model the electrons are supposed to be arranged in a series of spherical shells. It is also assumed that, in considering the average effect of the atom, we may take the effect of the electrons in each shell to be equivalent to a uniform distribution of diffracting particles over the whole surface of the shell. The radii of these shells and the number of electrons in each are as follows :— No. of No. of Chlorine. electrons. Radius. } Sodium. electrons. Radius. Istshell ... 2 0-12 tet shollyys con ne 0:40 2nd shell ... 8 0°41 2nd shell ... 8 0:67 ord shell ... 8 1:02 The diameters of the outer shells are those calculated by one of the authors * from crystal data. In the third model the electrons are supposed to be arranged on shells of the same diameters as in the second model, but to be in oscillation about their mean positions along a line joining them to the centre of the atom with a total amplitude equal to their distance from the centre. This extreme case has been chosen to illustrate the effect of such an oscillation of the electrons on the form of the curve. * W. L. Bragg, Phil, Mag. xl. p. 169, August 1920. eg ee ee ee eee ee ee ed ee ao Lhe Intensity of Reflexion of X-Rays by Rock-Salt. 333 The values of F for chlorine and sodium calculated for these three types of atom model are plotted against sin @ in figs. 4 and 5, the curves corresponding to the first, second, and third models being numbered I., Il., and III. The measured values of F are shown for comparison as a series of small circles. Of the three models chosen, the third type is the only one which gives diffraction curves of the same general shape as those actually observed. Both of the other models yield eurves which have maxima and minima. The actual values of F fall off more slowly than do any of the theoretical eurves, indicating that the distances of the electrons from the centre of the atom have been taken to be greater than their true value, both for sodium and chlorine. The curve for sodium becomes nearly horizontal at some distance from the vertical axis, and the curve for chlorine shows a similar tendency. This must be ascribed to the fact that the reflecting power for small glancing angles is diminished by the increase of the absorption-coetiicient, and that our values for (100) and (110) are too small. We know that this effect must exist, since measurement has shown that the absorption-coefficient increases by 15 per cent. in the neighbourhood of the angle corresponding to the (100) reflexion, and the increase at the exact angle of reflexion may be far greater than this. It is difficult to allow for this effect. In fig. 4 the circles on the dotted curve have been plotted so as to give that curve a more probable form passing through a maximum at10. The actual values for sin d=0-° : and sin @=0°'13 lie well below this dotted curve. In fig. the two greatest values of F have been increased by “the same amounts as for the other curve. This increase is of the order to be expected from the variation which was found to exist in the coefficient of absorption mw, but its exact value is, of course, merely conjectural. There is another striking feature of the curves for the observed values of F. The curve tor chlorine approaches the axis more slowly than that of the sodium. This is just the reverse of what would be expected from the relative dimensions of the two atoms, since we would expect the ‘electrons to be on the whole at a greater distance from the centre in chlorine than in sodium. Here, again, the third type of atom chosen as a model gives results which agree most closely with those actually observed. Although it is necessary to check the form of the curves for by measurements on other crystals before drawing any 334 Prof. W.L. Bragg and Messrs. James and Bosanquet : definite conclusions as to the arrangement of the electrons in the atom, the results so far obtained indicate that :— (a) The formula for the amplitude of the wave scattered by each electron holds good, at any rate for small glancing- angles. The values for F when sin@=0:1 are of the order to be expected if the curves for chlorine and sodium have maxima at about 18 and 10 respectively. (6) A uniform distribution of the diffracting points throughout a sphere cannot explain the form of the curve, for the theoretical form of the F function for such a model, which would coincide with the actual curve at small glancing- angles, meets and crosses the horizontal axis, whereas the observed curve falls away far more gradually with in- creasing 0. (c) the general form of the curve makes it probable that as @ increases, the outer electrons for some reason become less and less effective in diffracting the X-rays. The result of assuming them to be in vibration is that the corresponding theoretical curve for F falls rapidly at first, and then very slowly, with increasing @. It would seem necessary to make some such assumption, in order to obtain a theoretical curve approximating in form to that actually observed. The electrons have been taken to be in radial vibration in the model, but this has only been done in order to obtain a theoretical expression in which the effect of the outer electrons falls away with increasing 6. It may be due to the electrons having a form such as that of the ring-electron assumed by Coster * in discussing diffraction by. rings of connecting “electrons in diamond. Uhaless we suppose that the outer electrons become less effective with increasing 0, we must conclude, in order to explain the very gradual falling away of both curves, that the electrons are within a sphere of diameter smaller than is probable. By analogous reasoning, Debye (Phys. Zeitschr. loc. cit. p. 10) comes to the conclusion that in diamond the electrons are within a sphere of 0-43x107* cm. radius. This must not be excluded as impossible, or even very im- probable ; but if this is assumed, the difficulty remains of . explaining why the curve for sodium approaches the hori- zontal axis more rapidly than that for chlorine, as if the former atom were the larger. A uniform distribution of electrons throughout the volume of the atom will not account for the observed curves. A * TD, Coster, Proc. Roy. Acad. Sci. Amsterdam, xxi. No. 6, Oct. 1919. The Intensity of Refleaion of X-Rays by Rock-Salt. 335 model must be taken which has a greater concentration of electrons neur the centre than that in the case of a uniform distribution. An important distinction must be made between the diffraction of X-rays by a crystal and the scattering of X-rays by an amorphous mass of material. In the formula 9 a for the intensity of reflexion, the quantity ee represents the amplitude of a polarized wave diffracted by a single atom in various directions. This amplitude must not be supposed to be necessarily the same as that which determines the amount of radiation scattered in various directions by an amorphous mass of the same atoms in a random arrangement. It is justifiable to consider the scattering by an amorphous sub- stance as the summation of the intensities due to the separate atoms. If the electrons are in vibration, as has been supposed in the third atom model, their movements will be- slow as compared with the frequency of the X-radiation. In the case of a single atom which is scattering the radia- tion, the arrangement of electrons in the atom at any one moment may be a random one, and the displacements may be so large and arbitrary that we may simply consider the scattering as due to a random arrangement of Z electrons, Z being the atomic number. If the amplitude of the 2 scattered wave is I’ = ey FE’ may be nearly equal to SL for all angles of scattering, except for very small angles where all the electrons are in phase and “ excess scattering ” comes into play. The factor F’ will not be the same as the factor F in the case of the atoms of a crystalline substance. When examining the reflexion from a crystal, we have a large number of atoms diffracting waves which are exactly in : Conese , phase with each other. F —, is now the amplitude scattered me by what we may term the “statistical” atom. In this case the movements of the electrons ure allowed for by supposing that diffraction takes place, not at single electron points displaced from their mean positions, but from all over a certain region for each electron in which all its possible positions lie, due weight being given to each element of the ° ° cs Ro) . 2 . region. ‘This has been done in calculating F for the third t lel. This regi so large that the effect of atom model. 11s region may be so large that the effect of the outer electrons is practically zero for the higher orders, and this illustrates the essential difference between the two The Intensity of Reflexion of X-Rays by Rock-Salt. 337 eases. may become zero for certain values of 6. FE", de- pending on a random arrangement of electrons in each a atom, will have no zero values. 8. The anode of the atom which agrees most closely with oy Ried values of F for the large values of 6 would be one of type 3, in which, however, all ‘the figures given above for the distances of the elaine from the centre of the atom must be reduced in the ratio 3:2. Assuming this type of atom, it.is possible to calculate from formula (6) the re- flecting powers for any plane of the rock-salt crystal. The figures so calculated are shown by the continuovs curves in fic. 6, and can be compared with those acthally obtained (represented by points on the dotted curve). It is to be emphasized again that the comparison is not relative, but an absolute comparison of the reflecting powers actually observed and those calculated from the omnes. Summary. The absolute values of the reflecting power for different faces of rock-salt have been measured. The reflecting power has been determined for eighteen glancing -angles. over a range between 5° 30’ and 30° 0’. The values obtained have been compared with those calculated from the theoretical formule for reflexion deduced by Darwin and Compton, and it has been shown that they afford strong confirmation of the accuracy of these for mule. The oreatest care has been taken to make the measure- ments as accurate as possible, in order that they may serve as a basis for an analysis of the arrangement of the electrons in the atom. Possible arrangements are discussed. In order to confirm the results, the effect of temperature on the intensity of reflexion must be more fully determined. The authors intend to make a series of determinations at liquid-air temperature, in order to be able to extrapolate to the values at absolute zero. It is hoped to extend the measurements over a wider range of angles at low temperatur es. It is further intended to repeat the experiments with KC in order to check the formula in this case. Sylvine affords a simpler case for investigation than rock-salt, since the ions of potassium and chlorine will, in all probability, have a very similar structure. The authors wish to acknowledge very gratefully the kind assistance given them by Dr. W. D. Coolid ge, of the General Electric Company, Schenectady, to whom they are indebted for the gift of the Coolidge tube arith we hich the inv estigations were sarried out. Teel ) os K [Lt XXX. On the Colours of Mixed Plates —Part I. By C. V. Raman, J/.A., Palit Professor of Physics, and BHABONATH Banergl, W.Sc., Assistant Professor, University of Calcutta™. ‘Plates IV. & V.] 1. INTRODUCTION. hae colours exhibited by a mixed plate or film consisting of two interspersed transparent media were investigated by Thomas Young, and were ascribed by him to the inter- ference of the coherent streams of light passing through the media and enterging from the film having suffered different retardations. Later, it was pointed out by Brewsterf{ that the colours were due to laminar diffraction ; but his treat- ment was not very complete, and apparently did not convince later writers such as Verdet and Mascart, who dealt with this ease in their treatises as one of simple interference, and ignored the part played by diffraction. More recently Profs. Charles Fabry§$ and R. W. Wood|| have attempted to give an explanation of the colours of mixed plates from the standpoint of elementary ditfraction theory. On investi- gating the subject, however, we have found that there are several features in the observed phenomena which the treatments proposed fail to explain. For instance, when a mixed plate consisting of a uniform film of liquid enclosing a large number of air-bubbles of widely varying radii is held in front of the eye, and a distant source of white light is viewed through it, we should expect, according te the theory given by Wood, that the halo seen surrounding the source should be throughout of a more or less uniform colour complementary to the tint of the regularly transmitted light, and fluctuate as a whole when the thickness of the film is varied. As will be seen in the following section, this is very far indeed from being in agreement with what is actually observed. Further, the experimental examination of the subject which we have carried out has brought to light a number of interesting features which appear hitherto to have been overlooked. It is proposed, in Part I. of this paper, to give a general description of the observed effects. * Communicated by the Authors. + Phil. Trans. Roy. Soc. 1802, p. 390, and ‘ Elements of Natural Philo- sopky,’ vols. i. & 11. + Phil. Trans. Roy. Soc. 1288, p. 73. § Journal de Physique, vill. p. 595 (1899). || Phil. Mag. April 1904, and ‘ Physical Optics,’ 2nd Edition, p. 252. {] Loe. cit. On the Colours of Mixed Plates. 339 The subsequent instalments of the paper will contain a description of further observations and experiments on the subject and the detailed discussion of the theory of the phenomena. ) 2. Tue DIrFRACTION-HALOES DUE TO A Mixep PLATE The most instructive case is that referred to above— namely, that of a mixed plate which is held normally in front of the eye,a distant point-source of light being observed through it. The plate should have a ieleinees “us eouortn as possible, and to minimise the effects of slight unavoidable variations of thickness it should be brought up very close to the eye. In order to observe the phenomenon at its best, it is useful to work witha fairly powerful source of illumi- nation, which should be completely enclosed except for a small aperture through which the light issues and falls upon the plate held at a sufficient distance from the source. The observations should be made in a darkened room so that the fainter extensions of the diffraction-halo surrounding the source can be easily seen. A tungsten-filament lamp of, 400'c.p. is a suitable source for observations in white light. For observations in monochromatic light a quartz- mercury vapour lamp with green ray filter is the most convenient source to use, though good results may also be obtained with a monochromator illuminated by the electric arc, or with a bead of salt held on a platitium wire in the hottest part of the flame of a Meker burner. A mixed plate of uniform thickness may easily be obtained by spreading a few drops of saliva or of egg-albumen between two plates of glass 5 in. X 24 in. in size and 4 in. thick, and working up the material img a linn of mania consistency by circular sliding movements of the plates over each other. Examined under the microscope, a film of this description shows a thin layer of liquid enclosing a large number of air- bubbies widely varying in size, irregularly arranged and of shape often departing considerably from circularity (see fig. 1 in Plate IV.), but showing no bias towards elongation in any particular direction”. The air-bubbles can be distinguished from the liquid by the slightly diminished intensity of the transmitted light, and also by the presence of very minute drops of liquid on the surface of the plates enclosed within the bubbles. (These can be seen in the micro-photograph, * Cases in which the bubbles have distorted forms elongated specially in one direction will be considered below in a separate section. 340 Profs. ©. V. Raman and Bhabonath Banerji on fio g@.1lin PlateIV.) The laminar boundaries appear as black lines in transmitted light under the microscope. Using the method of observation described above and a freshly prepared plate, some remarkable effects are observed which will now be described. Observations in Monochromatic Light. Surrounding the source is seen a brilliant halo, which, if the source is sufficiently small and distant, shows especially near the centre a finely mottled or granular structure. The halo consists of asnecession of dark and bright cireular rings of which the number and position depend on the thickness of the film. With thick films fifteen or twenty rings may easily be obtained, with thin films a proportionately smaller number. ‘The most remarkable features of the ring-system are: jirstly, that the successive rings are closer together near the centre of the halo and wider apart towards the margin of the halo, where they are very broad and faint; secondly, the dark rings in the halo are all more or less perfectly black, except the two or three rings nearest the centre of the halo which are not so dark as the rest, and fluctuate in sharpness and intensity with their exact positionin the halo. Figs. 2 and3 in Plate IV. show the ‘ring-system above described for a moderately thin film. The photographs of the halo here reproduced were secured by using a short-focus wide-angle lens (Zeiss-Tessar I'/3°4 lens of focal length 5 cm.), the mixed plate being placed as close to the lens as possible. Fig. 2 is a light print showing the faint and broad outer rings, and fig. 3a deep print showing more clearly the part of the halo near the centre. When the mixed plate is moved in its own plane so as to bring a thinner part of the film in front of the eye, the rings seen in the diffraction halo move inwards, closing up at the centre so that the number visible in the field decreases and the rings appear wider apart. As each of the dark rings in the inner part of the halo contracts and moves inwards, it undergoes a periodic fluctuation of intensity, becoming alternately broad and diffuse, and then sharp and black, just as it is about to close up at the centre. Furtlter away from the centre, however, the rings merely contract and move inwards without any noticeable change in their ap- pearance. In fact, what is seen distinctly suggests that near the centre of the halo there is a second and fainter ring- system superposed on the first, with the result that in this region the dark rings vary in sharpness and intensity according as, ata given point, the two sets of rings are in or the Colours of Mixed Plates. d41 out of step. A similar closing in of the rings is observed if an electric are with monochromator is used as the source and the wave-length of the light incident on the plate is gradualiy increased. Observations in White Light. In this case the halo has a fibrous radial structure, and shows a series of coloured rings surrounding the source. The outermost ring in the halo is very bro Advan practically colourless or achromatic. The first few rings subsequent to this achromatic ring within the halo are strongly coloured even if a thick film be used, the succession of colours being not very dissimilar to that observed in passing from the achromatic centre to the coloured bands in the diffraction pattern of the Fraunhofer class due toa rectilinear slit. The colours near the centre of the halo are very weak and impure, unless the film is so thin that the total number of coloured rings in the halo is rather small. In the latter case the halo shows vivid colours, even near the centre. In all cases the source of light as seen through the mixed plates appears enfeebled, as is evidently to be expected in view of the fact that part of the incident energy appears in the scattered light forming the halo. When the obser- vations are made in monochromatic light, and the plate is moved in front of the eye so as to pass from a thick to a thinner part of the film, the intensity of the source suffers periodic fluctuations, being greatest when the innermost dark ring in the halo is just about to close in at the centre and least when one of the bright rings just surrounds the source. When similar observations are made in white light. the source appears to fluctuate both in intensity and colour as the plate is moved so as to alter the eee of the film in front of the eye. The part of the halo immediately sur- rounding the source and the source itself appear to be of complementary colours. 3. HALOES DUE TO OBLIQUELY-HELD PLATES. Tf the plate instead of being held normally to the light is eradually tilted, some very striking results are eed especially when the observat ons are Fores in monochromatic light. Asthe obliquity of the plate to the incident light is in- creased, new rings appear at the centre of the halo and move ards : “clive a dark ring first appears it is very clear and well-defined, but in moving ean it becomes faint and broad, aud then again sharper and blacker, and so on. These Phil. Mag. 5. 6. Vol. 41. No. 243. Coe TOD DORA 342 Profs. ©. V. Raman and Bhabonath Banerji on fluctuations, however, occur only when the dark ring is near ie centre, and become imperceptible when it has moved to the position of the third or fourth:ring from the centre, beyond which the dark rings are all more or less perfectly black. At the same time, the form of the rings undergoes alteration, becoming elliptical near the centre and of an oval form in the outer parts of the halo. With increasing obliquity these ovals assume unsy rmmetrical shapes, the curvatures at the two ends of the minor axis (which lies in the plane of incidence) being unequal. This effect is shown in fig. 7in Plate V. Indeed, on the flatter side the ovals may actually straighten out and even reverse their curvature— that is, become coneaye outwards; while on the other side, which corresponds to directions more nearly parallel to the plate, the rings remain convex cutwards. At the same time, the different parts of the rings in the halo appear very unequal in their illumination, the light becoming more or less com- pletely concentrated in the plane of incidence and the flatter side of the rings appearing brighter than the more strongly curved side (fig. 7 in Plate V.). 4, FILMS CONTAINING DistoRTED BOUNDARIES. Mixed plates freshly prepared by ee ee up saliva or white of egg between glass plates generally show, when examined under the microscope, that the air-bubbles tend to take up a circular form, or at any rate do not show a bias towards elongation in any particular direction. But if the glass plates enclosing a mixed film of white of egg and air are pressed together and continuously moved over each other in any one direction, the bubbles in the film become distorted, assuming elliptic or oval shapes of which the major axis is in a direction perpendicular to that of movement of the plates. Fig. 6 in Plate LV. is a micro-photograph of a film obtained in this manner. When the diffraction-halo due to sucha plate (normally held) is observed, it is found that the rings in the halo retain their relative position and circular shape, the only conspicuous effect of the altered form of the boundaries being to increase the luminosity of the halo in directions transverse to the preponderating direction of the boundaries. This is illustrated in fig. 5. This observation, taken along with those described in the preceding section, clearly shows that the essential feature of the halo, namely, the succession of dark and bright rings of widths increasing from the centre outwards, does not depend for its formation on the position, size, or shape of the air-bubbles in the film, the Colours of Mixed Plates. d43 and is determined only by the thickness of the film containing the bubbles. When glycerine, turpentine, fats, or oils are used for forming the mixed plates, the tendency towards elongation of the air-liquid boundaries in the film becomes excessive, in these cases occurring parallel to the direction of movement at any instant of the enclosing glass plates. With such liquids it requires some dexterity to prepare a film showing the complete circular haloes. Generally only a diametrical streak is obtained which shows different colours at different parts of its length, and turns round and round as the glass plates enclosing the film are slid over each other with a circular movement in front of the eye. 5. KFrect or PARTIAL OR CoMPLETE DRYING OF AN ALBUMEN FILM. When a mixed plate is prepared with white of egg between parallel glass plates and is allowed to stand for an hour or two, some very remarkable changes occur in the structure of the film and in the optical effects produced by it. The bubbles of air in the fiim which at first lie about indiseri- minately (tig. 1 in Plate LV.) soon draw together, coming into contact over a considerable portion of their edges (fig. 8 in Plate V.}, and this process continues gradually till the edges everywhere touch each other. (With very thin films, especially those formed between curved surfaces, this process may be indefinitely retarded.) Ultimately, the ‘edges which have joined up straighten out, and when after a day or two the film has completely dried up, it is found on examining the film under a microscope that the albumen is confined to a number of very fine ridges holding the two glass plates together, the form of these ridges being that of a number of co) irregular hexagons, pentagons, or quadrilaterals forming a network (fig. Y9 in Plate V.). The diffraction-halo seen round a distant source of white light when such a comp letely dried plate is held normally in in of the eye is entirely different in character from that due to a mixed plate freshly prepared ; in fact, the relative position of the achromatic and coloured portions ame the spacing of the ringsare completely reversed with the dried film. The halo in this case is much fainter. Itshows a broad central area which is achromatic, followed outside by rings of gradually decreasing width which are strongly coloured. The radial fibrous structure of the halo is exceedingly well marked (fig. 11 in Plate V.). Iie tact, the di ffraction-halo seen round the source when Be 344 Profs. C. V. Raman and Bhabonath Banerji on viewed through the completely dry film is remarkably similar. in appearance to the well-known diffusion-rings observed around the focus of a thick concave mirror with a dusted surface. The resemblance extends also to the case in which the plate is held obliquely in front of the eye. As the plate is gradually tilted, the halo runs out on one side, fresh fringes appearing on that side of the source ; and ultimately, when the plate is held at a moderate obliquity, the halo consists (at least in its brightest part) of a system of circular arcs which are unequally spaced, the are which passes through the source being achromatic and those on either side of 1t being strongly coloured in white light (see fig. 12 in Plate V.). [The oblique streak seen in the figure running across the circular arcs was due to an accidental circumstance. | In fact, the halo in its brightest part is very similar to the diffusion rings due to a dusted mirror when the latter is tilted. It should be mentioned, however, that in the outer fainter parts of the halo (which do not appear in the photo- graph reproduced) certain more complex effects are observed. The detailed description of these may be deferred for the present. When the film is only partially dry (as, for instance, in fig. 8in Plate V.), both sets of rings, that is those due to a freshly prepared film and those shown by a completely dry film, appear simultaneously, and owing to their superposition, the phenomena appear somewhat confused, especially near the centre of the halo. The effect of partial drying of the film is also clearly noticeable in the ditfraction-halo due to an obliquely held plate. The elliptic or oval rings appear traversed by a system of circular ares running transverse to the plane of incidence. These are seen (some- what faintly) in the photograph reproduced in fig. 10 ia PlateV. They become more and more prominentas the drying of the film progresses, and ultimately remain alone in the field (fig. 12, Plate V.) when the elliptic rings have disappeared. 6. Non-Unirorm KIums. Standing in close relation with the phenomena described above due to mixed plates of uniform thickness, are the effects. observed with mixed plates of variable thickness. The differ- ence between the two cases is principally as regards the method of observation. In the former case the film is placed close to the eye, and the diffraction-halo surrounding a distant source is observed. With mixed films of variable thickness, on the other hand, the most suitable method of the Colours of Mixed Plates. 345 observation is to focus the eye on the film itself, the latter being held at a suitable distance from the observer. As in the former case, it is necessary, if the effects are to be studied critically, to use a light-source of small dimensions and to place the film at a sufficient distance from it. A film of variable thickness may be readily formed between the surfaces of two lenses similar to those used for observation of Newton’s Rings. The thickness of the film at its centre need not necessarily be zero; and, in fact, if egg-albumen is used, there is considerable difficulty felt in forcing the lenses into actual contact at the centre. We shall confine our attention here to the phenomena observéd with /reshly pre- pared films. Observations in W hite Light : Normal Incidence. When the mixed plate is held ata sufficient distance in the line of sight between the eye and a distant source of light, vividly coloured rings localized on the film are seen (pro- vided its thickness is not too great), these rings being the lines of equal thickness on the film. In this case the light reaching the eye is that diffracted through small angles by the air-liquid boundaries which the film contains; and, indeed, itis these boundaries which appear luminous to the eye and not the whole continuous film*. The source of light itself appears coloured, and is of a complementary tint to the part of the film through which it is seen. When the eye is moved a little out of the direct jine between the plate and source of light, some remarkable changes occur in the appearance of the film. The colours become feeble and impure, and on moving the eye further out of the direct line, the colours reappear again vividly, the rings having simultaneously expanded and moved outwards. With still further movement of the eye the phenomenon repeats itself, but with much less marked fluctuations in the vividness of the colours ; and if the film were initially so thick as to show a coloured centre, fresh rings also appear and move outwards from the centre, until finally an achromatic centre develops and expands so as to cover the whole area of the film when viewed sufliciently obliquely. When the source of light is fairly powerful (as, for instance, when the film is held normally in the track of a * Fig. 4 in Plate IV. is a photograph (much enlarged) of these luminous boundaries in a dark field as observed by the method of the ‘* Foucalt test.” Each of the laminar boundaries appears as a pair of brilliantly coloured lines running parallel to each other and separated by a perfectly black line coinciding with the exact outline of the boundary. The theory of this effect will be more fully considered in Part IL. of this paper. 346 On the Colours of Miaed Plates. parallel beam of light from an optical lantern), this expansion of the rings with increasing obliquity of observation may be followed up till the direction in which the film is seen makes an angle up to 90° with the direction of the incident light. It isa noteworthy fact, that a film too thick to show colours when observed nearly in the direction of the transmitted light will show the coloured rings vividly when viewed at a moderate obliquity. Ultimately all films, whether thick or thin, appear practically achromatic in a sufficiently oblique direction. The thinner the film, the smaller the angle of diffraction necessary for this. The thinnest films, which may be obtained by forcing the glasses together till they nearly come into contact, scatter much less light than the thicker films. Hence the film which is achromatic when viewed obliquely shows a darker area near its centre. Observations in Monochromatic Light: Normal Ineidence. In this case, if the mixed plate is held directly in the line of sight between the eye and the source of light at a suffi- ciently great distance from both of them, a series of perfectly black rings (alternating with bright rings) may be seen on the film, even if this be fairly thick. Bringing the film nearer the source of light or increasing the dimensions of the latter has a very deleterious effect on the perfect blackness. and sharpness of the rings. Placing the eye a little out of the direct line also results in the rings becoming blurred in appearance; and, indeed, on merely bringing the film nearer the observer in the line of sight so as to increase the angle it subtends at the eye, the rings may appear broken up and blurred in parts. (The effect is the more striking the thicker the film and the larger its area.) Ifa particular dark ring be watched as the eye is gradually moved out of the line through film and source, it. will: be noticed that the ring becomes blurred and broadened, then again sharp and per- fectly black, but with its position shifted outwards in the film; this process further repeats itself, but with rapidly diminishing fluctuations in the intensity and sharpness of the ring. Viewed at a moderately large obliquity, the dark rings are always perfectly black, and their sharpness is much less dependent upon the use of a light-source of restricted area. The rings expand with increasing obliquity of obser- vation and move out of the film, until finally no rings are visible at all. A darker patch at the centre, where the film is thinnest, can be seen, as in the ease of white light. The Photo-Electric Theory of Vision. B47 Obliquely Incident Light—When the film is tilted rela- tively to the direction of the incident rays, the rings seen on the film contractand move inwards. As in the case of normal incidence, perfectly black rings may be seen on thie film, if it be viewed in HOnOCH Lonnie light very nearly in the direction of the transmitted rays, and the appearance of these rings alters with the obliquity of observation in much the same way. It should be remarked, however, that in the present case the effects observed vary not only with the angle between the transmitted pencil and the direction of observation, but also with the particular planein which the latter direction lies. The maximom permissible angle of observation varies with this plane. In the plane of incidence it is 7/2—a and 7/2 +. respectively on the two sides of the transmitted pencil, where « is the ang'e of incidence. If e is considerable, the rings continue to he yvistble whem the alm is viewed nearly along the surface of the plate on one side of the transmitted pencil, while on the other side the rings move out and ep from the film ata moderate obliquity, so that in white light the film appears achromatic over a wide range of angles of observation. Calcutta, India, 18th September, 1920. XXXII. The Photo-Electric Theory of Vision. By J. H. J. Pootn, M.A.* IN his, address: to the British “Association in 1919, Sir Oliver Lodge suggested that the light sensation in the eye might be caused by the action of photo-electrons excited by the incident light. He thought that, if expe- riments were carried out on the various substances present in the retina, some of them might be found to exhibit photo- electric properties when exposed to visual light. The same idea had occurred tu Dr. Joly some years previously, and in 1915-16 some experiments were actually carried out in this laboratory on thie possible photo- electric properties of the black pigment No evidence of the black pigment possessing any. photo-electric powers, when exposed to visual light, could however be obtained, aid as thus no direct experimental evidence in favour of the theory could be given, Dr. Joly refrained from publishing this view of the mechanism of * Communicated by Prof. J. Joly, F.R.S. 348 Mr. J. H. J. Poole on the vision. When, however, Sir Oliver Lodge independently proposed the same theory, it was thought worth while repeating the experiments more fully, and seelng if any definite evidence in favour of the theory could be obtained. It seems highly probable, it might almost be said certain, that the effect of light on the eye must ultimately be due toa photo-electric action of some kind ; but the real ; oint of issue is whether it can be shown that the photo-electron excited by the light ever gets free of its parent molecule as it does aie a metal is illuminated with ul.ra-violet light, or whether the action 1s less violent and may only result in disturbing the equilibrium of the electron inside the molecule, and hence altering the chemical properties of some light-sensitive compound in the eye. The latter view would favour such a theory of vision as the Young-Helmholtz theory, which postulates the existence of three definite and independent colour-sensations in the eye, which might be caused by three different light-sensitive compounds ; butif it could be shown that the former view were correct, a very much more simple and direct explanation of colour-vision can be given, at least from the physical point of view. ‘This explanation rests on the fact that it has been shown that the velocity of emission of a photo-electron is a simple function of the wave- length of the incident light—in fact, that the energy of the electron is given by the equation EK = hy—p, where fh is Planck’s constant, v is the frequency of the light, and p is the work donein getting the electron free from the molecule and will thus be a constant for any particular material. Thus for every different colour in the spectrum an electron of a definite velocity would be emitted, and it is only necessary to suppose that the manner in which the cones are excited depends on the velocity of these electrons, to obtain a possible view of the mechanism of colour-vision. It might perhaps be urged against such a theory as this, that on this view the eye ought always to be able to distinguish between a pure single frequency light and an impure mixture of two different frequency lights, which, as is well known, the eye is in certain cases incapable of doing : Cx Jos lit cannot dis- tinguish between spectrum yellow and a suitable mixture of green and red. However, we could meet this objection by supposing that the resolvi ing power of the cones for photo- electrons of various velocities was not perfect, and that they were only capable of separating the electrons into three main groups, which would correspond to the three primary sen- sations of red, green, and blue: Thus in the case of spectrum yellow, which lies between red and green, the resultant Photo-Electric Theory of Vision. 349 electrons weuld excite both the red and green sensations and hence such a colour would be visually indistinguishabl le from a mixture of red and ener. In fact, if we adopt this view of the resolving power of the cones ‘for photo-electrons of different velocities, we could explain all the facts which agree with the Young-Helmholtz theory vf three primary colour sensations. Before making any direct tests of the possible photo- electric powers of the retina, some preliminary indirect trials were made. It seems probable that the formation of the latent image in a photographic plate is really based on a photo-electric action due to the silver halides which are found to be vigorously photo- electric. Sensitisers are also photo- electric, their activity being dependent on the absorption of light of some particular colour. It is, therefore, perhaps permissible to infer that if either the black pigment or rhodopsin were really photo-electric to visual light, they would also be capable of acting as sensitisers for a photo-_ graphic plate, if treated in a suitable manner. Hxperiments on this point were accordingly made by exposing a plate which was insensitive to the red end of the spectrum to intense red light. One half of the plate was treated with the preparation from the eye, which it was desired to test. The untreated portion acted as a_ standard for com- parison purposes, and by putting a slight scratch with a penknife on the treated half it could always be sub- sequently identified. An ordinary carbon glow-lamp was used as a source of light. Before falling on the plate, the light passed through a picric-acid screen, ordinary photo- oraphic ruby glass, and also a special red screen, as 1t was found that the ‘ruby 2 class allowed quite a large proportion of the blue end of the spectrum to pass through. The first material to be dealt with was the black pigment. Various methods of applying it to the plate were tried, but in no case was detinite positive evidence of any sen- sitizing action obtained. In the first attempts the black pigment was simply removed from a freshiy dissected bull’s eye, and placed directly in contact with the sensitive film. The plate was exposed through the glass, as owing to the opacity of the black pigment no effect could be expected if the black pigment coating was between the film and the incident light. After exposure, the pigment was ca irefully washed off and the plate given a thorough rinsing with water to remove as far as possible any trace of grease. As already stated however, no sign of any sensitising action could be discovered on development, ‘and in fact 350 Mr. J. A. J. Poole oi the sometimes the reverse effect was observed, when a suffi- ciently long exposure to slightly fog the plate was tried. This slight fogging of the plate with long exposures was probably due to «a very small amount of actinic light getting through the screens, and the retarding action of the pigment observed may be caused by its slightly greasy nature impeding development. Other attempts were also. made by treating the pigment with various solvents, such us water, alcohol, chloroform, ete., and treating a plate with the resuitant fine suspension of the black pigment, as it did not appear to be soluble in any of the solvents used. No result was obtained. Another method adopted was to reduce the pigment to a very fine powder ly de- hydration and subsequent grinding in an w#gate mortar, the resulting powder being then dusted on to the plate; but again no sensitising effect was shown. The difficulty of obtaining good enough contact between the pigment wid’) the ‘sensitive: alm (Or the plate would probably si sufficient to account for the negative result in all thes cases, as it has been found that tor ordinary sensitising « ee to be effective, the dye must actually dye the silver-halide grain, and not merely stain the plate. It is more difficult to deal with the visual purple or rhodopsin in the retina, both on account of the much smaller quantity available, and also because the rhodopsin is only present in dark-adapted eyes. For this reason the animal from which the eye is to be obtained must be kept in the dark for some time before being killed, to allow the rhodopsin to form in the retina, and all sub- sequent dissections have to be carried out in the dark or in a very subdued red light, as the rhodopsin quickly fades when exposed to light. Professor Pringle was, however, kind enough to supply us with a solution of rhodopsin in bile salts which had been prepared from dark-adapted frogs’ eyes. This solution was tested on the plate in the usual manner. Again no sensitising action was detected : in fact, with very lone exposures which slightly fegged the plate, a small retarding effect was noticed. This eftect. was found to be due to the bile salts,x—a solution of them with no rhodopsin having the same slightly retarding effect. on the plate. It is unfortunate that bile salts have this. action on the plate, as it would completely mask any slight. sensitising effect which the very small amount of rhodopsin present might produce, and it is apparently not easy to. extract the rhodopsin from the retina by other means. These sensitising experiments thus lead to no direct Photo-Electric Theory of Vision. dol evidence in favour of the photo-electric theory of vision, but, on the other hand, it certainly would not be justifiable to conclude from them that the materials tested have no photo-electric properties, since the difliculties of getting intimate contact with the sensitive film are, as already mentioned, probably quite sufficient to account for the nul results obtained. After the failure of these experiments to show any positive effects, a direct method of testing the black pigment ete. was used. The usual plan of doing this was employed. The material to be tested is placed on an insulated conducting plate, and the rate of leak of negative electricity from the plate when it is illuminated measured with an electrometer. To facilitate , : Fig. 1. i | Lamp & Screerr | Peed AUC iene er zee, ieee [2 rs see LC | at et ee ] Bis BoTTery Llecitomeser L£ov7,4 it ||| -—— zon Bollery the escape of the negative electrons, an accelerating electric field is generally used. A sketch of the various electrical connexions is given. The Wilson Kaye Tilted Leaf Hlectro- meter was used to ineasure the electronic current. Its high sensibility to voltage changes and small capacity render it peculiarly suitable for this purpose. Fortunately it was found unnecessary to work with the instrument in its most sensitive condition, as the natural leak in the ionization- chamber limits the sensitivity at which it is desirable to work. Usually it was found that a sensitivity of about 30 or 40 scale-divisions to the volt was sufficient, though B)2 Mr. J. H. J. Poole on the on occasion sensitivities up to 130 divisions per volt were used. To maintain the charged plate of the electrometer at the requisite voltage, a battery of small Hver-Ready dry cells was employed. ‘his battery was carefully insulated from earth, the individual cells being stood en a layer of parafiin- wax, and its voltage was found to remain constant over long periods so long as no current was taken from it. ‘lo protect it in case of an accidental short-circuit due to the gold-leaf touching the charged plate, a water-resistance is inserted between it and the latter. A switch is also inserted for convenience in working with the instrument. For earthing the gold-leaf when requisite, a copper rod dipping into a strong CaCl, solution contained in the small copper cup supplied with tue instrument was used. When it was desired to isolate the gold-leaf, the copper rod could be lifted by a silk thread passing over a suitable arrangement of pulley-wheels. On the whole this key proved fairly satisfactory, but at times it was inclined to give too big an initial charge to the gold-leaf. Usually, however, after two or three attempts, a small enough displacement could be obtained. In order to conveniently measure the sensitivity of the electrometer, the copper rod of the earthing key was not directly connected to earth, but was connected to a two-way switch so that a known voltage could be applied to it by means of a volt-box. Thus the sensitivity of the electro- meter can be quickly determined at any time, which is essential as any small accidental shake of the instrument may alter it considerably, and for this reason it is necessary to test it fairly frequently. A separate sketch of the final form of testing-chamber is shown. The chamber was constructed of brass, and made in two parts so that it could be easily taken apart to insert the material under test. The joint was carefully ground with carborundum-powder and tallowed, in order to insure the vessel being air-tight when assembled. It could be exhausted through a side tubulure, which was connecied to a Geryk pump and manometer. Pressures down to about 3 mm. of mercury could be maintained in the testing- chamber. The black pigment or other material is placed on the small brass testing-plate. ‘This plate is carried on a copper rod which projects slightly from the bottom of the vessel and is connected to the electrometer by a fine wire. The wire is surrounded by an earthed shield to prevent any stray electrostatic effects due to the observer etc. affecting Photo-Electric Theory of Vision. 393 the electrometer. The copper rod itself is insulated from the main body of the chamber, which is earthed, by sealing- wax and a quartz tube as depicted. The sealing- WAX serves. to make the joint air-tight. About 3mm. above the testing- plate, a small quartz plate carried by a brass ring is fitted. The latter is supported by a small steel rod, which passes through the side of the vessel and 1s also insulated with a glass tube and sealing-wax. This plate replaces the ordinary arid of copper gauze or other form of grating usually employed to collect the negative ions escaping from the photo-electric surface. By applying a thin film of a fairly strong solution of either H.SO, or P.O; to the lower surface. of the plate, it can be made sir fticnenls: conducting for working with the high voltages and small currents met with in these experiments. The outer brass ring which supports. Zo Lilecirome/jser the plate also serves to make contact with the liquid film. To facilitate the escape of the electrons from the test plate, the upper quartz plate is conneeted through its steel sup- porting rod to the positive pole of a high- “voltage battery. At low pressures the potential gradient was high enough to largely increase the electronic current by ionization due to collision, but in experiments carried out at air-pressuie Ghis would not be the case. The use of the moistened quartz plate, instead of the usual form of wire grid, has some great advantages. The most obvious one lies in its superior transparency, but it also possesses a great virtue in that its shape is much. more constant than that of the wire erid. The importance of this fact has been pointed out by H. “H. Dixon and H. H. Poole in a paper on * Photo-Sy mthesis and the Electronic Theory ’ (Proc. Roy. Dublin Soc. vol. xvi. N.S. No. 5), in which they show that the change in capacity due to the sagging of a 354 Mr. J. H. J. Poole on the wire-egauze grid when heated by the incident light may produce apparently quite large photo-electric effects, which are really quite spurious. The quartz plate, however, will manifestly not suffer from this complaint on at least three accounts. These are: (a) The thermal expansion of quartz is practically zero; (b) Owing to its transparency it will absorb very little energy from the incident beam; and (c) Even if the plate did expand shightly it would ‘do so in its own plane and not alter the capacity of the isolated system appreciably. The chief drawback of the arrangement is that it cannot obviously be used at very low pressures owing to the evaporation of the liquid film. The test-plate was illuminated through a quartz window -at the top of the testing vessel. A quartz window was employed, and it is convenient to be able to use ultra- violet light on occasion to test whether the arrangement is functioning properly. As a source of light a "200-watt L-watt lamp was employed, and a suitable train of lenses used to concentrate the light. A cardboard screen was nineed round the lamp to protect the observer’s eyes, and the electrometer was completely shielded from the light by a large wooden screen. It was found that switching on this light had a slight effect on the electrometer, although the leads to the lamp were kept as far distant as possible. Tn consequence, a shutter was used by which the illumination could be controlled without turning the lamp off. In order to obtain some idea of the absolute current. to which a known rate of deflexion of the gold-leaf corresponds, it is necessary to measure the capacity of the gold-leaf and testing plate taken in conjunction. This capacity was deter- mined by comparing it with a second capacity whose value could be calculated approximately. The method adopted was that described by H. H. Dixon and H. H. Poole in their paper on Photo-Synthesis (loc. cit.). The capacity was found to be about 6 cims., and as the sensitivity is also known we can easily determine the value of the currents obtained. The value obtained for the capacity is probably not very accurate, but vreat accuracy 1s not requisite, as really all we require to know in these experiments is the order of magnitude of the currents dealt with. The method of procedure adopted was first to spread a thin film of the substance to be tried on a disk of pure lead- foil, which was then placed on the testing plate of the ionization chamber. The chamber was re-assembled, and the pump worked till the desired pressure had been attained. ‘The rate of leak of the electrometer with the light off and on Photo-Electric Theory of Vision. 359d was then measured in the usual way. The advantage of using lead disks to place the active material on is that it makes the manipulation of the latter easier, and also, when one preparation has been tested, the disk can be quickly removed from the testing vesse: and a second one sub- stituted. Lead is a convenient metal to use, as it is easily eut to the required shape, and is also very inert photo- electrically. The experiments carried out on the black pigment with this apparat 1s completely confirmed the nul results previously obtained with it. Various methods of preparing the pigment were tried. It was tested moist and after being dried in a desiceator, immediately after dissection and after the lapse of some time, but in no case was any photo-electric effect obtained. A est was also made on a complete retina, which was spread entire on one of the lead disks, but it also showed no photo-electric powers. All these experiments were con- ducted at a pressure of about 5mm. of mercury. There can, it seems, be very little doubt that the black pigment of the eye is not photo- electric when exposed to visual light. As regards the rhodopsin in the retina, it was thought best to test it in situ in the retina. This was done by testing a freshly prepared retina obtained from a frog’s eye. The frog, “pefore the experiment, had been kept in the dark, so that there was a certain amount of rhodopsin present in the retina. Professor Pringle kindly did the requisite dissections which were carricd out in feeble red light, the retina not being exposed to ordinary illumination at all till it was tested. As it was considered that perhaps the vacuum in the testing vessel might tend to render the retina inactive owing to its drying action, these trials were all carried out at air pressure. The result of these experiments showed that while the frog’s retina certainly possesses no permanent photo-electric effect, there is usually a very small effect for a few minutes after the retina is first placed in the testing vessel. In all eases such an effect was found, but its size was very small and irregular, varying from about 6x107° E.S.U. to ox 107° E.S.U. In view of the small size of the currents obtained, it is perhaps of interest to consider what would be the smallest number of electrons per second which the human eye could he expected to detect. If we assume that a candle is visible at night ata distance of 3000 metres, and that the effective aperture of the eye in such a case would be about 1 sq. em., we find that the minimum visible energy-How is equal to 396 The Photo-Electric Theory of Vision. about 4x 1077 erg per second, taking the energy emitted by the unit candle as. 5x 108 e ergs per second. Let us also assume that the average wave- length is about 5000 A.U. (this is probably too short, but is sufficiently accurate for this purpose), and we find that the energy of the corre- sponding photo-e'ectron on the quantum theory will be Egualsuo lpy—3-0'x 10 ye rene: If all the energy of the incident light is converted into energy of the photo-electrons, this would lead to about 10° electrons per second being the minimum number visible, and a current of 5x10-° H.S.U. being the minimum electronic current. In comparing this latter figure with the actual small current obtained from the frog’s retina, there are several considerations to be taken into account: 7. e., (a) a frog’s eye is very much smaller than a human eye; (b) the illu. mination was not the minimum, but was intensely bright. Also, in any case, we could not expect to get anything approaching the theoretically possible current owing to absorption of the electrons in the surface, loss of energy by reflexion, and also the fact that even under ideal conditions the total incident energy may not be entirely converted into that of the emitted electrons. As regards (a) and (6), the actual energy falling on the frog’s retina per second was approximately 10* ergs per second, or 2°D x 10 times that required to excite the human eye. This would give a theoretically possible current of 1°25 x 10° H.S.U., or about 2x10" times the actual current obtained. It is possible that absorption in the surface might account for the enormous discrepancy between these two figures, but certainly no other consideration could do so. In this connexion some results obtained by Elster and Geitel (Phys. Zeits. xiii. p. 468, 1912) are of interest. By the use of a very sensitive potassium cell, they were able to detect an amount of blue hght imparting 3 ‘100 vers per second per sq. em., and they found that under these conditions the energy of the emitted electrons accounted for sop part of the incident energy. As it has been found that the photo-electrie current for a substance is usually a linear function of the light intensity, it would from this appear probable that in all cases of photo-electric action a sensible proportion of the incident energy would be represented in the energy of the emitted electrons. Thus we are driven to conclude that since in the case of the retina only 5x10-" part is so represented, it cannot be considered to be photo-electric. The current obtained is active Deposits of Radium, Thorium, and Actinium. 357 so small that it may possibly be due to some extraneous effect due to heating ete. From these experiments we may therefore safely conclude that neither the black pigment nor the retina as a whole is photo-electrie to visual light. It is possible however that, as previously mentioned, ‘the nul effects obtained with the retina may be due to absorption of the electrons in the surface. It remains quite conceivable that the rhodopsin in the eye is actually photo-electric, but so immersed in ‘inactive material that. the electrons cannot escape. ‘Thus the photo-electric theory of vision, while still presenting taany attractions, cannot be said to be i in any way confirmed ‘by these experiments, but neither can it be actually refuted by them. ‘To obtain really conclusive evidence it would be necessary to isolate sufficient rhodopsin in a pure state to be able to test it directly. In conclusion I wish to express my thanks to Professor Pringle and Dr. Fearon for their kind aid in the preparation of the frogs’ retinas ; also to Dr. Joly, to whose suggestion and assistance the research is mainly due. Iveagh Geological Laboratory, “November 1920. DOXA The ieee of the Actwe Deposits vs Radium, Thorium, and Actinium in Electric Fields. y G. H. BRIGGS, BSe. ., Lecturer in Physics at the cara of Sydney * Il. Introduction. YYXHE experiments to be described in this paper were begun with the object of deciding whether, as E. M. Wellish ¢ had concluded, there is a definite limiting fraction of the recoil atoms fier radium emanation, positively eharged at the end of the recoil path, or whether: as GG. H. Henderson concluded, all the recoil atoms from thorium and radium eta nabions are positively charged the end of the recoil path. As the work progressed, i was found necessary to extend its scope. In his experiments Wellish used a cylindrical vessel with a central electrode. He obtained the parang results for the recoil atoms of radium emanation : 882 per cent. * Communicated by the Author. Tt Wellish, Phil. Mag. xxviii. p. 417 (1914). Phil, Mag. 8. 6. Vol. 41. No. 243. March 1921. 2B 358 Mr. Briggs on Distribution of Active Deposits of | initially positively charged in air and hydrogen, 78°9 in carbon dioxide, the remainder being neutral, and in ether vapour all the deposit atoms were neutral. The term “initial” refers to the instant when the recoil atom has reached the end of its recoil path, and the values quoted represent the fraction of the recoil atoms which possess a positive charge at the end of this path before either volume or columnar recombination has had a chance to become operative. The chief advantage of the type of vessel used by Wellish is that a delicate test of the presence of negative deposit atoms can be made if the central electrode is positive. In his first paper Henderson * de- scribed the results of experiments on thorium active deposit. The electrodes were parallel plates with a guard ring, and were 3 cm. apart. In the air the cathode activities at 120 and 12,000 volts were 98°6 and 99°8 per cent. As the latter value differed from 100 by less than the experimental error, Henderson concluded that all the deposit atoms from thorium emanation are initially positively charged in aur. When ether vapour was added to the air the percentage decreased, becoming zero for pure ether vapour. In two succeeding papers, “however, Henderson T described expe- riments on radium emanation, and cencluded from the results he obtained when using a new form of parallel plate exposure vessel and strong uniform electric fields, that all the deposit particles from radium emanation in air, carbon dioxide, and sulphur dioxide are positively charged. In eolumn D of Table I. are given values calculated from those of Henderson, who expressed his results as the ratio: of the cathode activity to the total activity. As the values were one increasing at the highest potential used, he concluded that there is no limit to the percentage cathode activity, and hence that initially all the deposit atoms are positively charged. Il. Experiments with Radium Hmanation in Strong Fields. Fig. 1 shows an apparatus designed by the author for use with strong fields. A and B are two circular brass disks cemented to an ebonite ring P. The inner surface of the ring was curved, as shown, to prevent sparking across. 2 , Mees this surface, and it projected beyond the brass disks to prevent brush dischar ge and sparking from their edges. * Henderson, Trans. Nova Scotian Inst. Se. xiv. p, 1 (1914-15). + Henderson, Trans. Nova Scotian Inst. Sc. xiv. p. 128 (1916); and Trans. Roy. Soe. Canada, x. p, 151 (1916). . T e / . oJ e ° . Radium, Thorium, and Actinium in Electric Fields. 359 The central portions, C and D, of the disks were removable, and could be made air-tight by running soft wax into channels formed by brass rings on either side of the junction-line. The diameter of the portion of the disks Fig. 2. Sia TW exposed on the inside of the vessel was 10°6 cm., and that of the removable disks 5°6 em. The plates were 2 cm. apart, and their internal surfaces were cleaned with fine emery paper. Radium emanation was introduced mixed with the gas under observation, or was allowed to accumulate over-night from a small quantity of radium placed inside the vessel. A Wimshurst machine was used for high potentials. The gases were dried and filtered. At the end of an exposure, which was always longer than three hours, the alpha-ray activities of the eentral disks were compared by means of a Dolezalek electrometer. Wellish * * Loe. cit. 2B 2 |G e Y 360 Mr. Briggs on Distribution of Active Deposits of showed that the anode activity was due to the diffusion of neutral deposit atoms: hence if ¢ and d are the cathode and ae ae é c—d anode activities respectively, —— ae a of the RaA atoms positively charged, if, as assumed by Wellish, the recoil of RaB from the electrodes is negligible. This assumption will be shown later to be incorrect. In c—ad ) c+d culated from the anode and cathode activities for radium emanation mixed with air. No evidence is shown of a continual increase with voltage, the limiting value being practically reached at 60 volts per cm. The mean value 89°6 is in good agreement with that found by Wellish when using a different type cf testing vessel. would give the fraction column A of Table J. are given the values of cal- TEAR IgE le: Ape CEs ane. c+d Pressure. Volts perem. A. B. C. D. 20 cm, 60 89-2 Bi ks 89°5 a £. 804 Atmospheric. 150 a as ont 89'8 a 1000 ae on ns 92°4 3 2000 oe ee ae 2-8 ¥ 4000 on fe oS 95°8 a 6000 90°7 ts 94°7 a 7000 &8°9 Bas 95°4 5 < 89-0 : 8000 89:3 + a 88:7 a 9000 ae 90°5 ey of aa 90:6 : 10,000 90°6 21-9 is 12,000 E9°4 91:3 i 14,000 91-0 918 are meh WNT pVy Radium, Thorium, and Actinium in Electric Fields. 2371 Let us consider the activity accumulating on the anode. It receives by diffusion half of the neutral A atoms, 7. e. s l—a r ah Of these, g—-4) free themselves by recoil, and es) of them are initially positively charged and are drawn to the cathode. The neutral atoms return to the anode by diffusion. Hence the total number of atoms br oy ad / left on the anode is 3(1— ) =a). The rest of the atoms resulting from the decay of the emanation reach the cathode. It is readily found that Ais 21 = br 2—br Since } is obtained by multiplying the experimental values. given in column 3 by 0°93, and r is obtained by dividing the value found for the efficiency of recoil found in air by the same number, then br and hence the above equation are independent of the experiment in which 93 per cent. was. obtained for RaB in air. The values of a calculated from this equation are given in column 5. VIL. Vhorium Emanation. To test the theory given for the distribution of the active deposits, exposures were made with thorium emanation mixed with various gases, using the vessel shown in fic. 1. The relative difference in mass of the isotopes of RaA or of RaB is so small that we may assume that for each group of these isotopic atoms the percentage positively charged, the coefficient of diffusion, and the efficiency of recoil are identical. Hence it is easily shown that po, the value of c—ad a 7 for thorium emanation, 1s given by c+¢ 2 a a+b—a)(1—s+ NO sr ak It is to be noted that p 2, owing to the terms involving s, depends on the size and the type of testing vessel used and also on the pressure of the gas. The values of p. found by experiment and ‘by means of this equation are given in columns 6and 7. It was assumed that the coefficient of dif- fusion ot the A-atoms is inversely proportional to the square “ 372 =6Mr. Briggs on Distribution of Active Deposits of root of the molecular weight of the gas through which diffu- sion takes place. The experiments with radium and thorium emanations were made at a pressure of 20 cm., the time of exposure being generally greater than 16 hours, and at. a voltage of 720, except in “the case of hydrogen sulphide, for which consistent results for either emanation or in the experiments with RaB were only obtained when voltages as high as 10,000 were used. Zinc electrodes were used for ithe g@ases ammonia, acetylene, and hydrogen sulphide. Bach result given in Table If. is the mean’of cesenal expe- iments. It was found necessary to liquify and fractionate the gases methane and ethylene to free them from the vapours of organic compounds produced during their pre- par.tion. Hydrogen sulphide was obtained by heating a solution of magnesium hydrosulphide, and was also fractionated. VILL. Actinium Active Deposit. The experiments of Kennedy ”*, Lucian +, or of McKeehan ¢ might be used to test the theory given above for the case of -actinium emanation. Lucian’s experiments are by far the most suitable for this purpose, since direct recoil on to the electrodes was negligible in his but not in the expe- riments of the other two. Lucian’s method was very similar to that used by Wellish for radium active deposit. In -section V. it has been shown that the number of neutral -atoms of AcA diffusing to the walls of a vessel with parallel plates 2 cm. apart, before transforming into AcB, is 1°8 per cent. at atmospheric pressure. The number so diffusing, in ‘the vessel used by Lucian, which was a cylinder of 4:9 em. diameter with a central rod as cathode, will be of the same order of magnitude and may be neglected. By means of a formula given by Lucian, the amount of AcA positively charged which reaches the eathode before transforming into AcB may be calculated. Taking 1:2 for the mobility -of AcA, the fraction of the AcA positively charged reaching the cathode in Lucian’s vessel, before transforming into AcB, ‘is 14 per cent. at 600 volts, 17 at 980 volts, and 20 at 1700 volts. Let us consider 100 molecules of actinium -emanation. ‘The first transformation produces 100—82°4 -or 17°6 initially neutral atoms of AcA. Practically all of these break up before diffusing to the walls, and 100—93 -or 7 per cent. of them, 7. e. 1°23, become neutral atoms * Kennedy, Phil. Mag. xvii. p. 744 (1909). + Lucian, Phil. Mag. xxviii. p. 761 (1914). t McKeehan, Phys, Rev. x. p. 473 (1917). Radium, Thorium, and Actinium in Electric Fields. 373 of AcB. Of the original 82°4 positive A-atoms, 86 per cent. at 600 volts break up before reaching the cathode, and therefore 82°4x0°86x0:07 or 4:96 become neutral B-atoms. The total number of neutral B-atoms is then 6:19, and since the half-transformation period of AcB is 36 minutes, they all diffuse to the walls and remain there at the @-trans- formation to AcC. Lucian found, for the smallest quantity of emanation used, 6°3 per cent. neutral, volume recombination being apparently not entirely absent. At 980 he found values ranging from 5°3 to 5:0, while the value calculated as above is 5°9; at 1700 he found 5:1 per cent. neutral, while the calculated value is 5°8. It is to be noted that the neutral B-atoms resulting from the disintegration of positive A-atoms have been carried some distance towards the eathode, and hence the cathode receives a greater percentage of the neutral atoms while the field is on than when there is no field. The values found by Lucian are calculated on the assumption that the cathode receives the same percentage of neutral atoms in both cases, and hence they are too small. Lucian’s results are therefore consistent with the general explanation advanced for the distribution of the active deposits. IX. Discussion of the Results. These experiments confirm Wellish’s conclusion that there is a definite limiting value to the percentage of the active deposit initially positively charged in a gas, and that the anode activity is due to the diffusion of neutral uncharged deposit. The latter conclusion is at variance with that of Ratner *, who maintained that the anode activity is due to the carriage to 1t by an electric wind of deposit particles which have lost their charge. It is difficult, however, to see how there can be any wind involving the motion of neutral gas molecules or neutral deposit atoms in an enclosed parallel plate vessel such as used in the present experiments. ‘The gas-pressure acting on the two plates is certainly increased as a result of the electric force acting on the positive and negative ions; but there will be no wind unless these forces are unbalanced, as, for example, if the plates were perforated, or as in the case of discharge from a point. While there is no doubt that many of Ratner’s experimental results were due to an electric wind, quite a different explanation is suggested for those made in the glass tube with disk electrodes, and which have most bearing on the present subject. It is possible that the darge amount of emanation used per c.c., which was * Ratner, Phil. Mag. xxxiv. p. 429 (1917). Phil. Mag. 8. 6. Vol. 41. No. 248. March 1921. 2C 374 Mr. Briggs on Distribution of Active Deposits of about twenty times that at which Wellish* showed that agoregation set in, together with the additional strong source of alpha-ray ionization, resulted in the production of aggre- gates of deposit atoms which are known to be able to take positive or negative charges. They are likely then to assume a positive or negative charge according as to whether they are in a region in which there is an excess of positive or of negative ions. The additional source of alpha-ray ionization, together with the separation of the ions by the electric field, would produce a distribution of positive and negative ions in the gas which would account for the results on this theory. On searching for a general law connecting the resnlts obtained and the properties of the various gases, it was seen that the percentage of either RaA or RaB positively charged was generally high for those gases whose heat of formation was large. However, a closer relationship appeared to exist between the results and the stability of the gases for increase of temperature: this latter quality, of course, depends partly on the heat of formation. In the table, data estimated from the results of various observers are given concerning the temperatures at which the various gases begin to dissociate. For the gases investigated, the percentage of recoil atoms positively charged decreases with the temperature of dis- sociation of the gas. It is well known that alpha-rays decompose many gases, and as the initial energy of a recoil atom is of the order 10'° times the average energy of a gaseous molecule at 0° C., it is to be expected that a recoil atom will dissociate many of the molecules in its path, even in the case of elementary gases such as hydrogen and oxygen, and that a part at least of the observed dissociation is pro- duced in this way. The dissociation of gases by heat is usually a bimolecular process: whereas if dissociation by recoil atoms occurs, it will probably consist in the disruption of single particles. Hunter t, for example, has shown that the dissociation of nitrous oxide by heat is represented by the equation 2N5On— 2INe Oo and Wourtzel t gives reasons for believing that in the case ot alpha-rays some of the gas is dissociated according to the equation N5O = NON: If a bi-atomic molecule, for instance one of carbon monoxide, * Wellish, Phil. Mag. xxvii. p. 417 (1914). , + Hunter, Zeit. Phys. Chem. li. p. 441 (1905). t Wourtzel, Le Radium, xi. p. 333 £1919). Radium, Thorium, and Actinium in Electric Fields. 375 is dissociated by a recoil atom, a free atom of oxygen and one of carbon will be produced. The heat of formation of bi- molecular oxygen from atomic oxygen has been estimated by Lewis* to be about 14 x 10+ calories per gram-molecule. The work done, therefore, in dissociating single molecules will be much greater than when the dissociation is bi- molecular. It is considered, therefore, that the dissociation of a gas by heat is only an approximate guide to the stability of the gas under the bombardment of recoil atoms. A qualitative explanation of the results obtained may be given if the following assumptions are made :— 1. Ata collision between a recoil atom and a molecule the former may become positively charged if neutral before the collision, if the relative velocity of the two is greater than a certain value, which depends on the nature of*the recoil atom and of the gas. 2. If the relative velocity is greater than this amount, the recoil atom has a large chance of being positively charged immediately after a collision in which the gas molecule is not dissociated. If positive before such a collision, it generally retains its charge. 3. At a collision which dissuciates the gas molecule, the recoil atom has a large chance of becoming neutral. According to these assumptions, the last collision causing dissociation paey leave the recoil atom neutral, and may reduce its velocity to zero or to an amount which is too small to allow the recoil atom to ionize itself and become positively charged. Hence some neutral atoms are likely to be present in all gases except those which are monatomic. If the minimum velocity at which dissociation can be effected is much greater than that at which the recoil atom can ionize itself, then its chance of being positive at the end of its path is large. If, however, the former velocity is less than the latter, all the atoms would be neutral if we were to assume that dissociation occurs at every collision, provided the velocity is sufficient and that the atom is never positive after such a collision. These two assumptions are not at all likely to be correct; but if the former velocity is very much less than the latter, values approaching zero would be expected from the assumptions previously made. In support of assumption (3), it is suggested that on the disruption of a molecule one or more electrons may be freed which are liable to be captured by a positively charged recoil atom. Also, a head-on dissociating collision will probably be co) inelastic, the recoil atom penctrating the molecule and * Lewis, J. Chem. Soc. Trans. exy. p. 182 (1919). 2C 2 376 Active Deposits of Radium, Thorium, and Actinium. thus having a large chance of neutralizing itself. Assump- tion (1), besides appearing necessary in order to explain the present results, is a natural deduction from Werten- stein’s * observation that RaD recoiling from RaC€ in very low vacuo is uncharged at the moment of expulsion, and that the rate of acquisition of a positive charge increases with the pressure. The ionization potential of the elements increases generally from left to right in the periodic table. From this it is to be expected that the ionization potential of RaA is greater than that of RaB. If a similar relationship holds for ionization by collision with molecules, then the theory given would require that the percentages for RaA should be less than for RaB, as is found in all the gases examined, though it is quite possible that the effect is due to a difference in dissociating power of the two. Summary. 1. Wellish’s conclusion that a definite percentage of the active deposit atoms from radium emanation is initially positively charged at the end of the recoil path, the remainder being neutral, has been verified ; and the error in Henderson’s experiments, from which he concluded that all the deposit atoms are initially positive, pointed out. 2. The importance of the part played by the recoil of RaB from the electrodes has been indicated, and the per- centages of RaB initially positively charged in various gases have been measured; and from this the percentages of RaA positively charged calculated from experiments with radium emanation 3. An explanation of the difference in distribution on the anode and cathode of the active deposits of radium, thorium, and actinium has been given on the assumption that the percentages of ThA and AcA and of ThB and AcB initially positive in a gas are identical with those of RaA and RaB respectively, as is to be expected from their isotopic nature. The distribution of thorium active deposit was experimentally determined for eight gases In a parallel plate vessel, and found to be consistent with the theory, as also were Lucian’s results for the dis- tribution of actinium active deposit in air in a cylindrical vessel. 4, An apparent connexion between the percentage of recoil atoms initially positive in a gas and the temperature * Wertenstein, C. R. clxi. p. 696 (1915). Magnetic Field of Circular Currents. StL at which the gas begins appreciably to dissociate has been indicated, and a theory, based on the assumption that the molecules of a gas are decomposed by the deposit atoms during their recoil, has been given to explain the behaviour of the deposit atoms. I have to thank very sincerely Mr. E. M. Wellish, a suggestion of whose led to this work being undertaken, for his continual interest and advice. The Physical Laboratory, The University, Sydney, N.S.W., October 13, 1920. XXXII. Magnetic Meld of Circular Currents. By H. Nacaoka, Professor of Physics, Imperial University, Tokyo*. [Plate VI.] Ne years ago If have shown how $3-functions are suited for calculating the strength of the magnetic field due to a circular current. By means of the same functions, the inductances of circular coils can be easily expressed by formule developed in g-series, which are rapidly convergent. In the present paper the expansion in g-series is applied to obtain expressions for the magnetic force of a single coil and of Gaugain-Helmholtz coils, at points not far distant from the axis, in powers of the coordinates. It is usual to have recourse to expansion in spherical harmonics for expressing the magnetic force of circular currents ; but as each harmonic contains different powers of coordinates calculated from the centre of a single coil, or from the middle point of the axis of double coils, the calculation of the deviations of the magnetic force from that at the middle point cannot be easily expressed in power series of the coordinates. By the expansion in q- series, we can arrange terms giving the deviations according to their importance in disturbing the field, and map out the distribution of these perturbations. The study of the curves of equal deviations will be of practical use for finding to what degree of accuracy the field can be assumed as uniform. * Communicated by the Author. + Phil. Mag. vi. p. 19 (1903); Bull. Bureau Stand. xiii. pp. 269-393 1911) ; Phil. Mag. xxxv. p. 13 (1918). 378 Prof. H. Nagaoka on Magnetic It is found that deviations of different orders are positive in one sector starting from the mean point, followed by that of negative sign in alternate steps, and the curves of constant deviation resemble hyperbolas. Recent advance in accurate measurement of current calls forth the necessity of deducing some formule, which will easily give the magnetic field accurate to about a millionth part. For this purpose, the ordinary method of expansion in spherical harmonics 1s not suitable, as the convergence is not sufficiently rapid. The following cal- culation was ‘made with the view of meeting such needs in practical problems. ~ Let U be the Newtonian potential of a homogeneous body of rotation about the y-axis ; referred to the axial co- ordinate y and the radial coordinate p= ee Laplace’s equation can be written Un ol fou € 9 e = () e . . . Bt oy Op? pp Op ) or oT 5 7 At (ery Pane foyo Pa.) and Sy by pay? Be) Opoy poy’ Op/” If we put cons | J el ces, Ae DRA den Ue? k a k? k? ? ra NEES |e ln) YY ti a OWE E qa Le are ] he ar (Z-1)-2 > dE) ic (11') 2/aké LRNE “dk S* 382 Prof. H. Nagaoka on Magnetic For the first coil we have to put the value of f, in (9), and for the.second coil £, in (9"') for calculating K and HE; the value of 7 is positive for the first coil, but negative for the second. This can be easily distinguished by adding the sufix + or — to X and Y. Thus we may conveniently write for these components 2) Ge. 0. Ge Y Yor od pa (12) We shall in the first place confine our attention to the evaluation of the values of X and Y for the single coil. Hvidentiy k, K, and K—E can be expressed by means of 3-functions as given in every text-book on elliptic functions, so that by expansion in terms of. g we obtain easily an E{ j,B-2(K—H) } = 192 7gi(1-+2042+2259!4...); ) (13) jab = 32mqgi{1 +309 +4239! + 3986 9° + ._.), These expressions are of great importance in calculating the value of the magnetic force. Thus for a single coil, X = 96 7—_ gi(1 4 209? + 225q'+...) ; ™ apts q gt.) Y= 16 —1 gh {(1430q'+423q'+ 3986 9°+ > (1.) Wie 5 my | | | —69(1+20¢°+2259'+...) } : bs j and for a double coil referred to the middle point as origin, a stn) : Ki 96, gt (1 209? + 22594 4 1), roe: + + a : (1+ 309% +423¢4 + 398698 +...) | Ne == 67, jah | CEhs | — 69, (1+ 2097 + 22594 + a) } | Putting «=£&, y=” in (8), we find for the case of a single coil fe dak ; Crier Py Field oj Circular Currents. 383 whence we find, by expansion and the known relation [Boe [se Pale Ls ee 1) SoS ee the following powers of g expressed in terms of the co- ordinates. Ag they are of great convenience in various calculations of similar nature, ct will not be out of place to give them te the 7th power of E and 7. ey Sen VIE Ea? er , S045E! EO Go? Age 5 19a5 6a + 90484" 30989? , 9E°n fr, i 51 2a" 8a’ = 4a’ ; 2 ER aa oe fe eee 3& ee 15? 27& 1? 3& 4 = is DNGae) 6has am Ob Ga? soe es y. er poe aan Y ~ 256at" 25608 64q° (14) _ ee 40960" = 1024G" meSile = Tn a Ef = n 154 _ 138% 2 nt 8a? 2a?" 256at —16at = 160755" | 551E%y n° 115&y 4 sie! 2048a° 512a8 " 64a8 mune | | | ; as | ag From these expressions we easily find, by ae in the expansion in terms of gq, the following formule _ for X and Y: ease 157? a 5258n? | 105 y* . ah wen at cee 64 at 16 at 8 oul 1) iw = qieves AD Eni, om, OL toe ee eee a a 82 ce 525 Ent — 35 9° v aa vent cae aa Cenc Bae So. (B) 384 Prof. H. Nagaoka on Magnetic Thus the values of the magnetic force, as we proceed outward trom the plane of the circle by &, 7, are given by the formule (A) and (B). The process seems at first rather tedious, but when once the value of g is known in terms of the coordinates &, 7, the expansion in ascending powers of the variables is but a simple process of addition and multiplication. The usual process of expansion in zonal harmonics of different orders is looked upon as a practical guide to the solution of such problems. If we try to treat the problem such that the magnetic force at the point &, 7 is expanded in different powers of &, 7, the solution in sperical harmonics becomes very tedious as each harmonic involves different powers of &,7. A practical calculation will convince the reader of the facility of the g-series. Of course the knowledge of the expansions of the integrals in q-series must be presupposed. The above calculation was checked by expanding the component forces X and Y in zonal harmonics. ‘The advantage of expansion in power series of the eoordinates referred to the centre and the plane of the circle lies in finding the deviations of the magnetic force from the value at the mean point as we recede from it. We can group terms of different powers and discuss their influence in disturbing the field, just as we discuss the aberrations of a lens, by expanding them according to powers of the so-called Seidel’s coordinates, and class the deviations according to the orders of the coordinates involved in the aberrations. Radial Component X for a Single Coil. Reverting to the expression of magnetic force X given in (A), we find, by assuming the radius of the coil a and the axial component of force at the centre to be both unity, that the deviation of the second order is given by. (Plvae tin =G. . . « » (eae which is represented by rectangular hyperbolas. By con- sidering ¢ as a parameter, and giving small fractional values to it, we obtain loci of points at which the radial component is ¢ times the force at the centre. Thus the distribution of the radial deviations of second order is given in fig. 1; they are mapped out for equal difference of c except for curves near the centre, in a square of sides equal to the radius, in a plane through the centre perpendicular to the plane of the circle. Field of Circular Currents. 385 Since the deviations of odd orders vanish for a circular coil, we have next to discuss the deviation of fourth order, which is given by 16 (B£°—4En*) =e, . (16) (Hig. 2.) as is evident from (A). The curves of constant deviation have different branches, all of which resemble hyperbolas, with asymptotes inclined to the axis of & at angles of +40° 53’ 48'' and the normals at the vertices inclined at +20° 58’ 32" and +66° 7' 18'.. The distribution ot the deviation about the centre is shown in fig. 2. The deviation of sixth order is given by rae (Oe 7 — 2087 + 8£q°) = ¢, (17) (Fig. 3.) The curves of constant deviation are more complicated than the preceding, but have branches resembling hyperbolas, of which the asymptotes are inclined to the axis of & at ammelesobed 24. 51 347" and 4-567 7% 8. The normals at the vertices are inclined to & at angles of +14° 8’ 56”, pea ll 73) 3044") The inspection’ of itis: 3 will at once show how the deviation of this order is very small near the centre, but increases very rapidly as we recede from it. For practical purposes the deviation of higher order will be generally negligible, and if necessary the calculation will not entail much labour. Axial Component Y of a Single Coil. As to the axial component Y, the discussion of the deviations can be made in the same manner as already indicated, so that it will be only necessary to tabulate them. The common characteristics of the curves of constant deviation are that they all resemble hyperbolas and are separated by asymptotes, which form the boundaries of positive and negative deviations, as is evident from the figures at a glance. Deviation of second order for Y-component :— pee eee ey 6 (18) (Rig. 4.) Asymptotes : +35° 15! 52". Normals at the vertices : 0°; +90°; 180°. 386 Prot. H. Nagaoka en Magnetic Deviation of fourth order for Y-component :— Ga (3&4 — 24F'n? + 8") = c. . (19) (Hig. 5.) Asymptotes : +19° 52’ 33”; +59° 26’ 40”. Normals at the vertices: 0° ; 180°; +40° 53’ 36”; 250", Deviation of sixth order for Y-component :— zee (123525 —630E'n? + 840£%n' — 112°) =c. (20) (Fig. 6.) Asymptotes: +69° 41' 34”, Normals at the vertices: 0°; 180°; +90°. Gaugain-Helmholtz Coils. For the calculation of the double coils at distance equal to the radius of the coil, we have j2 = fag 9008 ee 9 a ‘ (a+ 6?+(5+7) where + sign refers to the first coil and — sign to the second. Hxpanding k° and expressing g according to the formula connecting g and sf, we obtain the following sets of formule :— 4(—31E + 280? — 1528 Ss 87 oy 625a4t on <= ed 7% = on fier +o, = Seat (FEE TN) os Sort, } 2 (eee oon. 3 aL prea Ee 5a bar) 25a" 5 ie & C es { 164 i 1 = 6254! ear Kae y | Ee ae oy OG 0 ae eta ee eee | OH 2 a5qo + (FE nn") oes : 1 +(—31Et+ 20089! 80!) gome t Field of Circular Currents. 387 whence we notice that = 4 f ‘ > 9 9 ¢,+@= S135" T 4. (—31E!+ 280E2y? i Cae: V9 .4Vq_ = 2a) = L157, + (—B1E + 2008%? = 809!) get ee }. it will not be out of place to illustrate the advantages of g-series (II.) in the calculation of X and Y in the case of double coils. Jf we use the formule (II.) for the ealculation of X,, X_, and Y,, Y_, the second term in X contains q??, anol dine fied meen, an g4?, Take for eee the case E=0°2, n=0°2 ; then «= Ue and ga —) 0:04 nearly: so that gi? = 0:00000016 and g%® = 0-000000057, and = (000000005 and g!/?= 0-0000 00024. Thus three terms are generally sufficient to attain the limit of practical accuracy. g must, however, be calculated to SIX or seven significant figures. Substituting the values of Ghee Gi) Many ait expression for x, Gas “and Ne Yeas or vene tor (a) cimele cor: in (A) and (B), we easily find that 23047 2h ee A eC M I a teN NG fala te wk CS age Ta (V) anil S27 18 } | f= —— 4 1—— __ (3 — 2489? + 8n'* Mectau ee mi ae 1258 PE cep n') p+ (D) showing that at the middie point of the coils D) C= OS > lenge NG —— ag BA Wiha te hs oh rare ea 55, ja 388 Magnetic Field of Circular Currents. The deviations of the second order vanish for the double coils here considered, so that we have only to diseuss those of the fourth order upward. The beauty of the arrangement lies chiefly in the elimination of this most important deviation, and we at once notice how terms of higher order are insigni- ficant and make the use of the arrangement for obtaining uniform field very important in some practical applications. In the following, deviations of fourth order for X and Y components are given in tables. Deviation of fourth order for X-component :— jos G7'E 398) =c.. . . COG Asymptotes: +40° 53! 48”. - Normals at the vertices: + 20° 58’ 32”; +66°7' 18”. Deviation of fourth order for Y-component :— —y'g5 (38! — 248° + 8y) = c. . (25) (Fig. 8.) Asyimptotes : +19° 52' 30" ; 59° 26' 40” Normals at the vertices: 0°; 180°; +90°; + 40° 53’ 48”, The comparison of the figures 7 and 8 with the corre- sponding deviations of fourth order for a single coil in figures 3 and 4 shows how the deviations in double coils are small, and especially the gradients. The use of double coils in exact measurements of magnetic field, as for example of the terrestrial magnetic force, is strongly to be recommended. The discussion here given refers to single layer coils. By winding the coils in different layers, we have to choose a proper section of winding layers. For a rectangular section, Maxwell shows that when the ratio of the depth to the breadth is as 6 to 5°57, an important correction due to the tinite section of the winding is made to disappear. As before discussed, the deviations of magnetic force due to terms of different orders have an analogy to the aberrations of lenses and mirrors, and will be of great advantage in designing galvanometers and other electric instruments, in which high order of accuracy and the uniformity of the field are a principal aim. My thanks are especially due to Mr. 8S. Sakurai, assistant in the Institute for Physical and Chemical Research, for verifying the formulz and for constructing the diagrams of deviations of different orders. Physical Institute, Imperial University, Tokyo, July 1920. PR | XXXIV. Integration Theorems of Four-Dimensional Vector Analysis. By H. T. Furnt, Lecturer in Physics, King’s College, London ™. Introduction. HE four-dimensional vector analysis developed by Sommerfeld (Ann. der Physik, vols. xxxii. p. 749 sqq- and xxxill. p. 649 sqgq.) evidently bears the same relation to a more general analysis as do scalar and vector products and the theorems of Gauss and Stokes to vector or quaternion analysis. In attempting to work out some of the details of the restricted principle of Relativity in terms of the more general notation, it was observed that such expressions as the scalar and vector products of four vectors, combinations of four and six vectors, together with the generalized div and curl and the operation lor (Lorentz operator) of Minkowski’s Calculus are parts of more general expressions. It is tne object cf the following pages to set forth this analysis. It will be obvious to the reader how much is owed to quaternion analysis, in particular to Joly’s *‘ Manual of Quaternions,’ ch. xviii., of which the notation is adopted here, and to Sommerfeld’s papers mentioned above. The applica- tion of the notation to the theory of Relativity has already been pointed out by Professor Johnston and discussed by Sir J. Larmor f. In the first few sections the notation is explained before the main object of developing the integral theorem is reached. § 1. In treating space of four dimensions it is usual and convenient to adopt a language and notation similar to that with which we are familiar in three, though it is difficult and perhaps impossible to form corresponding geometrical pictures. Thus a straight line through the points (a, 0; ¢ d;) (ay bo ¢y de) 1s the locus : B@—a yo-by 2-4 u—d, = a SS eee A, —A»o b, — be Cy — Co ad,— dy (wy 8 @ + “eee (2°8) But in this case the relation V,P,P,'#0, as in the case of the product of vector areas when these are normal. We shall return to this point later. It is to be noted that we can always express P, in the form : Po=khV oe 8 +k! | GIS k’ gives the additional arbitrary constant. This should be compsred with Sommerfeld’s discussion, § 1 of vol xxxil., already referred to. If a vector area be split up into elements so that each element projects into rectangles on the coordinate’ planes, we write for it: | dp =igisdy dz + igidz2 dex + tyinda dy + iaytyde du + isigdydu+isidzdu. . (2:9) This element is a vector like P,. of Four-Dimensional Vector Analysis. Joe §3. The laws of multiplication lead to certain relations between products of the 7’s of which the flowing are illustrative : iylaly = —Ulyly = be, a ee: (1) tgtoly = — 71]. A change of position of any vector by an odd number of places in any product changes the sign, unless the product is scalar, or contains a scalar part. Thus aBry = —Bay=Bya. Products of the type i223 are not reducible. In multiplying three vectors it is convenient to regard the product as composed of reducible and irreducible parts. Thus P1P2P3= Vspipop3+ Vipipopgs - - - (8°2) V3Pipop3 Contains terms- in 2g23t4, 23742, etc., and Vj p,pop3 contains terms in 2, 7,73 and 74 which are the result of such reduction as is illustrated in (3:1). If p, and p, have the values in (2°3) and 3 = Aly + Ogle + Cals + dst | the expansion of the product V3p,p.p3 gives : V 00123 = la/gta(OyCods) + Ugtyti (Cyd ode) + (142322) (dyaob3) + ty2ot3(Ayboc3), - + (3°3) where (A, B,C3)= | A, By C, | A,B, Cy | A; B; CG, | The determinants are recognized as three-dimensional volumes. They are the components of the direeted volume V.0\P203 On the various sets of three dimensions that may be chosen from the four. The coefficients of these irreducible products are minors of the determinant (a,b.csd,). and in writing the components V, V. V3 V4 we shall adopt the rule of signs: Vi= (by cod), Ve= — (cia), V3;= (d;asb3), 3 Vi — (Ayh9¢3). 394 Mr. H. T. Flint on Integration Theorems Thus for any directed volume V we have: V — dntata NG nay Lath V5 ae duties Vig ae Tose Ne ° (3-4) The volume element dt may be divided into elementary components. So that: AT =iszigdz dy du—isig, dy du dw t+ tyyty duda dy — tyigtgdadydz. (35) The remaining part of the complete product p,psp3 is V101p2P3, and on expansion it is seen to be: — 1,0 Ddgdg +21 {by (dads) + ¢y (ocx) +d, (Agd3) } + similar terms. | (3°6) The coefficient of the second 7, is the OX component of the vector product of a four- and six-vector employed in Minkowski’s Calculus. It may be represented in our notation as: Vi (91 Vepaps):. |. ee § 4. The product cf four vectors will consist of terms in dylalgla, others in 723 which have arisen from such products AS lolglgl, and so on, and finally there will be terms inde- pendent of 2’s arising from such products as ?9/3t913 which reduce to Ds leOslie a piped a ibe Thus P1P2P3P4= V 101P2P3P4+ V2p1p29301+ Vopipopse1. (4°11) On expansion we find : V4P1P2P3P4 = Vylolsta (aybor3d 4) 5 : : . (4:2) Thus the tensor of V4p,p.p3p4, or TV 4p1p2p304 a8 it will be written, is the volume of the four-dimensional parallelepiped bounded by the four vectors. It is not possible to write 7,2,%3/,= +1 by analogy with the corresponding wk=—1, for this would lead to incon- sistencies. Suppose we write Desist — oe Then 1ylylolsly = — 1 and isa —1;. The first gives: les Uh the second Iplsly = — he of Four-Dimensional Vector Analysis. 395 Acain we should have: oD DOr OR — —1yly or lal — Orne The product 7429232, will be denoted by I. When [I enters into any expression it obeys the ordinary rule of sign, 7.e. an odd number of displacements of I ina vector product changes the sign, while an even number leaves the sign unchanged. laBy=—a2lPy=«Bly=—«Byl. Also Je a BO and JO Oh GG Thus the factor —I will change a product V, into its reciprocal and V3 is changed into a vector of the type V,. We have evidently TV3;a0y= Vi leBy. Solis WN OReH IAs (4°3) § 5. The definition of the component of a vector p along any direction denoted by the unit vector n is similar to that in ordinary vector analysis. We define the component to be — Vopn. et the case of a plane we define the component in the plane of a vector of the type V, in a similar way. The component is — V oP.v, where v denotes a unit plane. Thus the component of P, in the plane yz is We cannot define a component along a vector normal to the plane since the normal is indefinite ; we may, however, define a component in the reciprocal plane. Let V,«8 denote a unit plane, and V.’e8 its reciprocal. The component normal to V, may be defined as —V,P,V.'aB, or what is the same thing : VoP. (LV.«8) : This is equal to VoIP,V.e8= —V oP,’ Voa8. Thus the component of a vector in the reciprocal (or normal) plane is equal to the component of the reciprocal n the plane. 396 Mr. H. T. Fhut on Jntegration Theorems It is to be noted that Vo(V.«8)(V,'e8)=0 on account of the relation (2°5), so that the two planes may be said to be normal in the same way that two vectors p, and pe are normal if Vop;po=0. et Wolbses S20 although the relation V Poke ke. stilleholds: Sommerfeld has defined the components of a six-vector with respect to a plane. We can very easily obtain his result in our notation. The component is —V )P2(V2a6). We must make V.¢8 a unit plane and naturally shall choose « and @ as two unit perpendicular vectors. W rite a= 1+ Mylo + Nyl3g + pts, B= lots + Moly a Nols + pols. Since « and @ are unit perpendicular vectors, Se >/ — 1 and >i,l,=0. Then —V,P.V.28 =—V, (S2—73P,2) (Ztgt3(m 72) ) = Py, (mn) + Pzr(mle) Pry (hme) + Pru(hipe) + Pyu(mpo) + Prukripe) . (ol) This should be compared with Sommerfeld’s result (Ann. d, Phys. xxxii. p. 760). For a vector A of the directed volume type we have: As = lotgta Ar 131A, + LglyloA- = dylots Ay => I (2, A, + 1A, + 3A, “+b ZAG) = TA, (say). The component of A along any direction n inay be written : | —TV,A.n or —V,(IA3)n or —VoAjn. The component of A; along Ow is by this definition A,, and similarly A, A, A, are the components along the other axes. A; is normal to any direction n if V)(IA;)n=0. If four vectors p; p2 p3P4 bound a four-dimensional volume of Four-Dimensional Vector Analysis. 397 it is convenient to adopt a rule of signs for the three-dimen- sional volumes that may be chosen from them. The rule adopted is to define the volumes as : : + Vsp0p3p4, — V3PsPsP1, + V30sP1P2, — VsPi1Pops- This is to be associated with the rule of §3. The volumes corresponding to the vectors 2,da, i,dy, i3dz, 14du are: iglslgdydzdu, —istyidzduda, i,ytodududy, and = Widols ax dy dz. § 6. A vector operator D, defined as OME SLO ds OO Os, ee tp (Le pes plays an important part in four-vector analysis just as the Hamiltonian V does in three dimensions. The suffix D, serves to show that it is of the type of a vector V,. Any function o of p will undergo a small change do, on account of a variation dd given by the equation do=—V dp D,. 0. ° . . . (6°1) (6:1) is the same result as is expressed in Cartesians by : NOG OY e), O@ rover do= ae a a y+ Aye ie du. This method of introducing D, appears to be the simplest. VY is sometimes defined by the operation : : dv . K() vim ee Sit, . where dy denotes a divided element of area in two dimensions bounding a small volume v, and F is any function of p, the vector from the origin to dv. If this method of defining an operator be extended to four dimensions we shall have: Lim \dr¥(e) pe “9 ek age ea OB) (Gre) where dris the directed element of three-dimensional volume and v a small four-dimensional volume. 398 Mr. H. T. Flint on Integration Theorems This mode of arriving at D gives to it the inconvenient canonical form : EAE Go SACS rear ty : isis 2 ae + i4lylo Ae TE: A SE $7. The circuital theorems of ordinary vector analysis \ dp F(p)=)\ VV) . F(p) and \\ @F(o)=\{\ V F(p). dv are generalizations of Stokes’s and Gauss’s theorems. We proceed to establish the corresponding circuital theorems for four-vectors. They are three in number and the operator V is now replaced by Dj. § 8. In order to establish the theorem for integration along a line we require the relation : + 01 V 9@t— ay V oa,e=V{(Voejao. a). . » (81) Jf we write: = ay =24a, +256, $23¢) + 24d), | thy = 11g + Igdg +1302 + lye, a =A +b +ige + 24d, we find that the coefficient of 7, on the left is Ay (Ady + bb; + cc, + ddy) — a4 (aay + bby + cg + dd), while on the right the coefficient is — b(ayb2) + ¢(c,a,) —d(aqdy), and these are of course identical. §9. Let M be at the extremity of the vector p and construct about it as centre a small parallelogram bounded by the vectors dp; and dp, as in the figure. of Four-Dimensional Vector Analysis. 300 If the value of a function be F(p) at M then along the side marked dp, the value will be F(p)—4Vodp,D,.F(p), . - . - (91) in accordance with § 6; while along the opposite side we shall have the value F(p) q= £V odp, D, ° INGo). . 5 : a (9-1) and there will be similar expressions for the other two sides. Thus on making the summation }dpF(p) and making the circuit in the positive direction, we find ~dp Fp) =dp, Vodp, D, . F(p)—dp.Vodp, D,. F(p) ) = Vi (P,dpidp: . Di). F(p) = V,(dv D,). F(p). ei (Oo) If F(p) is regular throughout a finite surface we have by adding up for each element of which the surface is composed : : dp F(p)=[fVildvD,) Fie). =. . (9°38) In the summation all the edges of the elementary area except those round the boundary contribute twice to the summation with opposite signs in each case, so that finally the integral is taken round the boundary. A particular case is obtained by choosing F(g) in the form 7X +2,Y +7;Z+i,U, when we obtain: \ (Xdv+ Ydy+Zdz+ Udu) Me (OZ TOMY ah co O45 00) =WN(Sr Sa) avert Ge + Sn) oY a SS dedy + ee ae en oe ane =| du du OUL ov aU a) i Re a. ou) by writing dy in the form (2°9) and taking the scalar product of each side. 400 Mr. H. T. Flint on Integration Theorems If F is of the special form given above since dv Di, = V3dv D,+ V,dvD, i and WROVG GD ID) iSO. we have: V {V,dvD,.F}=V,(dv D,F)=V,(avV,D,F), Vo J dp F(p) =|\Vo{(Ve DF) . dv}. The curl of a vector is detined by the equation Vo \dp F (p) )=\fVo( (curl F. dy). Thus Cul oh == ee ht: § 10. The second theorem obtained by performing the operation Sd7F(p) through a sinall four-dimensional ele- ment dv connects a three-dimensional to a four-dimensional integral. By a similar treatment in this case as in the last : Lda ho) | — Vsdpodpsdps. Vode, D,. F + V3dpzdpydp;. Vodp.D,.F —V3dpsdp,dpo. Vodp;D,.F + V3dp,dp2dp3. Vode, D,. F. This is simplified by observing that: V 401 h0t304.4= — Va anes a4 V yao + Vises 4 ot Vora — V3, ec. Voe3a+ Vs Ob | by Oe. V eae. Zdz. F(pj= Vidpidpodpsdp,. D, F(p) lide <1 hi(o): JNJ ar -Ke)= \\\S 1D, F(e) dv, eae sj) dr. B(p)= ay D, F(p) dv. The same limitations as before are to be imposed on F throughout the finite region. By taking the SCallaes product of each side and employing again the special form for F(p) we have : in (Xdydzdu+ Y dzdudx + Ldudady + U dedyde) =(\(V(; oF +e +e 4 Or )dadydedu. Sommerfeld’s extended te F is equal to — V)D,F, or to: Vari —V,\Idr a) dv—>0 Ov of Four-Dimensional Vector Analysts. 4AO1 § 11. The last of the three theorems occupies a place inter- mediate between the other two and connects an integral in two dimensions with one in three dimensions. Fi 6) ae or oC Proceeding as in the other two cases, if p defines the centre of the element bounded by the three vectors dp,, dp2, and dps : Soy. hp) = Vidpzdp; . Vodp; D,. F(p) — Vedpsdp, . Vodpe2 D; . F(p) — Vedpidpy.Vodp3 D,. F(p). It is easy to show that Vo(V 3c e903) c= + Vo as. Vor a+ Voaza,. Vou x + Veo 1c, 6 V oat3% (lala so that Sdv. B(p) = —{ V2 Vsdpidp,dp;) . Di} F(p) = + V,(drD,) . F(p). Thus by extension as in the previous cases to regions of finite extent : ide Wiel Wi VadaD eRe). 6 i4(112) If the summation Sdv'F(p) is required, since dp'= —Idbp, {\dp' F(p)=—J\{1V.(drD,) lp). : (11:3) Sommerfeld defines the component of the vector divergence normal to dt of a six-vector P, by the operation : tian» ay \ | Vo (av' Ps) Cte OM Oe) t° we have: where |'é7 is the magnitude of the three-dimensional volume ov, This expression is called the component normal to ét. 402 Mr. H. T. Flint on Integration Theorems We may also define Minkowsk1’s lor P, from (11°3), this expression denoting the Lorentz operator, equivalent to the vector divergence. Lor P, is defined by the equation : + [J Vold'P)=— FJ} Volor Petar) 43 ae j = — {iI Vol. Voldr Dy) .P2 > (11-4) ARPA =—({{{Vo{ldrD:Ps}, J since the part Vid7D,, which with V.d7D, makes up the product dz D,, cannot add to the scalar product. Now Idr is a product of the type V, so that Vi (ldr s iD; 4 Ee) = Vo (Ldr = Vi Pea lor les sae MG IOHIES ae Since by § 4, Vo(dv P.) = Vo(dv' P.’), we have by (12-4) and (12-2) 3 (| Vo(dvPs)= | (V,(dr' P.')= — {\\ Vo (lor P.’.La@r). (11-5) Thus in integrating the normal components of P, over a surface, the operation lor P, occurs, while for tangential components (2.e. components of P, along dy) lor P, is replaced by lor P,’. This is important in the electromagnetic equations. 9 12. The field equations of the electron theory as expressed by Minkowski may now be regarded as a generalization of Gauss’s integral of intensity over a surtace. Putting these integrals in their usual form we have: j Da = —\pdv or js Ddv= | pdv, \B,dS=0 or \SBadv=0. If now we form two six-vectors : P,=(—2D,) 923 + (—7tDy)isi: + (—7D- iin + Hii, + Hist, + Heist, Os 787th, hee gill + Ht, + See where i= /—1. D, ete. denote the components of displace- ment as usual, and H, B, and E have their usual significance. * Cf. Cunningham, ‘Theory of Relativity,’ pp. 101-2. of Four-Dimensional Vector Analysis. 403 If S is the current four-vector and equal to PUrty + P Uyle + putzts + Pt tag, where pis the density of charge and the w’s components of velocity of the charge, the general equations may then be written : Lor iP, = S, Lor = 0. and for the case when the permeability and dielectric con- stant are both unity Q,=P,’. Thus from (11:4) and (11°5) we may summarize the theory by stating that the normal component of the six-vector P, over the surface of a three-dimensional volume is numeri- cally equal to the amount of 8 within the volume, while the total tangential component vanishes over the surface. By the amount of S is meant the numerical value of | V\Idr.S. It is the value of the component of S along the four-vector Id. § 13. The four-vectors of Minkowski’s Caleulus are limited in that they are subject to a linear transformation. The vector p becomes a new vector o where o=¢p. @ is a linear vector operator and can be expressed as: pp = — (a, Voe2p+BiVo Pop +NVoyept o1Vod2p). (13°1) Thus ¢ is a dyadic and associated with it are certain invariants just as in the case of three dimensions, and whereas in the simpler case @ satisfies a certain cubic relation so here ¢ satisfies a quartic. In three dimensions the ratio of the two scalar products SdrAgdudv to Srpv is independent of A, u, and v; to this corresponds the invariance of the ratio TV,¢drA dudvdr to TV,Apv7r, which means that the ratios of the four-dimen- sional volumes after and before the operation of @ are a constant. When ¢ denotes the Einstein transformations this ratio is unity. The linear transformation referred to is analogous to that occurring in the theory of elasticity in the case of homo- geneous strain, and since a further condition to be satisfied by the four-vector is that of unchanging tensor such a trans- formation is analogous to a rotation. ° 404 Integration Theorems of Four-Dimensional Analysis. Gibbs has shown that a rotation may be expressed by pp=t' (ip) +7 (jp) + (kp), where (7o)=the scalar product of 7 and p; and we may also express a rotation in four-vectors similarly, op = _ (a! Voap + B'V,Bp =F y' Voyp + 8’V dp), 2! i'k', 27k. are unit vectors and each group is mutuall je, 9 sit vectors and each group y perpendicular. Similarly a@' B'y'd and @ By6 are unit mutually perpendicular vectors. The Kinstein transformation is a particular case of this in ; Pp which 2 = (0 — 100) Oa B=2=f, yS5=05 6’ =kh(iv, +14), where k=(1—v?)~?,1= 7 —1; v the arbitrary constant in @ is the velocity and may be said to define the strain. The notation suggests that the restricted principle of Relativity may be summed up by stating that a fundamental four-dimensional medium exists which may be subjected to a strain of the type (13:1) with the values of the vectors given above and that phenomena described by the unstrained vectors will be described by the strained vectors in exactly the same way. It is natural to generalize ¢ and give up the demand that it should be linear. The strain is then heterogeneous and the principle becomes more general. For small regions there is still a linear relation, for if o is a function of p we have : do=—V,dpD,.c=¢dp ; but @ now ‘contains p in its constitution and the linear relation is true only for small changes dp in the neigh- bourhood of p. This appears to correspond to the “‘ naturalness ” of small regions in the theory of Relativity. Space-time is Galilean for infinitesimal regions. In addition (dc)? is invariant or (do)? =(dp)?. XXXV. Does an Accelerated Hlectron necessarily Radiate Energy on the Classical Theory 2? ByS. R. Mitner, D.&Sc., Acting Professor of Physics, The University, Sheffield *. HiS question is of fundamental importance in modern theory, and it would, | imagine, at the present time ‘receive an affirmative answer from most physicists. It is true that certain adverse experimental results, such as the normal absence of radiation from electronic motions in the atom, call urgently for theoretical explanation ; but it seems to be accepted that the necessity in the classical theory for radiation from accelerated charges is so firmly based that it can only be removed by far-reaching and revolutionary changes, such as the quantum theory supplies. Some apology seems necessary for attempting to open the question again at this date; and I should not have ventured to do so, but for the result of the consideration of a certain concrete ease of accelerated electronic motion which is amenable to an accurate mathematical treatment. This example shows. that it is possible to obtain, even on the classical theory, a solution for a particular case of the accelerated motion ol charges, which satisfies completely both Maxwell’s equations and the mechanical laws which characterize a conservative system, without any irreversible radiation of energy. A study of it enables us, I think, to prove that a certain step made in the deduction on the classical theory of the general necessity for radiation is not invariably a valid one, and to show that a comparatively minor modification of the boundary conditions of the solution is sufficient to do away with radiation in at any rate one case of accelerated motion. A remarkable solution of Maxwell’s equations for a particular type of accelerated electronic motion has been given by Schottt. It is remarkable in that it is the only case, other than that of uniform motion, which up to the present has been solved in finite terms. Consider a point-charge moving along the (positive) axis of «, and, & being its distance from the « origin at time ¢, let it move in such a way that ote acre eS cy CD) where & is constant. Att=—o the charge is at £=+0 moving towards the origin with the velocity of light ¢, it * Communicated by the Author. + ‘Electromagnetic Radiation, pp, 65-69. Phil. Mag. 8. 6. Vol. 41. No. 243. norte G21, 2h 406 Prof. S. R. Milner: Does an Accelerated comes to rest at £=+h when t=0, and moves back towards f= +0, acquiring again at t=+o the velocity of light. The motion is that which would be produced, according to the ordinary principles of mechanics, by the action, on a particle of mass Me is PBSa ar i dé m= (AAP where B= nape of a constant force F, such that ey Pel”) ol eee k To express the electromagnetic field associated with a point-charge e moving in this way, at any point let y be the angle included between two lines, lengths 7, and 79, respec- tively drawn from it to the instantaneous position of the point-charge and to that of its image in the plane =O, and let w=logr,/7,. Then yx, y are related to the cylindrical co-ordinates 2, y by the equations os eae ae Esiny Ae cosh w — cos x cosh w— cos y The third co-ordinate ¢, the angle through which the plane containing the point has rotated about the axis of « from a fixed position, is the same in both systems. The electric and magnetic forces at any point of the field are given by 1—” har— cos yx)? : ee H=8siny. Hi. (4) The nature of the field will be snadlz clear by a reference to figs. 1 and 2. These show meridian sections at the ie : moments CS and ¢=0, when the charge is at the distance “2k and k respectively from the origin. The lines of force in each case form the ares of circles (.~=const.) passing through the charge and through its image in the median plane =0; in the figures they are drawn so that x of the total flux of induction is enclosed by adjacent lines. The changes which the field undergoes can be pictured by supposing that each line of force in fig. 1 att = — * is moving c normally inwards with a velocity @csiny. The velocity gradually decreases until the line comes to rest momentarily Electron necessarily Radiate Energy ? A407 in the position of the corresponding line of fig. 2 at t=0; it then moves outwards, passing the position of fig. 1 again aut — ++ . The magnetic force is directed along the parallel of latitude, and, passing through zero value, is reversed in sign at the moment when the electric lines change their direction of motion. The dotted circles, y=const., ortho- gonal to the lines of force, are the lines of the Poynting energy-flux. Oe ADS Proie o. y. Milner : Does an Accelerated The field, as given by Schott, is limited by a moving boundary, formed by the plane e+ct=0 (AB at t=—A/e, AUB! atud= i kc, fio. 1 1; OY, fig. 2). The oundaryiicis transition layer (of a thickness comparable with the radius of the electron) in which the electric and magnetic forces vary from the values given in (4) on the right of the boundary to zero bey ond it. Fora point- charge it forms a layer of discontinuity, in which the lines of force emanating from the charge suddenly change their direction, and thence- forth lie in the plane. It will readily be seen that this field gives an irreversible radiation of energy. The direction of the energy- -flux shows that, except at the moment ‘=0 when the moving boundary crosses if, no energy ever passes through the median plane. Thus the field energy which, at positive values of ¢, exists to the left of the origin, must, along with the boundary, constitute a permanent loss by radiation from the system. A very simple modification of Schott’s solution eliminates from it the presence of irreversible radiation. Consider the equations (4) for Hand H, and let them new be valid over all space and all time, the moving boundary being dispensed with. The electromagnetic field thus expressed possesses the following properties :— (1) It satisfies Maxwell’s equations E=ccurlH, H=-—ccurl E£, div E= 47rp, div a= 0), at every point of space and time from —o# to +”. (2) The third equation is satisfied in the sense that div E=0 everywhere except at the points «= +é, where it becomes % in such a way that JE, dS round any closed surface surrounding the point is equal to +4ae. The solution thus forms throughout all space and time an electro- magnetic field which can be associated with two point- charges, +e and —e, moving with the particular type of accelerated motion defined by (1). The two charges start with the velocity of light at t= —« from positive and negative infinity of a» respectively; they move symmetrically along the axes inwards towards the origin, come to rest at t=0 at the points z= +h, and then move outwards to infinity, ultimately acquiring again the velocity of light. in Hee ot force are as in figs. 1 and 2 except that the image is now a real charge and ‘there is no boundary, the lines “extending from one charge right up to the other. Electron necessarily Radiate Energy 2? —. 409 2 (3) There is no irreversible radiation of energy from the system. This is evident, for the field at t= +1, 1s identical throughout all space with that at t=—t, except that. the sign of H is reversed. Hence,.whatever flux of energy outwards from either charge may have occurred at t= —t,, it will be exactly annulled by an equal inward flux at f= + ty. (4) It will be useful to express the energy and the momentum of the system. The volume of the. orthogonal element comprised between the spheres y and y+dy, yy and rtd, and the planes ¢ and + d¢ is pe E sin yx dy dvr dd ~ (cosh y— cos y)*’ and the total energy of the field is cca note \ gq (EP + IP)AV. The integration extends from d=0 to 27, y=0 to a, but with regard to that for yr, the space occupied by the nucleus or charged surface of the electron must be excluded from the integration. The result will depend on the shape assumed for the nucleus. If we integrate from y=0 to the surface given by smhy 1 1— ? ) 5) Ee ecient | we get for the field energy of either charge in the region external to the gs This is identical with the field energy of a Lorentz electron of the same e and a in uniform motion with the same velocity. The surface (5) is not precisely the same as the spheroid which forms the surface of the j.orentz electron in a state of uniform motion; but it reduces to it in two cases: when the acceleration is zero, and when the radius of the electron a is indefinitely small. In each of these cases &/sinh y (the radius of the x sphere passing through a given point) reduces to identity with the radius vector. to the point drawn from the centre of the nucleus. There is a difficulty in dealing with the problem for an electron of finite size, ben) partly mathematical if the Lorentz spheroid is assumed, and = 410 Profeis: Whiner: Does an Accelerated partly because we cannot say @ priort what is the exact shape which must theoretically be ascribed to the nucleus of an electron in this type of accelerated motion. The theory of the Lorentz spheroid only applies strictly to uniform motion. There is no need, however, to discuss this difficulty here, as for the present purpose it can be set aside by our assuming, as is now done, that the electrons in the problem are of infinitely small size. The surface (5) then becomes identical with a Lorentz spheroid bounding the charge, and it can be taken to represent the surface of the nucleus without any difficulties being encountered. It is true that the assumption that @ is infinitely small makes the energy and the momentum of the system formally infinite ; never- theless, they are definitely evaluated in the limit, and the essential feature of the solution, the absence of radiation from the system, is not affected by the assumption in any way. The expression (6) now represents the total electro- magnetic energy in the external field of one of a pair of Lorentz electrons moving with the given type of motion, in the limit when they are of infinitely small size. In order to produce agreement between the electromagnetic and the mechanical schemes, the nucleus. precisely like that of the uniformly moving electron, must be supposed to possess a store of internal energy of the amount Le ae : = (1—3})!. ne Then the total energy of each electron in the system becomes eG | 5 —(1—f’)=. Mees se 3a eS (8) (5) The electromagnetic momentum parallel to @ is given by U= ‘EH sin 0 Aare dV 9 @ where @ is the angle made by the line of force, y=const., with the positive axis of 7. On substituting sinh yf sin x coshy— cosy ~ sin @= and integrating with respect to @ and y as before, but with respect to W up to the surface cosh 1 1— y E Ga 7 Electron necessarily Radiate Eneryy ? All we find the xz-momentum associated with the positive electron external to this surface to be 2 9g=3 —RO—A?). er Abn cack This is precisely the same as the momentum of the Lorentz electron of the same e and a in uniform motion with the same velocity. Jt must be noted that this result has only been obtained by making the surface (10), to which the integration in was extended, a slightly different one from that (5) used for the determination of the energy. The difference, however, disappears when, as is the case here, the surfaces are of indefinitely small size, as in this case (10) as well as (5) reduces to identity with the Lorentzian spheroid. There is a distinction between what we may call “ ficti- tious” and “real” eases of electronic motion, which is brought into evidence by this example. Maxwell’s equations, as is well known, are of great generality, and can be satisfied by cases of motion which cannot really exist. Theoretically a solution for the field of a charged particle in an arbitrary state of motion can be obtained. But from a physical standpoint the purely arbitrary motion of a charge 1s a condition impossible to produce. We know of no way in which an electron can actually be set into motion except by the application to it of an electromagnetic field (excluding possible effects of gravitational fields). Not only is this not entirely arbitrary, being subject to the fundamental equations, but also its inclusion alters the problem in an important respect. The problem becomes now, not that of finding the field of an electron whose charge is imagined to be moved about in a given way by external non-electromagnetic means, but that of finding a solution of, the fundamental equations which will represent the real time-history of a given electromagnetic field which contains an electron. But for this problem Maxwell’s equationsareinsufficient. Being satisfied by the simple superposition of two electromagnetic fields, they do not reveal the way in which, when an electron is present in a given field, the two fields interact. An additional theory is required for this, such as that of the Lorentz equation. Leaving this aside for the moment, we can consider in a general way the conditions which a system of two superposed fields must be expected to satisfy for it to represent a real case of electronic motion. There is undoubtedly the law of AY Prof. 8. R. Milner: Does an Accelerated the conservation of energy. The electromagnetic system must, in fact, be a conservative one, capable ef existing without the introduction of energy by imaginary processes from outside. (Radiation, 1f present, is of course part of the electromagnetic system.) We may, perhaps, further expect that the energy and momentum of each part of the system which is in motion should be related to each other in accordance with the laws of mechanics. From this point of view the solution of electronic motion which has been discussed is a fictitious one, as it does not obey the conservation of energy. As the electrons are approaching each other the energy of the system gets smaller, and it increases as they move apart. By a simple sothiodkon. however, the system can be made conservative. Superpose on the field 1 (4), which forms one solution of Maxwell’s equations, another solution in the form of a uniform electric field X of infinite extent and parallel to the axis of 7 Let Xe=F, so that X is the field whieh “is required, on the basis of Lorentz’s equation, to produce the actual motion which the electrons possess. Then, since for 9 ~_ e~ a Lorentz electron we have m,= 5 -3-, we have by (2) 3 ca yy a 3 ak , The resulting field is now given by H,=EHcos@+X, H,=Hsin0, H=@siny . H, where E and X are given by (4) and (12), and it forms of course a solution valid over all space and time. | These equations represent a system of two electrons of opposite sign and Lorentz mass moving in a prescribed way in a uniform electric field. We shall, however, as before consider only the case in which a is infinitely small, when the electrons become point-charges, still of Lorentz mass, and the field X, though of infinite strength, is the limit of the definite value required to cause in the point-charges a finite acceleration. Although the total energy of the system is infinite, the question of its.conservation can be examined by considering the flux of energy in the field. For mathematical purposes we can regard such a field as a solution of Maxwell’s equations which extends over all space and time, except Electron necessarily fadiate nergy ? 413 that it is limited by certain moving boundaries, which form the surfaces of the electrons, and inside which E and H are : 9 Wan te °. 5) prescribed to be permanently zero. Weknow by Poynting’s theorem that, if the quantity Pa +H’) be identified with the energy density, the conservation of energy will hold throughout the whole field except for the space included within the boundaries; for the Poynting flux is equivalent to the transfer of this quantity froin one part of the field to another without net loss or gain. The only places, there- fore, where energy can enter or leave the electromagnetic system are the boundaries, so that if we calculate the total flux of energy which is passing through them at any time, we shall obtain the energy which is being introduced into or is disappearing from the electromagnetic system per unitof time. I¢ must be noted that thisis not necessarily zero tor a conservative system. Hiven in the simplest case, that of the uniformly moving electron, considerations of the continuity of the flux *, as well as nase based on dynamics and on relativity +, necessitate the postuiation of a definite store of internal energy, of the amount given by (7), within the nuclear boundary. When the internal energies of the electrons are included in the scheme, conservation in the system will be characterized by the net flux of energy out- wards across the boundaries being equal to the net rate of decrease of the internal energies of the electrons. Confining attention to the positive electron and to the positive x half of the field, we shall take the boundary to be the spheroid given by the limit of (5) when a is infinitely small. Let P be the Poynting flux outwards through a fixed boundary momentarily coinciding with (5), and let v be the velocity outwards of any point of the Snes surface, then the net flux outwards from the whole moving ania is given by \ (P—Wv)nds, where n is a unit vector outwards normal to the surface, and i - ee (H? + H?+ 2H cos?.X+X?) . . (13 is the energy density of the field at the point. Hxpressed in * Milner, Phil, Mag. Oct. 1920, p. 494. iGh Lorentz, “Theor y of Electrons,’ § 180. 414 Prof. 8. R. Milner: Does an Accelerated Cartesians, the flux becomes aan dy \ {Ea cos 0+ X)H+W Qary dex ¢ ak +{{7,Bsind. HW 27ry dy. ee (14) Here x and y are the co-ordinates of a point of constant y on the surface (5), which is bodily moving through space and at the same time altering its shape since & and 8 are functions of t. To effect the integration we substitute Melee Ov ov Ax ies (Ow On ow dé H ‘ae if . i —— - + ) 7 5 \OX dt \OE OW 0&/ dt etc., and use the transformation equations (3). On writing for E and H, X, W, and @ their values (4), (12), (13), and (9), and substituting for wp, a oy everywhere their values in terms of y and & obtained from (5), and then expressing by (1) & in terms of & and 6, (14) becomes a determinate function of y, which, integrated from 0 to 7, cives the net flux outwards from the boundary. On finally taking the limiting value when a is indefinitely small, it reduces to “Be e?(1 A {" (1 any) cos” x — IS sin? x) sin x dy Aak (1 —? sin? x)? _ Bee lB) singe 3ak Jo (L—? sin? ye The first term represents the flux corresponding to the electronic field E, H alone, the second the additional flux due to the superposition of the field X. On integration the net flux outwards becomes ‘ Lee ag), . | 6 ak The internal energy is given by (7), and its rate of decrease with the time, in the existing state of motion (1), is identical with the expression (15) for the net outward flux. It follows that the total energy of the system is conserved. As the electrons are being brought to rest, a centinual process of conversion of magnetic energy into electric is going on in the field, and at the same time field energy is being transferred into the nucleus, where it shows itself as — a= Hlectron necessarily Radiate Energy ? 415 the increase in the internal energy of the electron, When the electrons, having been brought to rest, begin to move away from each other, the stored internal energy comes out into the field, and at the same time the excess of electric energy which has accumulated in the field is reconverted into the magnetic torm. EEE | If we distinguish that part of W given by —_—_ as the T Hecos@. X irony tie <° mutual.” 79 Aar gn? energy densities, the equations (8) and (11) show that the total electronic energy w and momentum g fit into a mechanical scheme. In fact, since 3) “ electronic ’ and the 2) “external field,” dg ts dw —~ —Xe and — =—Xe. Bc dt dt es their rates of increase are identical with those which would be created by a mechanical force Xe acting on the moving nucleus. This shows that the system formally satisfies the Lorentz equation F=p(E+-[vil]), as applied to a point-charge e in a uniform electric field X *. There seem to be only two respects im which the solution fails to represent a real case of electronic motion. (1) It refers to a limiting case in which the electrons are concen- trated to a point and are moving in a correspondingly infinite field. (2) Extending over all space and time, it does not represent the initiation of the motion, and does not correspond substantially with a practical case, such as is given when an electron, passing through a hole in a charged plate, comes into an ultimately uniform field. Neither of these points, I imagine, has any significance with regard to the question at issue. (1) clearly does not affect the radiation from the system; and with regard to (2), electro- magnetic solutions, like those of other differential equations, in general only represent steady states under idealized conditions, and without reference to the manner of their initiation (cf. the solution for the uniformly moving electron). * It may be noted that the example does not reveal whether a term of the order e?/4é, expressing a mutual action of the charges, should be included, as, if present, it would be negligible compared with the infinite Xe. 416 Prof. S. R. Milner: Does an Accelerated In the light of this solution it seems desirable to examine with some care the proof of the presence of radiation as a necessary accompaniment to the accelerated motion of charges. The proof, as given oy Larmor in ‘ Alther and Matter’ (Chapter XLV.), and by Lorentz in ‘The Theory of Hlectrons’.(§39), is based on the solution in terms of retarded potentials for the field of a point-charge in a pre- scribed state of motion. This solution shows that the alterations in the field at any point can be considered as due to disturbances which emanate from the charge at each point of its path, and are propagated outwards with the velocity of light. Outside a moving boundary, marking the farthest points to which the disturbances have travelled, the previously existing field is unaffected by the motion of the charge. It is now found that in the resulting field, at a sufficiently great distance from the charge, there is an bul yard Wusd lot lencs gy proportional to the square of the acceleration, which clearly seems to represent an irreversible loss by radiation. Let us first test this general conciusion by applying the method to Schott’s solution, as originally limited by. the moving boundary. The point law of retarded potentials gives for the field of a point-charge moving arbitrarily * (B.—"){@R)+e—0} OO atualete ia Alien ORR H=([R,E|, where K= ( s =). C E, H are here the electric and magnetic forces which exist at the time t=7+ s at any point which is at a distance R from the point where the charge was at the timer. For simplicity take r=0, so that the disturbances considered are those emitted at the turning-point A, (4,0), of the charge, whence in the formula we shall have y=0 and V=Cyk parallel to 2. Then at the time t=R/c, and on the spherical surface of radius R described about A as centre, E and H will be given by sin i B=e(—[P+ 701), H=(REI, where @ is the angle between the axis of # and R, and Py, RB, * Schott, ‘ Electromagnetic Radiation,’ p. 23. Electron necessarily Radiate Lnergy ? ANT are unit vectors perpendicular to and parallel to R. This is consistent with (4) as it should be. -When R is so large that the second term is negligible compared with the first, E is perpendicular to R, and there isa flux of energy outwards Gale igh the surface of amount eras rer Gin Nines an, EH .27rR’sinada=,, 7 ee CLO) This agrees with the usual formula, and apparently verifies the presence of radiation in the solution. Nevertheless, an examination of the lines of energy-flow in figs. 1 and 2 shows that, as regards more than half of this flux—namely, that through the part of the surface on the positive side of the origin th ie energy concerned is not being radiated to infinity irreversibly at all, but is travelling along curved paths which in the course of time inevitably bring it back again to the nuclens of the electron ; and, in fact, it forms a ‘permanent part of the electronic field. The reason for this is that the proof of the existence of the flux (16) does not apply to any large sphere surrounding the point A, but only to a specified sphere at a given time. If we apply the calculation to a sphere of different radius R’ at the same time ¢, this new sphere is not centred on A, but on the position of the charge at the time t—R’/c. The instantaneous lines of ener oy-flow, which are normal to both spheres, are therefore necessarily curved lines, however far out they may be traced, and in thie ease their curvature is such as to keep the energy permanently in the electronic system. In the negative half of the field, the energy, although here also it is not travelling omirande indefinitely, but in curves related in a similar way to the image, is apparently true radiation, since, as is shown by the direction of the flux- lines, no energy ever comes back to the electron through the median plane. Nevertheless, a curious point may be noticed in this connexion. If we investigate the flux of energy from the field into the moving plane, tet 0. we find that itis invariably negative; consequently the boundary, considered as the limit of a thin transverse field separating the electronic field from zero, is always engaged in laying down field energy behind it as it advances, ‘and nev er, even at t= — %, in receiving any energy from the electron. The solution therefore premises an initial intrinsic energy in the boundary, apart from that of the electronic field. Now the field energy on the negative side of the median plane is clearly laid down by the boundary, since no energy ever 418 Does an Accelerated Hlectron Radiate Knergy 2 passes through the plane except at the moment when the boundary crosses it. We consequentiv see that the only radiation which the solution gives is, strictly speaking, not from the electron at all, but is to be attributed ultimately to the moving boundary which is postulated to be the limit of the field. The limitation of the solution bya moving boundary seems to be included, although perhaps tacitly, im the proof of radiation; but a comparison with the unlimited solution discussed in this paper raises the question whether the boundary is essential in every case for the representation of real motions. The arbitrary motion discussed in the proof makes the case considered there fictitious in the sense pre- viously explained; but this point is only material in that it makes it clear that an additional conception is involved in the complete proof, which must necessarily concern itself with a charge set in motion by an electromagnetic field. Lorentz’s equation is applied, and interpreted by assuming that the electron will move in the same way as it would do if it were a particle of given mass acted on by determinate mechanical force. The retarded potential solution corre- sponding to the resulting motion of the charge, superposed on the external field which gives the motion, then forms the complete solution of the problem, and the boundary is present in it as before. The question whether the boundary is necessary or not seems to be largely a question of the physical interpretation made of the point law and of Lorentz’s equation. The conclusion that it is necessary is based on the conception that the charges or nuclei of the electrons are first set into motion by the operation of the field in their immediate neighbourhood, and that the resulting changes in the field are actually propagated outwards from them. But it does not follow, from the mathematical fact that the changes in the field at a point are the same as if the disturbances were propagated from the charge, that the propagation is a physical fact. The field variations at a point can also be described in terms of the differential coefficients of the field at a point in a way which does not bring in the charge at all. Moreover, if we regard the nucleus of the electron from a mathematical standpoint as a small closed surface limiting the field and characterized by the constancy of the flux of force over it, once the field is known at all points, not only the field variations but also the motion of the electrons is uniquely determined. It seems from first — principles as logical to consider the motion of the charges On determining Frequencies of Lateral Vibration. 419 and the changes in the field to be two connected aspects of a single solution regarded as a whole, as to suppose that one is antecedent to the other. The adoption of this point of view would not affect the character of the field of being expressible in terms of retarded potentials, but it would permit other boundary conditions than those of a simple moving boundary to be applicable for the representation of real motions. The example which has been discussed in this paper seems to show, in one case at least, the legitimacy of a solution with less prescribed boundary conditions. October 10th, 1920. XXXVI. On a Graphical Method for determining the Fre- quencies of Lateral Vibration, or Whirling Speeds, for a Rod of Non-Unijorm Cross-Section. By R. V. SouTHWELL, M.A, Fellow and Lecturer of Trinity College, Cambridge™. [Plate VII.] r¥*HE determination of the normal modes and frequencies of lateral vibration for a rod of varying cross-section is a problem which has attracted the attention of many elasticians, and several papers on the subject have appeared in this Magazine+. It would seem that a complete solution, * Communicated by the Author. + References to the investigations of Bernouilli, Kuler, Kirchhoff, Sturm, Liouville, and others will be found in Lord Rayleigh’s ‘ Theory of Sound,’ or in the ‘ Dynamical Theory of Sound’ of H. Lamb. The following is believed to be a fairly complete list of recent papers bearing on the problem :— W.J.M. Rankine, ‘ The Engineer,’ vol. xxvii. (1869), p. 219. S. Dunkerley, Phil. Trans. Roy. Soc. (A), vol. clxxxv. (1894), p. 279. C. Chree, Phil. Mag., vol. vi. (1904), p. 504, and vol. ix. (1905), . 132. | C. ACB. Garrett, Phil. Mag., vol. viii. (1904), p. 581. J. Morrow, Phil. Mag., vol. x. (1905), p. 118; vol. xi. (1906), p. 354; and vol. xii. (1907), p. 233. A. Morley, ‘ Engineering,’ July 30th and August 13th, 1909. P. F. Ward, Phil. Mag., vol. xxv. (1918), p. 85. A. Fage, ‘ Engineering,’ July 20th, 1917. J, W. Nicholson, Proc. Roy. Soc. (A), vol. xciii. (1917), p. 506. H. H. Jeffcott, Proc. Roy. Soc. (A), vol. xcv. (1918), p. 106. W. L. Cowley and H. Levy, Advisory Committee for Aeronautics, Rh. & M. 485 (1918). J. Morris, Advisory Committee for Aeronautics, R. & M. 551 (1918). G. Greenhill, Advisory Committee for Aeronautics, KR. & M. 560 (1918). F. B. Pidduck, Lond. Math. Soc. Proc., vol. xviii. (1920), p. 893. 420 Mr. R. V. Southwell on a Graphical Method applicable to rods of any given shape, is beyond the range of exact analysis: at all events, mathematical investigations have in general been restricted to rods of fairly simple form, the case most frequently discussed being that of a rod with one free and one clamped end, in which the flexural rigidity at any section varies as some power of the distance from the free end. If it be conceded that the interest of this problem lies principally in the quantitative results obtained, then a reasonably accurate method of solution which is unre- stricted in its application would appear to possess advantages over any isolated analytical solution, however rigorous. No great mathematical interest can attach to exact results when these are based upon a theory (of thin rods) which ad- mittedly is only an approximation to the truth, and a slight decrease in accuracy will be more than compensated by ability to take account of any specified end conditions, and to deal with rods of which the cross-sections oy in any specified way, or with continuous rods supported at several points. ‘This is more especially true in relation to the engi- neering applications of the theory. The “ whirling speeds” of a rotating shaft, as has frequently been pointed out”, are identical with its natural frequencies of vibration, and in many cases in which their values are of practical importance the variation of flexural rigidity along the length of the shaft is incapable of expression in mathematical form, whilst the bearings impose constraint upon the direction of the centrai line at two or more points : no great accuracy is required in the result, but it is essential that the method employed shall not break down in any particular instance. by reason of purely mathematical difficulties. The graphical processes now to be described constitute a simple extension of methods which I have recently pro- pounded f for finding the critical load of a strut of varying cross-section. ‘They seem preferable, as regards ease and quickness of application, to any general method of solution which is based upon the use of infinite series {; and the -accuracy with which they reproduce the results of exact calculation in examples which can be treated analytically * Cf, e.g. C. Chree, Phil. Mag. vol. vii. (1904) p. 504; the ex- pressions obtained in this paper for “the frequencies of loa ‘vibration in a rotating shaft have, however, been shown by Pidduck (Joc. cect.) to be erroneous. I hope shortly to publish a paper dealing with this point at greater length. iene Aircraft Engineering,’ April 1920, pp. 113-114. t Cf.,e.g., the discussions of W. L. Cow ley and H. RY and of J. Morris s, quoted above. jor pee Frequencies of Lateral Vibration. 421 suggest th: ate draughtsmanship, and assuming the validity of ane theory of thin rods—errors will not ex- ceed 1 or 2 per cent. in any practical problem. Stated mathematically, our object is to determine the forms of certain curves of deflexion (the “normal modes” of vibration) which are associated with any given shape of rod. These curves form a tamily which is defined by a certain differential equation, and they are subject to certain “conditions of constraint” (expressing the effects ot clamps, journals, etc., in the system under consideration), which have the common feature that they leave the magnitude of the deflexion unrestricted, whilst defining the manner of its variation along the length of the rod. In the problem of ‘ whirling,” each curve of deflexion has the property that it can just be maintained, when the shaft is rotating at some definite and appropriate speed, by the centrifugal forces acting against the elastic restoring forces : in the vibrating bar, the instantaneous deflexion at every section of the rod is compounded of one or more “normal modes of vibration,” any one of which may be regarded as a curve of deflexion which varies, in absolute magnitude, as some appropriate simple harmonic function of time. The differential equation which governs the curve of deflexion may be obtained as follows :— Let p be the line-density of the rod at any section, in pounds per foot run, (=H) be the flexural rigidity at the same section, in pound- (foot)? units, and « be the distance of the section from one end of the rod, measured in feet (so that p and % vary in some specified manner with x) ; and let y be the instantaneous deflexion of the shaft, in feet, at the section 2, and n bethe number of vibrations—or of revolutions, in the problem of the whirling shaft—per second (so that our problem is to determine the value of n and the manner in which y varies with 2): then the elastic restoring couple M, which acts at any section of the deflected shalt, is given by the usual equation M+B5 5) ORs Si iokhi sat) Aaya Ne) Phil, Mag. 8. 6. Vol. 41. No. 243. March 1921. 2 F 499 Mr. R. V. Southwell on a Graphical Method and its restoring effect is equivalent to an intensity of latera] loading w which is a by the equation — a2 the direction ant w being opposite to that of the deflexion (and so tending to restore the straight configuration), and its intensity, as given by (2), being measured in pounds per foot run. In the vibrating bar, this ‘‘ effective lateral loading ” produces the required acceleration of the bar towards the central position: in the “whirling” shaft it produces the normal acceleration towards the axis of rotation. In either case, the magnitude of the acceleration is given (in foot-second units) by the equation a=47r'n’y, «woh Sei (3) the instantaneous deflexion y being a function of x only in the case of the whirling shaft, and represented in the case of the vibrating bar by the expression y= sin 27, |...) ee wp =O es where Y is a function of x only. We have therefore, in both problems, the equation pa An’n’o LSS = —— Mees ES ae (5) and by eliminating w and M from (1), (2), and (5) we obtain the differential equation An’n’p 0. (anew Gl? bow ABS which reduces to @(@Y\ 4n’n2p aAS aa) k=O, |) both when y=Y simply, and when y is given by (4). Hquation (6) governs Y as a function of x, and thus defines the curve of deflexion. The arbitrary constants in its complete solution are determined in any particular instance by the special conditions which define the constraints: thus, at an end which is “simply supported” (as by a swivelling bearing, in the problem of the whirling shaft) we have g=0; -M=0:.-). 2. ee for determining Frequencies of Lateral Vibration. 423 at a completely free end (where the resultant shear, as well as the resuitant bending-moment on the section, is zero) we have OM eae and at an end which is “clamped” (as by a fixed bearing in the problem of the whirling shaft) we have fe) y=0; 5 =O: Tdi e (G9) It will be convenient at this point to introduce a new quantity M, defined by the equation PY ee Gl) M +3 —so that M is equal to M, simply, in the problem of the whirling shaft, and to M sin 27n¢ in the problem of the vi-- brating bar. Then we can express the foregoing conditions of constraint in terms of Y and M, as follows :— ia asimply supported end, Y=M> =0); at a completely free end, Wee - =(); Ta) and at a clamped end, Y= = () Corresponding relations can be written down to represent the conditions imposed by other types of constraint, e. g., the effects of large concentrated masses, spinning masses which introduce gyroscopic couples, ete. For present purposes it is only necessary to add that the existence of ‘“ simple supports” at intermediate sections of a continuous rod requires that the values of Y, of a and of M shall be continuous at such sections. Whatever be the specitied conditions in the problem under consideration, it will be found that a solution of (6) is ob- tainable by graphical methods, when we have assumed any value for the frequency N, which satisfies all but one of them ; but that they cannot all be satisfied simultaneously unless n has certain definite values, known as the “natural frequencies ” of whirling or of vibration. It is these frequencies with which we are primarily concerned in practice, and the 25 2 424 Mr. R. V. Southwell on a Graphical Method method of determination which is now te be described pro- ceeds by constructing a curve of deflexion which satisfies equation (6), and by modifying this (on “trial-and-error ” principles) until it satisfies all the constraint conditions simultaneously : to each modification corresponds a change in the frequency n for which (6: is satisfied, and thus we arrive finally at the required solution. | Our graphical construction employs the principle of the “funicular polygon.” We construct two diagrams (figs. 1 and 3), the first representing the deflexions Y, and the second representing the corresponding bending-moments M, defined by equation (10). Hither end of the rod may be taken as a starting-point, and constraint conditicns of the type (11) will determine the values at this end of some pair dM 3 ie ay, of the quantities Y, ee? M and —— . By assuming values du for the remaining two quantities, we are enabled to proceed with the construction of the curves, making use for this purpose of equation (10) above, and of equation (6) re- written in the equivalent form 2 gee =Q.)) 02/0. 020 re da? where the symbol W is substituted for g/4a°n’p, a positive quantity of dimensions [LJ [T]?/[ Mj, which varies in some. specified manner with @. In the engineering problem at any rate, it will often be desirable to vary the length of our diagrams, and we must therefore give careful consideration to the question of scale. Let us for the moment replace « by the non-— dimensional variable z, defined by the equation wale, in which J denotes the distance between any specified pair of sections (for example, two adjoming sections of con- straint). Then equation (6) may be rewritten in the form desir i) Ne (3 Gey Pa 52 whence it is evident that the natural frequencies will be obtainable from expressions of the type =0, . 3 rin \ Ann? i So g a lap ype ee (14), —po, Bo and K, denoting the values of p, 53 and K at some a Oe ls ie i for determining Frequencies of Lateral Vibration. 425 specitied section, and % being a non-dimensional coefiicient. From this point of view, our problem is to determine pos- sible values for A, and we are free to change our scale of length in any convenient manner, whilst the results will be applicable to a whole family of rods, differing in actual length, and specified simply by the variation of the ratios 3/2, and p/py with the ratio w/!: the natural frequencies for different members of the family will vary inversely as the square of the length of the rod. in the light of these remarks we may now proceed to de- velop our construction. In fig. | (Pi. VII.) we choose axes of Y and 2,as shown, and take OA to represent /, the length of the rod (or the distance from O to the next section of con- straint, if the red is continuous); and we draw vertical lines to divide OA at the points 1, 2, 3,.... into any convenient number, n, of equal parts (n=10 in the diagram shown). The values of 33 are assumed to be known at the points of subdivision, and lengths bj, b,.... are taken, as shown in Hier proportional.to:3,,3,,--).. etc. Similarly, in fig. 3 we choose axes O’m and O'e# for a curve of which the ordi- nates M are to be made proportional to M, and continue the vertical subdividing lines; and in fig. 4 we take lengths ky, ky,.... etc. proportional to the values oF K which corre- spond to the points of subdivision. It should be noted that the actual scale on which M represents M is for the moment left undetermined. The construction then proceeds as follows :—Oa and O'a' are drawn to represent the values of Y, M, and = me at the end «=0 (figs. 1-4, Pl. VII. are drawn for an example in which it is specified that both Y and M vanish at this end: the slopes of Oa and O/a' are for the moment immaterial) ; and the polar diagrams (figs. 2 and 4) are begun by drawing pO, parallel to Oa and p Ox parallel to O'a’, the points 0, and QO,’ being distant by amounts b, and ie respectively from the verticals through p and p.. We then make pq in fig. 2 equal to a'l’ in fie. 3,and p’q’ in fig. 4 equal to al in fig. 1; join gO, and q'O,’, producing them to O, and O,! as shown, and continue figs. 1 and 3 by drawing ad and a’b! parallel to gO, and g’O,/ respectively*. The processes just described are repeated at ev ery section of subdivision: viz., gr is made equal to b'2' and g'r’ to 62, be and b’c' are drawn parallel to Ogr and O,’7" respectively, * Dashes have been accidentally omitted from the letters # in fig. 3 and O,, O2,... etc. in fig. 4. 426 Mr. R. V. Southwell on a Graphical Method and so on: it is not easy to explain the method in few words, but its principles should be easy to grasp, if the reader will construct a diagram for himself by means of the fore- going directions. We have now to show that the curves which would be obtained by drawing continuous lines. (10) and (12). Considering, for example, the portion alt of fig. 1, we see that the tangents at the middle points of the ares ab and be will, if these ares are small, be very nearly parallel to the chords ab, bc: hence, to a first approxima- 2 tion, the value of _ expression : OA oy aa Fea = (slope of bc) —(slope of ab), = (slope of rO,) —(slope of qOs), by construction ; at the point 6 will be given by the — - TOY UIs 740 2 m e ae = ——’, since gr=b'2’, by construction. Dy ( 2 A similar relation obtains at every section, whence it is evident that we may drop the suffixes, and write GPE s06l = a e e e . e 15 n dx? 7 b 0 (2) (since OA in fig. i is the quantity denoted by Z) ; and by a similar investigation we see that the ordinates m of a smooth curve drawn through O'a’b’c’.... in tig. 3 would satisfy the equation am a som ou) jis . oe . (16), Eliminating m from (15) and (16), we obtain the relation Ce CRS ON a ie SG ia Te dis and comparing this with (6) we can see that the ordinates. of fig. 1 (PI. VIL.) will satisty the differential equation of our -). | for determining Frequencies of Lateral Vibration. 427 problem if the scales of the diagrams are such that bo n’Ko 3, ral or—by (14)—if me) he lene a Se oe bo and ky being the actual lengths in figs. 2 and 4 which represent YB) and Ko, the values of 33 and K at the specitied section. Equation (18) therefore gives us the value of A—and so, by (14), of the frequency n—for which the ordinates of fig. 1 represent the deflexions Y. Comparing (15) with (10), we see further that the ordinates m of fig. 3 will represent the corresponding values of M on a scale given by ie eee) 100 NET A 6) cet this last result, however, has little practical importance. Having thus shown that the ordinates Y of tig. 1, for some value of Nl, satisfy the fundamental equation (6), we have now to consider how far the conditions of constraint at the section A may be satisfied : it is evident that we shall not in general satisfy either of them at our first attempt. For purposes of illustration, we shall assume that the specified conditions are that Y and M also vanish at A; 7. e., that A is the other end of the rod, and is simply supported. At the beginning of the instructions given above for constructing figs. 1 and 3, it was stated that Oa and O’a' could be drawn at any convenient angles, their actual slopes being im- material ; if we now change the slope of one of these lines, und repeat the construction, we shall obtain different values for Y and M at the end A. Then by ‘synthesis of the two solutions thus obtained—z. e., by writing Na, +aY, \ (19) and M=M, + «M.,, where Y, and M, correspond to the first, and Y, and M, to the second solution,—and by suitably choosing the con- stant a, we can evidently obtain a third solution of the fundamental equations which will satisfy both of the im- posed conditions at the end O, and one of the imposed conditions (say, the condition M=0) at the end A. The 428 Mr. R. V. Southwell on a Graphical Method second condition will not also be satisfied, unless by a lucky accident, because we shall not in general awe assumed for the frequency Nn one of its possible values. It is at this point that we have to introduce “ trial-and- error”? methods. Equation (6) shows that the curvature of figs. 1 and 3 will be increased by an increase in the value of. n, and hence we can see that if the result of the synthesis ee by (19) has been to make Y positive throughout the span OA when « is so chosen that M vanishes at the end A, then a higher value of n must be assumed in our second attempt. The necessary procedure, in the case ofa rod constrained in any specified way at two sections only, should now be obvious :—Assuming any value of 1, we can obtain two graphical solutions of the differential equations, each of which satisfies the imposed conditions at the starting end : by combining these solutions, we obtain a solution satis- fying all but one of the imposed agntitions of constraint, and we can calculate the “error,” or amount by which it fails to satisfy this last condition (e. g-» In the exe = just con- sidered, the “error” is the value of Y at the end A, since in the correct solution Y,=0): repeating the process for a secund assumed value of n, we obtain a second resultant error, and by plotting the ‘“‘error”’ against N, as in fig. 5, we can construct a curve which enables us very quickly to make a correct guess at the required value of n. In practice, labour may be saved by making the length scale Ale variable quantity in our successive “attempts at constructing the correct diagrams which correspond to figs. 1 and 3:. the leneths h;, bo, .... ete. in fig. 2 and lng kee ete. in fig. 4 are then a eanamed once for all, and only the spacing of the equidistant vertical lines in figs. 1 and 3 needs to be varied. Hquation (18) shows that taking these lines further apart is equivalent to assuming a greater value for (or N), and hence we can conveniently nroceed by plotting the resultant error against the assumed value of AB (or of lL), instead of against nN as in fig. 5; then, having determined by interpolation the value of 2 which will result in zero error, we can calculate the corresponding value of n directly by means ot (14) and (18). It remains to discuss the procedure which is required for dealing with continuous rods or shafts. We have seen that at sections where these are simply supported there must be continuity in the values of Y, a and M, but not necessarily d in the value of ——, since the resultant shear will in general dx — for determining Frequencies of Lateral Vibration. 429 have values differing by a finite quantity on opposite sides of the support. If therefore we are given that A is a section of support, but no longer an end section of the rod, and that Y vanishes at A, then this condition ean be realized by combining two solutions as before, and we obtain definite values of aY and of M wherewith to begin the portion Ax of the curve which lies to the right of A. In our first attempt at determining the curves for the next unsupported span we may assume that does not undergo a change d da of value at A: if, then, at our second attempt we start from A, i UNG : d taking zero values for Y, daa and M, but a finite slope ae QA’ Ak in fig. 8, and proceed for the span on the right of A by the methods which have been described above, the -eurves so obtained can be combined in any proportion with our previously determined curves for the double span, and thus we can satisfy the conditions at the next point of support. It is not until we reach the other terminal section that trial-and-error methods (in which N is varied) have to be introduced. The labour is naturally increased by more frequent supports, but need not be regarded as prohibitive. The diagrams (Pl. VII.) which accompany this paper have been prepared from actual drawings, wherein the foregoing construction was applied to a rod of variable cross-section for which the mathematical solution was known. It is easily verified that if 33 and p are given by the expressions sin ne ] l Slee 228 it TP eee ee | / oO diel: sin l | Sy PR AND) ee 372 10 gee Rin and p= iy Po 1+ She \eak ei on —so that 3, and po are terminal values,—then equation (6) will be satisfied by assuming that ) Na 1988 La Qe Ya (sin Sia i: l 27 hee 430 Mr. R. V. Southwell on a Graphical Method provided that 997? Bo a 20n*l* 9 idee whence we obtain N= 9/5 X w*=175°36 approximately. To compare with this result, a solution was obtained graphically. The length of the rod was divided into ten parts, and after values of 3 and p had been calculated for the points of sub-division, suitable lengths were chosen OI, Dyula We meena etc., and k,, ky, k3, .... etc., aS Shown 1m the table below * :— TaBpiEe 0: HE = 0 0-1 0-2 0:3 0-4 0-5 S$ -— 639 682 831 1134 1611 1920: p ‘= 668 676 - -700 732 7595 ‘770 K(x l1/o) oe 1-498 1-489 1-429 1366 1317 46299. k(ininches)= 5992 5920 5716 5464 5268 5196 b (in inches) = 3195 3-41 4-155 567 8055 —«- 960 A value for > was then guessed roughly, by estimating mean values of Y and p, and thence calculating the whirling speed from the ordinary formula. Substituting this value for X in (18), it was found that the corresponding assump- tion for the length OA in the diagrams would he about 5°71 inches, and this length was adopted at the first applica- tion of the construction. Figs. 1 & 3 were then constructed with initial slopes of 30°, as shown, for a first attempt, while in the second attempt fig. 1 was started with zero slope, and fig. 3 again with a slope of 30°: this second attempt gave diagrams of which the general nature is indicated in the left-hand bottom corner ot the Plate. The purpose of starting tig. 1 with zero slope in the second attempt was to enable the boundary conditions for M to be more easily satisfied by synthesis, without altering the initial slope of the corre- sponding Y-diagram : it is clear that the error will be pro- portional to the initial slope, and hence, when constructing * The values here given for %, and p, are representative of a steel shaft of diameter 3 in. at its terminal sections. For finding A, only the relative values of 3 and p at different sections are, of course, material. for deternining Frequencies of Lateral Vibration. 431 the “error diagram” (PI. VII. fig. 5) it is necessary to have all errors referred to some standard value for this slope. The diagrams illustrate an obvious simplification which can be made when, as in the present instance, a shaft is symmetrical about its centre. It is only necessary to continue the construction as far as thie first point of sub- division past the centre, since the slope of the lines df and d'7’ gives a measure of the error involved : obviously, if the complete diagrams had zero error they would be symmetrical about their centres, and so df and d’7" would be horizontal. It is convenient to take as a definite measure of the “ error ’ the total rise of these lines over the horizontal distance ae so that when the diagrams obtained in the first and oe attempts have been combined in such a way that d'/”’ horizontal, the total rise of df over this length may be eee as the ‘error,’ and plotted against the assumed length of OA. In the first application of the construction (length OA taken as 5°71 inches) the line df was found, after correcting the bending-moment diagram, to fall by 0-044 inch over the total length. A fall indicates that the figure assumed for l (or N) was too small, and hence at the second application a length of 6 inches was assumed : this gave an error of the opposite sign and of amount 0:28 inch. Plotting the corresponding points on the error diagram (fig. 5) and joining them by a straight line, a length of 5°75 inches was obtained as the estimated length corr esponding to zero error. This was tried, and an error of the same sign as in the first attempt was obtained, the amount of the error being now reduced to 0:004 inch. Plotting this as a third point in the error diagram, and estimating again, the value of / for zero error was given as 5°755 inches, and the corresponding value of 7% was obtained from equation (18) as 173. The correct answer, as shown above, is 175°4, approximately, and hence the method has resulted in this instance in an error of 2°4 in 175—slightly under 1:4 per cent. From (14) it follows that the whirling speed N is given correctly to well within 1 per cent., and it is clear that prac- tically as accurate a solution could have been obtained by dispensing with the third application of the construction, and accepting tle estimated value of 5°75 inches for /. Trinity College, Cambridge, June 4th, ‘19: 20. ge ya XXXVIT. On Systems with Propagated Coupling. By ALFRED -W. Porter, .D.Se., F.RS., Fdnst Pavan REGINALD EH. Gisss, B.Sc., A.lInst.P.* N the majority of electrically coupled circuits, the two circuits are so close together that the time of propa- gation of the mutual action can be neglected. For great distances of separation or for very rapid frequencies this would no longer be true. When the coupling is of a meclianical nature “oread of being electrical), and the actions are propagated with the velocity of sound, the entry of time would have to be considered for even quite small frequencies at moderate distances. For example, with a frequency of 1000 per second, through air, even 30 em. separation would change the phase of action through a complete period. A fairly simple case in which acoustic coupling comes into play as mentioned above, is that of a microphone transmitter and a telephone receiver ‘coupled electrically through a valve set, and simultaneously acoustically through the intervening air. It is well known that a transmitter and receiver can maintain each other in vibration as in the case of the singing telephone ; the introduction of the valve set is merely to amplify the action and allow of its study at greater distances of separation. To maintain continuous oscillations in any system, definite relations must exist between the phases of the forced and forcing systems, as is often stated by saying that they must be “in ee ? with one another. In the case under consider- ation the electric coupling in the valve is constant, whilst the acoustic coupling between the transmitter and receiver is a variable quantity dependent on their separation. Thus at some distances this ‘‘ phase condition ” will be realized, at others it will not, so that a periodic variation of intensity with separation is to be expected, while experimentally it is found that if the receiver be moved gradually away from the transmitter in any direction, a series of maxima and minima of mutual action can be obtained. The whole of the room in which the experiments are con- ducted can be mapped out in this way into surface-loci of maxima and minima alternating with one another every quarter wave-length. ‘The first impression, that these surfaces represent nodal and ventral surfaces of a system of stationary waves, cannot be true, for the conditions were not suited * Communicated by the Authors. On Systems with Propagated Coupling. 433 to the creation of such waves. Experimentally it was found easy to follow the variations for separations of over twelve ards, whilst diffraction effects could be illustrated using a plane ‘edge (a door) or by reflexion from, or transmission through suitable zone plates. | The system used was too complicated to deal with satisfac- torily when treated mathematically, but other simpler cases. can be considered in illustration. Case (i.) below is given merely as an illustration of the propagation of mutual induc- tion ; but no importance in practice can be attributed to it.. Gi.) Electrical transformer allowing for propagation. Assuming the two circuits to be separated by a distance wand an alternating e.m.f. in the primary, using the ordinary notation, we have Ly +R \G + Mee? = Boi, dt at Meier of (Ie 5 Seay 2)C. ==(()), dt Gh 2 The variation of the phase of the mutual action with z is. adequately expressed by the introduction of the factor: exp(—7q@). Hence ‘a : . é MEA gs my ak aarieakle ah Ne ipt. | Lup + Ri + +e loo Ke Let M?p?[R, cos 2gv—Lyp sin 2ga| | eke , 9 5 R =R,+ R2+ Len ) YT,’ ede wey Meg [ Ro sin 2 age cee oP eos 2g Rk? at L,? Moe 2 Lp Lip Lp tan &= By tan Ah ae : ee ae | donee tl pea 1en C, R’ af Lip _ Heos (pt—a) , Wy ees Sy be ae ; The denominator passes through maxima and minima as x increases. ‘The value of M itself decreases with increase of distance. At near points the change through the distance of a wave-length is very great, but at distant points its variation produces a small effect compared with that due to 434 Prof. A. W. Porter and Mr. R. E. Gibbs on change of phase. Attention may therefore at first be con- -centrated on the latter change alone. Differentiating R’?+ Lp? with regard to «, one obtains finally 29@ =a, +a,+Par as the condition for maxima and minima: the maxima are separated by half a wave-length and the minima are midway ‘between them. ies! The above result can be shown diagrammatically as a vector diagram (fig. 1). The triangle A represents the operation ber +R, the triangle B the operation ie + Ra, whilst C is an allowance for the mutual action. The maxima an! minima of Y occur when M’p? (which may tirst be considered constant) is in line with X, 7.e. when Qqu=a,t+ a4+Pr. If M is not constant, then with centre O a helix (shown dotted) is constructed such that its radius vector represents M?p? for each value of z Y is a maximum now, for those values of 2 for which it cuts the helix perpendicularly. (ii.) Simple Dynamical case illustrating maintenance of ‘vibrations. Whereas in the previous case the frequency was fixed, in that of the reacting telephones it is not so: the problem, in fact, becomes one of the stability of chance vibrations that may be excited. Such problems have been discussed by the late Lord Rayleigh who, taking an ordinary dynamic equation of the second order Oy OU aa ae Wert et 7 inquired the effect of disturbing terms arising in phase with either y or dy . In order that a small disturbance if dt Systems with Propagated Coupling. 435 excited shall increase with time the, resulting coefficient r must be negative ; the motion will then inerease until the equations (which ‘are only approximate ) no longer hold good and would need to be modified. A much more com- plete treatment is to take the disturbance as itself arising from a mechanical system for which the full equations can be written down and then to solve the equations for the two systems simultaneously. As an’ example, consider a telephone membrane placed at an adjustable dis- tance z from the coil which is connected to a battery. The current at any instant is C, and the displacement of the mem- brane y. An alteration of C will alter 7 ae y and vice versa. If a chance alteration is produced, will it persist ? The equations including a mutual action proportional to di the current and to 7 are, a igs 2: d CUTE Caen x aC+(m Fa a + K ly =0, which on elimination of y gives {f me + (rL + Rm) oe +(KL+ ey +RK \ C=EK ’ at? at? at i Putting d E , R 1 K ae f — SSO — =0 — =C = = dt aR aR Patsy ian am ; the auxiliary equation is B+ (a+b + (ct+ab—f)E + be=0, or B+o92+hE+j=0, where g and j are essentially positive, whilst may be either positive or negative, If the roots are &,, &, &, then £+&+&=—g9, ££ + &€,+ &8=h, E,f.€5= —). 436 Prof. A. W. Porter and Mr. BR. E. Gibbs en We are only concerned with the case of two roots complex, say &=A-+ iB, £.=A—iB, Hence & $i 2A= =>. eee A? +B 4+ 2A8 =h, 9... ese : a) ae eo (iil.) From (iil.) & is of the opposite sign to 7 and is therefore essentially negative. €=—P,, 2A is positive when P,>g. From (i.) A passes through zero when €,=—g, and at the same time B’?=h from (ii.), thus / is then positive. Further, &,B? is then equalto —7 from (iii.) and therefore bo J =p). d) If A>0, £, is more negative than —g. Interms of a and b, for A to be positive Tae —&=P\>a +h> x + i . It follows that if A=0, &=-—g, and j=gh, so that his positive. Hence the solution for the current is, C= i + Fe-#+G cos (Bt—n) R where b= 4) = Vlg « Also 2% 4 Fle-% + G! cos (Ba! VEST nrg Cet cos ( t—7). : Rie ak he eelucioh has a KYL Kjm Lae kK ‘ | (Te ey Hence a disturbance such as in the above critical case if once excited will be maintained constant. It will be noticed. that ah is the frequency-constant of the free undamped Systems with Propagated Coupling. 437 vibrations of the membrane, say vo, and therefore Vi pe a ————— ee rl» Rm = 1p) pees! 1 4 where ¢.=time-constant for electrical circuit, Sew arate = 55 membrane ; in each case when isolated from the other vibrating systems. It €, is more negative than (—g) the vibration will grow, whilst the equation for y is then Pe + T'e-Gt+Pt+ Gle2 cos (Bit- 7’), Ra ae) RK where P=positive quantity. The value of B’ is given by hence for a small positive value of A it is less than j/g and will always be less than the free frequency of the membrane. For all positive values of A, y will increase up to a value at which the assumed equations are no longer valid owing to the higher order terms (which have been ignored) acquiring increasing importance and ultimately controlling the maxi- mum attainable value. So far the time of propagation of the mutual action has been taken as zero ; the change necessary to allow for it is, as in the previous example, to introduce into the value of a aytaeton en”: The auxiliary equation becomes mL& + (rb+Rm)&+ (KL+rR—a%e 2" )E+ RK =0, or B42 +hE+j=0, where h=r+ip KL y= 2e0s2qa and ee KL +a ee ed ee a? sin 2qw nel The critical case for which maintenance will begin is “Phil. Mag. S. 6. Wolk oe No. Poh Mareh 1921. 2 G 438 Prof. A. W. Porter and Mr. R. E. Gibbs on that for which two of the roots are given by +ip. This condition gives the two equations p=, gp’ + wp —j=9, whence eliminating p, BO | a se Vf Whatever value of w satisfies this equation, other values will also satisfy it, provided that 2ga is greater than at first by 2c times an integer, at least for points so distant that a® can be treated as constant. pP=rA=)/9—(u/g) V2- But in any usual case 2 and therefore w will be very small compared with KL+7rR, so that P&=jl9—(ulg) Vlg. 2 LL =I —— iG 9g where the second term on the right is small ; and a? : Se SI AED mi and therefore p” is subjected to a small harmonic variation (dependent on 2) about its average value nj’. overt The equations become somewhat modified bit at the same time more symmetrical if the electrical circuit possesses a capacity S. FKiach of the systems is now “ elastic’ and possesses a free vibratory period of its own. The equations are, for the circuit EueeE ay (LE. tRS+ sot 9! = i, and for the membrane as (metre +K)y =). leading to the auxiliary equation (Le + Res < ) (met +r E+ K) otf? = =). q a Systems with Propagated Coupling. 439 If any solution of this is a pure imaginary zp, then (- Lp? + = 5) Came" +K)—rRp?+ «?p?=0, ov Rie (Lp + 5)=0, whence for the critical case at which self-maintenance just begins — mp 74K = tp [ah —¥ sor with the condition ~ 7 fore one cf the conditions for maintenance. >r if pis to be real, which is there- Thus p? differs from a by a quantity proportional to mM p which is in practice small. If the factor of p is denoted by F, then os ! a Me ae ine ae i Gee mv) 2a It should be noted that if «? is sufficiently small it is impossible for self-maintained oscillations to exist. As before, to pass to the case of propagation a? is replaced by ae%qr, which (if «? is eliminated instead of ( — Ly? + a) as before) leads to the equation p sin 2g i= Lip? + >) (—mp?+ Kk) — rRp? | =p" cos 2qu | R¢ — mp" + K) 4-9 (—Lp' tr a ) It would be unprofitable to consider this fully ; special cases can be worked out when desired. Gi.) ‘lhe above cases differ widely from the actual con- ditions of the experiments referred to in the first paragraph. They are given simply to illustrate a general mode of examining the maintenance of vibrations, both with and 2.G2 440 On Systems with Propagated Coupling. without allowance for the influence of time of propagation upon the coupling. : The experimental case-—except for the presence of the valves—can be represented by the diagram following (fig. 3). Fig. 3. In this diagram M is the mutual inductance between the electric circuits, a; and «, are the mutual coefficients repre- senting the magnetic coupling between the coil and diaphragm in the respective telephones, and yw the corresponding coefh- cient for the acoustic coupling between the diaphragms. The equations are a i dC, dy, Res, +R, |C+M dt ae dt me d eee: Ayo | Lae - R,| CaM ai eames i? dl i ai aC; 1 | om a ate Ne is Ki |yte on =), d? if i dy ate + | ms an LS af Ke lyta— =0, leading to an auxiliary equation of the sixth degree. This case is too complicated to justify further account being given of it here. | The general nature of the effect of propagation is sufh- ciently well represented by case 11. oe XXXVI. On Vapour Pressures and the Isothermals of Vapours. By J. H. Saaxsy, B.Se., Lecturer in Physics and Director of the Viriamu dienes Research Laboratory, University College, Cardiff *. a LARGE number of empirical formulee have been proposed to express the vapour pressure of a fluid in terms of the temperature and certain “ constants” for the substance in question. Many of them, e. g. those of Dalton, Roche, Biot, Kirchhoff, and van der Waals, are of logarithmic or exponential form. This at once recalls Dieterici’s Equation of State ea ee Se oRD o—b ‘ Pe in which p is the external pressure upon a fluid of specific volume v at absolute temperature T, R being the gas con- stant and A and ¢$ constants for the particular substance. In obtaining this equation Dieterici makes use of a quantity A’, the work done per unit mass against the forces of cohesion in removing molecules from the fluid ; A’ is pro- portional to the density of the fluid and is replaced by oa On similar lines, let us examine the relation between the total internal pressure, II,, in the liquid phase of any sub- stance, and that, [!,, in its vapour phase. ‘The work done in the transfer of unit mass from | liquid to vapour may be equated, following Dieterici, to twice the kinetic energy lost during this peaneracmi cio of high speeds (of the “liquid molecules which escape) into average speeds of these same molecules on their arrival in the vapour. This loss of kinetic energy per unit mass is 3s?, where s is the critical value of the component normally towards the surface of the velocity of a liquid molecule, 7. ¢. the smallest value com- N, SN, y where N,, N, are the numbers of molecules per unit volume in liquid and vapour respectively. Thus, in the case of a substance which is not dissociated either in liquid or in vapour form, patible with escape into the vapour. Now, s?=2RT log 2 = 2RT loge. ty” where d,=density of liquid, d, of vapour. * Communicated by Prof. A. W. Porter, F.R.S 442 Mr. J. H. Shaxby on Vapour Pressures s2 2 Thus MWe Rr =0,% : ay But a fundamental relation connecting II, and H, is II,(v, —6) = RT =I, (2, — b), where v, and vz are the volumes of unit mass in the liquid and in the vapour. So 1_, dy Ui D dy TE Vo—b af , : ee whence = - dy +d, The quantity (dj+d,) diminishes linearly, and fairly slowly, with rise of temperature : this fact is well known and is embodied in the so-called Cailletet-Muathias rule. The rule is nearly exact for substances which are not associated, but does not hold for such substances as water, for which the changes in (d,+d,.) areless regular. [For most substances the rate of decrease of (d,+d,) is of the order of ‘001 per 1° C. Hence 0 is not a true constant, and this is known to be the case from comparison of Dieterici’s equation with experimental data. We find, then, that Oi ae acl ee Ca dee die v% dtd,’ “ +—b d b ds Thil U2 =—= = i — = aa U9 dy =F do and Ug ditd,. That is to say, the “free” space per unit volume in a liquid at any temperature is equal to the “ occupied” space per unit volume in its saturated vapour, and conversely. Equal volumes of liquid and vapour are, as it were, comple- mentary in this respect. § 2. We may write Dieterici’s equation in the form piv—b) se ical Thus for a liquid and its saturated vapour at temperature T, if p is the saturation pressure, ee p(ty—4) = =e Te RT, RT and the Isothermals of Vapours. 443 Hence (oy x Cee) RT RI | pyr _ (ano Or (Fy) ay CF wi 6) ° ae 1 1 1 i Substituting = for v,, = for v,, and ———~ for 6, this dy ds d, +d, becomes d, +d, d. d,-—dy p=RIG,+4)(2) Scoala We thus obtain an equation for the saturated vapour pressure in terms of the densities of the liquid and its saturated vapour, the absolute temperature, and the gas constant. The equation contains no adjustable constants whatever. The constant b of Dieterici’s equation is here replaced by 1 Dieterici’s A now becomes , which varies slowly with the temperature. Similarly, also a function of temperature. or the critical temperature, when d,=d,-=-d,= =, we find c pea 2 ae 2 ‘The equation, like Dieterici’s, thus gives the satisfactory Waluel>e — aoe) LOE = at the critical point. The agreement of the values given by the equation (1) with these directly observed has been tested by calculation for the substances * whose densities in liquid and vapour states are given in the tables of L. Graetz’s article on vapour pressures in Winkelmann’s Handbuch der Physik, 2nd ed. vol. iii. pt. 2, pp. 962-1086. The approximation to actual values is about the same for all these substances, whether associated or non-associated. Table I. is typical ; * Ammonia, Carbon Dioxide, Water, Pentane, Hexane, Heptane, Octane, Isopentane, Di-isopropyl, Di-isobutyl, Hexamethylene, Ethyl Ether, Methyl Formate, Ethyl Formate, Propyl Formate, Methyl Acetate, Ethyl Acetate, Propyl Acetate, Methyl Propionate, Ethyl Propionate, Methyl Butyrate, Methyl Isobutyrate. 444 Mr. J. H. Shaxby on Vapour Pressures the first seven columns are as follows :—(1) temperature, . t®? C.; (2) density of liquid, d, ; (3) density of vapour, d, ; Ay 636. >, G) Apt omen lon eee (4) b, i. @. aReee (5) A, 2. e. ea ee (6) calculate saturated vapour pressure; (7) observed saturated vapour pressure. It will be seen that the equation represents the facts with fair accuracy. ‘lhe calculated value is usually too high at low pressures, then becomes somewhat low until the temperature is within a few degrees of the critical ; at the latter temperature it is again too high. The corresponding Equation of State thus becomes 2 log 7 7 pre 7 1 oF PERU, } PN : p(o- zg) = tare) /(d,—d,)v=RT (2) 2 |(hi-d,)w q uh. (ae The Isothermals of Isopentane, Hexane, Ethyl Ether, and : Water have been calculated from this equation for the specific volumes quoted in Winkelmann, vol. iii, pt. 2, pp. 1114-1135. Table II. gives the result for Water and Isopentane ; those for the other substances are quite similar. Column 1 gives the Specific Volume, column 2 the calculated pressures, and the /ast column (5) the actual pressures, as given by experi- ment. It will be noted that the agreement is fair for low temperatures. . At the critical temperature the equation assumes the form 1 anes p(r— a = RT-.e dv = For temperatures above the critical this same form has been used, except that d, has been replaced by gate and the value of the latter obtained by extending the linear relation of Mathias to the given temperature. For the present we cannot assign any physical meaning to (d,+d,) for tempera- tures above the critical; we return to this point later. The table shows that the equation still represents the facts moderately well for the smaller pressures, but completely breaks down at the higher, 7. e. when the specific volumes are small. Since this paper was wriiten (and in part read at the Cardiff meeting of the British Association, 1920), Professor Porter has called my attention to a paper by F. H. Mac- Dougall (J. Amer. Chem. Soc. xxxviii. no. 3, 1916, p. 528) and the Isathermals of Vapours. 445, on the Equation of State, in which Dieterici’s equation is studied. In this paper MacDougall develops the same equation as (1) above for vapour pressures. He reaches it by assuming that & a where v,; and v, have the een U3 Same meanings as in the present paper and vr; is the volume corresponding to the same pressure (the vapour pressure) on the unstable (James Thomson) part of the isothermal. ‘Thus. = is equal to 6, and the equation b= heeds of course follows. immediately. The grounds for the assumption are given :— (1) v1, =v2.=v3=2b at the critical temperature, and v varies slowly as the temperature falls, so that it is plausible to assume v=2) for all temperatures. (2) The Cailletet- Mathias relation becomes d;=d,(1—a#T); a small and linear temperature coefficient suggests a real significance for the quantity d;. (3) The saturation pressure is the geometric mean of the maximum and minimum pressures on the isothermal, if v= 20. MacDougall also notes and illustrates in tables the approxi- mate constancy of the product Ab isee § 3 below), but draws no further conclusion from this. He does not appear to have noticed the remarkable relation (§ 1 above) that liquid and saturated vapour are complementary as regards free and occupied volumes per ¢.c. The thermodynamic relations of Dieterici’s equation and the calculation of the latent heat of vaporization are discussed in detail, and the whole paper is of great interest. § 3. The matter may usefully be examined from a slightly different point of view. The work done in transferring unit mass of a substance isothermally from liquid to vapour is {Ptav=nel ER bee ORT log », V—b vy —b ay dg? the symbols having the same meanings as before. This work Seu) : may be equated to A,—A,, where A, is the work required to overcome the cohesive forces of the liquid, per unit mass, and A, is the similar quantity for the saturated vapour. These quantities, A, and A,, as mentioned in § 1, are pro- portional, at any assigned temperature, to the densities d, and d, at that temperature. Hence we have d log d dy re Yee RT i Ls Su. Tinie ? : log ds A,(1 ay dy — dy ; ) or A= 2RTd, 446 Mr. J. H. Shaxby on Vapour Pressures Similarly, log? As 2Rhte, ee i from either of which equations it immediately follows that log? A=Zi0 pes a as before. Before we can proceed further it becomes necessary to consider the meaning of the quantities A and 6. A may be termed the Cohesive constant of a substance, and is the quotient of the work per unit mass against the forces of cohesion divided by the density Jay ENS a“ a; dy” If A, and A, depended simply on the density, we should have A the same for all temperatures, which is not the case. A diminishes with rise of temperature, indicating, what one night expect, a decrease in the forces of cohesion as the temperature is raised. The quantity (d;+d,) which plays such a fundamental part in the equations deduced above, may be regarded as a measure of the closeness of molecular packing which would result from the cohesive forces alone in the absence of molecular motions; for, if there were no “free”? space, v=b. Thus the density would be (d,+d,), and we may regard “free” space, in the sense of the equations of the kinetic theory, as a result of the molecular movements, which prevent the molecules from being as closely packed as they would be under their mutual attractions if they did not possess kinetic energy. If this be so, we can regard the quantity (d,4-d,), which we will term the Cohesive Density, as a measure of the intensity of the forces of cohesion. We should thus expect (d,;+d,) to depend in some simple manner upon A. ‘The two quantities, as a matter of fact, prove to be almost exactly proportional to one another. That is to say, the product Ad is nearly constant, as is shown in column 8 of Table I. There is a clear tendency of the product to increase slightly with rise of temperature to a maximum value, and then to decrease again as the critical temperature is approached in all non-associated substances. This lack of exact constancy in Ab may be and the Isothermals of Vapours. AAT : A : regarded as due to the fact that the relation ta = is : 1 2 only approximately true for any assigned temperature. ene If we write — or for A, we have eae dy d, A Ay As A,—A, EE de mana) — did tds) dt ds” and these expressions should also be independent of tem- perature. For an actual substance, let us suppose that A,=rd,, and that A,=r(1+«)d,, where rand « are of course not independent of temperature. Then A Pilg A,—A, Pe | 1 -*,) doen — ds res d?—d2Z ~ \d,i+d, d,?—d,?/ With rise of temperature the first term inside the bracket increases slowly ; the second, apart from a, also becomes greater, and very rapidly so as the critical temperature is approached. Hence, if we may regard r and @ as not varying very rapidly with temperature, it is easy to see that for small positive values of « the quantity Ab will at first increase slowly to a maximum and then decrease, as the temperature rises. § 4. For an “ideal” substance, then, we may suppose that RE log! A is proportional to (d,+d,), i. e. that yeaa is inde- 3 ie pendent of temperature. For actual substances this is not exactly true, but we may none the less suppose that there is for them also a relation, Ab=constant, expressing the con- stant relation of the cohesive forces to the cohesive density. In this, however, either A or 6 or both differ slightly from 2RT log | ’ the values ( Re ? and aes respectively ) which we have hitherto assigned to them. To test this point we must modify our original equation for the saturation pressure d, +d. p=RT(d,4+ dz) (7) means f ad, and the most simple way of doing this in the required way 448 Mr. J. H. Shaxby on Vapour Pressures is to substitute a constant K for the quantity d d 2RT log —* log —* Pd: Thus Ss dy _ K(d,+dp) dy—d,? ~ d,=d, | 2RT | C and panne HT, Merri (o. 2), where C is a new constant (independent of density and temperature), and b has its modified value, nearly but no longer exactly equal to d, + ds , In the same way the Equation of State becomes RE == = vb £G , [ig sik: Me ie), In applying the equation (1a) to actual substances, we may make our adjustments most readily as follows : C L624 The term log’ is small and not very different from log p= log RT— log b— iL ee aoa so we may replace i by the latter without : C serious error and so. evaluate Typ using experimentally determined values of p. Again substituting (d,+d.)? for we obtain the general 1 by’ magnitude of ©, and finally select from the array of slightly varying C’s a value which is seen to give vood agreement (for its own particular temperature) with experimental fact, i. e. the C for a temperature at which the original unmodified formula (1) gave a satisfactory value of p. From the adopted value of C the quantity b is recalculated for each temperature, and the new value of p then obtained from the equation (1a). The results of this semi-empirical method are given in column i0 of Table I., column 9 giving the recalculated values of 6, upon which they are bused. It can be seen that a much closer approximation to experi- mental fact is thus attained. ‘The lack of agreement now seldom amounts to much more than 1 per cent., and in the best “fitting” part of each table is very slight indeed, but a systematic deviation is still apparent. The closeness of fit for any particular substance depends on the selected value of C, and no very great care was taken in most cases to. and the Isothermals of Vapours. 449 secure the best value ; a likely value was “ spotted,” and the resulting magnitudes of band p worked out. It was evident, in several cases, that slightly better agreement could be bbtained but this was not worth the time it would have taken, in view of the facts that the formula can be modified and made somewhat less empirical (see the next section), and that a systematic error exists which requires a re- examination of the premises ; this has been done, and will form the subject of a further paper. The new values of } determined as above have also been applied to the recalculation of the isothermals of Table II. Galina 3 of this table shows the resulting pressures. Here ilso greatly improved agreement with experimental data is found. At temperatures above the critical the agreement 1s as good as at lower temperatures, whereas the former equation broke down in these cases. § 5. We have thus found that equation (1 a) for saturation pressure, and equation (2a) for the isothermal relation between pressures and the corresponding volumes yield close agree- ment with the results of experiment. In these equations, however, © is an adjustable constant, and 6, though it has a physical interpretation, is not directly given by independent data, whereas in the original equations (1) and (2) only separately measurable quantities occurred, viz. temperatures and densities. We may to some extent restore a physical meaning to C, and thereby give it a determinate magnitude for each substance. At the critical temperature equation (1a) becomes mn ne Cc Pe= ae on be ( : : ; Ts while equation (1) reduces to p.= ae ce If, then, we identify these two values of p,, we have Cea abe and therefore : é Toe = aa é ir 2d,2Tb2 Fs A és (1 b) The equation of state similarly becomes AT 2 T. Vole ote)) eal aul) Wo ed od WC The values, calculated from (20), of the pressures alone I $ theivarious meOthouauls are given in ela 4 of Table II. 450 Mr. J. H. Shaxby on Vapour Pressures 27 4b ,2 § 6. The equation p= aioe eS T+ can readily be thrown into a “ Reduced” form: at the critical temperature Es = a ers oe Tb,2 p_ Tbe -2 ao} Paw teens), al b Write a for 4 oe O for ; TT and y for i Dae Then T=—e \ve oy Raplaciie) bye and b : and writing 6 for 2 d,+dz, i Nila 2 d, + dy 2h , we obtain 2G) 7 =06e 4 § 7. The critical density of a substance is difficult to measure accurately, and its magnitude is commonly estimated by using the Cailletet-Mathias linear relation. It may also be calculated as follows : d,?—d,* ia Oita : Let d;=qd,. Therefore log. g L. ik —= =. T ny 2 vad As 2 logi0g | 4343(¢? =e Thus d,= ee writing @ for the last square root. gis a purely numerical quantity’; to facilitate calculation a logarithm can be tabulated for a series of values of log, 2 say from 0 to 3 in steps of ‘1, with difference columns allowing of estimation for intervals! of ‘O01. The values thus found for the critical densities of Water and Isopentane are given in the last column (11) of Table I. The Cailletet- Mathias method gives 0:2344 for Isopentane and fails for Water. 451 ‘apours. ay e « and the Isothermals o i TABLE I. Water, p. O=38085. CO} ah Ghee b, AXx10-9, Cale, Obs. Abx 10-9, Recale. b. Reeale. p, d.. 8) ‘9999 0.4737 1:000 30 89 4-478 4°51 30°89 "952 3h - Spee 60 9833 '0,1303 1:016 27°91 1498 148°7 28°38 1:009 149°7 ee: 120 ‘9441 ‘0011388 1:058 25°86 1529 1496 20°36 1:069 PACD oes ee ee 144°2 923 ‘002187 1-081 25°27 38089 3047 27: 2F 1:093 DUGG? = .aF as, ee 180 "e881 "005216 1-119 24°33 7754 7520 28:09 1:132 1437 "0289 202-2 ‘S61 ‘007976 Wile 24:07 12140 12160 28°16 ay, 12090 3261 240 ‘S117 ‘01632 1 208 23°26 25210 25093 PAS) 1:205 25130 3239 270 ‘7701 ‘02766 1:253 22-47 42080 42235 27°71 1251 42300 "3236 Tsopentane. p (mm. He), C=4287. e°C, d,. ne b, A. Cale. Obs. Abx10-°°, Recale. d. Recale. p. d 20 6196 0024 1:608 6:085 584'8 726 ‘978 1614 GyAOHG = 0°2332 40 “D988 ‘0045 1:658 5943 1141 eee ‘985 1:659 1130 "2324 60 “5769 ‘0078 eZ) 5811 2025 © 20385°6 “994 1707 2039 ‘2314 80 5540 0128 1764 5671 3346 3400'8 1:001 1:758 3414 2304 100 5278 ‘0203 1824 5°526 52385 5854 5 1:008 1-812 5389 "2297 120 ‘4991 ‘0311 1:886 5379 7766 8039°9 1014 1873 8098 "2288 140 4642 0473 1:955 0°222 11090 11620 1021 1:940 11700 "2384 160 "4206 0729 2027 5 037. 15380 16285 1 021 2018 16350 2283 170 3914 0932 2064 4919 18030 19094 1015 2-062 19110 -2290 180 3498 *1258 2:1038 4-881 20250 22262 1-027 2°110 22190 oo 185 3142 "1575 2'123 4659 23420 23992 ‘989 2°135 23850 "2320 187 "2857 . 18384 2131 4597 24500 24713 ‘980 2°146 24550 °2329 187°8 "2344 "2344 2°138 4°516 25300 25005 964 2°150 24800 Mr. J. H. Shaxby on Vapour Pressures TaBLeE II. Isothermals of Water. 149-91 C. v. 39941 8051] 992-650} ‘798645 395380 2421-19 1981°36 1690°46 182°°9 C. “2204-262 1021-215 531-649 256348 218314 195°316 187-622 2022-21 €. 1720°812 941-536 515°864 226-418 133°169 125°372 d,+d,=9991. P}: Po: 7-12 fle 12-41 12°41 d, +d, ="9597. 1613 161°3 3251 325-1 529-7 529°8 646°3 646°5 756°5 7567 d,+-d,='8891. 7423 744-2 1589 1598 3009 3041 6046 6180 7021 7204 7784 8009 8074 8320 d, +d,= "8688. 947-9 947° 1729 1729 3130 3129 6958 6955 * 11480 11480 12130 12130 Tsothermals of Isopentane. ETS. C: d,+d,=5941. Dic Des 811-9 811-7 1157 1156 1511 1511 d,+d,= "5115. 839-0 839°5 1925 1928 3753 3764 5772 5798 7720 7771 588 9677 10180 10280 11070 11200 b= 7a: De== 1a aor Corrected b= 967. _ oi Pe Exp. p. TA (ile: 12°41 12°34 Corrected =1:045. 161°3 159°61 325°1 322°26 529°7 525°47 646°3 640-78 706°5 74912 Corrected 6=1°131. 738°7 708°7 1572 15181 2949 2903°1 5802 5990-7 6689 6952°2 7374 7888'8 7633 7971-4 Corrected 6=1:157. 947-9 943°6 1729 17148 3130 3111-4 6958 691071 11480 11592°6 12140 12181°1 Corrected 6=1°683. Mes 812-2 1157 1512 Exp. p. 814 1158 1514 Corrected 6=1°940. 839°5 1928 3768 9806 7785 9699 10310 11230 858 1933 3808 5914 7977 9993 10640 11595 and the Isothermals of Vapours. 433 TABLE I]. (continued). 187°°5 C. (Crit. temp.). d,+d,=-4688. Corrected )=2°150. v. Peo De. (e. Fxp. p. 19-41 14870 14800 14844 15190 14-40 17970 17240 17920 18430 10°67 20980 20810 206900 21480 6°970 24240 23910 24080 24430 3872 25310 24780 25110 5030 280° C. d,+d,='3840. Corrected 6=2°49. 540'8 872°2 879°0 879:0 876 263°0 1766 1795 1795 1786 125-70 38579 3698 3698 3680 52°34 7884 8525 8531 8459 24°43 14310 16900 16930 16870 11-40 21810 31110 31210 31240 5810 24860 49380 49680 495350 5172 24880 53560 539380 53760 Summary. 1. Consideration of the internal pressure in fluids leads to the relation that in “ideal” substances the ratio of the occupied volume (co-volume) to the total volume of a liquid is equal to the ratio of the unoccupied volume to the total volume of its saturated vapour. ‘lhe two phases, in equi- librium with each other, are complementary in this respect. The relation is nearly true for actual fluids. 2. The equation d, +a mn Le NRE p=RT(d,+d;)(7)" De hor (1) l is obtained for the saturation pressure of a vapour in terms of the temperature and the densities of the two phases at that temperature. ‘This leads to an Equation of State p(v— pect fe )=n0(2)r=. epheart) ereng)) P di +d) a a . 3. It is shown that the quantity (d;+d,) may be regarded as a density factor which is a measure of the cohesive forces per unit mass. 4, On this hypothesis that for actual substances there is a ‘“‘density ” proportional to the cohesive forces, nearly but not exactly equal to (d,+d.), moditied forms of equations (1) and (2) are obtained. These equations express the experimental facts with considerable accuracy for all temperatures and pressures. Pint. Mag. S. 6. Vol. 41. No. 243. March 1921. 2 H ADA Prof. W. A. Jenkins on 5. **Reduced” forms of the vapour-pressure equation are obtained and an equation for the calculation of the eritical density of a substance. ‘This gives consistent values, even in cases such as that of water when the Mathias rule does not hold. I wish to thank Professor Porter for his kindly criticism and for the interest he has taken in this paper. XXXIX. On the Determination of “H.” By WALTER A. JENKINS, M.Sc., Professor of Physics, Dacca College™. Le the Philosophical Magazine of October 1913, a new method of determining the Horizontal Component of the Earth’s Magnetic Field was described by the writer. The method, suggested by Dr. Hicks, was the creation of an artificial magnetic field exactly equal in intensity to twice the Earth’s field. Equality of the two fields was determined by the equality of the times of swing of a suspended magnet placed in both fields. It was there shown that the method was capable of giving an accuracy of one part in 10,000 and was as efficient a method as the Kew Magnetometer one. But in both the Kew Magnetometer method and the Solenoid method previously described, tle chief part of the experi- ment is the determination of a time of swing, and the limit of accuracy of the methods is the accuracy with which the time of swing can be determined. In the previous method difficulty was experienced in obtaining a suspension fibre sufficiently fine, strong, and short enough to allow the magnet to oscillate for the period required for making accurate observations and at the same time to conform to the rest of theapparatus. ‘Two methods will be described in the present paper, both of which obviate this difficulty and make the determination of “ H” a short and reliable experiment. First Metaop.—The apparatus used is essentially the same . as that designed for the former experiment and the principle of the method much the same. In this method, however, the equality of the fields is determined by measuring the angle of deflexion of the suspended magnet when under the influence of both the Harth’s field and the Solenoid field. The Solenoid field is placed exactly at right angles to the * Communicated by the Author. the Determination Ofendt.? 455 Yarth’s field, so that if 6 is the angle of deflexion, 2) sin’ @ = 2 il eos 0, El= cot 0. A determination of F and @ therefore gives the value of “‘ H.” As before F is equal to 4a7nOcosa, where n and @ are eonstants of the Solenoid and the measurement of F reduces itself to the measurement of the current C. Diagram 1. « < @ PAS < *S Diagram 1 illustrates the method adopted for measuring the angle of deflexion. Oe is the coil accurately placed at right angles to the Harth’s field. A B is the undeflected magnet. An image of the scale E is reflected from the mirror surface C into the telescope G. Suppose the zero of the scale to be at the intersection of the telescope cross-wires. The current is now switched on and the magnet deflected as shown in the lower part of the diagram. The strength of the current is then adjusted until the zero of the scale is reflected from surface D to the intersection of the cross-wires of the telescope. The angle of deflexion is then the angle between the mirrors Cand D. This isa constant quantity and can be accurately determined. © and D are silvered microscopic cover-glass mirrors attached to the sides of an octagonal elinath sienna framework which is hollowed out for the sake of lightness. Similar mirrors are attached to all sides of the octagon in order to preserve a symmetrical distribution of the weight. The framework is attached to the underside of the magnet holder. 29H? A456 Prof. W. A. Jenkins on Details of the Experiment.— The tube, mounted as before on a board capable of rotation, was adjusted until it was exactly parallel to the Harth’s field. The method used for this—that of adjusting the position of the tube until, when a large current is sent round the solenoid, no deflexion of the magnet occurs—was described in the previous paper. The solenoid tube was then rotated through an angle of 90°. This was done by means of a telescope. and scale and two mirrors at right angles to each other, mounted in a suitable position on the board carrying the solenoid. The actual angle between the mirrors was 89° 25/ 30”, but so long as the angle is definitely known the fact that it is not 90° makes no difference to the experiment. The distance of the telescope and scale from the mirrors was 210 em., and as a rotation of 1° gave a motion of 75 divisions of the scale, an accuracy in the determination of the angle of rotation of 12” was obtainable, for a motion of + of a division could easily be followed. The source of light, plane reflecting glass, scale and telescope were then arranged as shown in diagram 1 and an image of the scale reflected from face U obtained. A current was now sent through the tube and adjusted until the deflexion of the magnet was such that the mirror-face D reflected the same mark of the scale on to the cross-wires of the telescope as did the mirror C. ‘The current strength was then measured by means of a Kelvin Balance which was in the circuit, and the current was found to be ‘01945 ampere. The balance will measure the current to ‘00001 ampere, and is a very convenient instrument for carrying out the experiment to a fair degree of accuracy. W hen high accuracy is desired, an electrical method similar to the one described in the previous paper can be used. The angle between the.mirrors D and C had previa been found by mounting the system on the table of an accurately calibrated spectroscope. It was found to be 44° 29' 15". Thus we get H = F cot @, or allowing for the fact that the angle of rotation was not ‘90°, cos 442 DOT is ST yy elena = BOO 209: =i where F = 4anC cosa, « being half the angle subtended at the Determination of “HH.” 457 the centre of the solenoid by the diameter of the end of the helix. Substituting the values n = 13°362, cos « = = SUS) we get H=°3236. Accuracy obtainable.—The povowne sources of error were discussed in the previous paper : 1. The magnetic axis of the solenoid not being horizontal. 2. The magnetic axis of the solenoid not being parallel to the direction of ‘** H.” 3. Irregularity in the winding. 4, The longitudinal displacement of the magnet from the centre of the tube. 5. Residual torsion of the fibre. 6. Inaccurate determination of the period of oscillation. 7. Inaccurate determination of the current. 8. An error in the determination of n. 9. Heating of the coil by passage of the current. It was there shown that with the exception of 6, 7 and 9, the sources of error were negligible. In the present experiment 6 is not present at all, while as the current passing is only one half of what it was un the previous method, the heating effect is diminished to 1 of its previous value. As by taking proper precautions it was previously negligible, it is more sonow. ‘The determination of the magnitude of the current still remains one of the chief factors limiting the accuracy obtainable, but with standard resistances and a good galvanometer an accuracy of 1 part in 10,000 is not difficult to obtain. Error 5, due to the torsion of the fibre, is negligible when an extr emely fine fibre isused. Inthe present experiment, however, when prolonged oscillation of the magnet is not only not required, but is undesirable, a thicker suspension fibre possesses considerable advantages. Consequently a moderately thick suspension fibre was used. A stronger, heavier magnet system was also inserted in the holder, and on putting into the fibre 1440° of torsion a deflexion of sin-! °012 was observed. The following equations give the maximum allowable value of the angle ‘of deflexion due to the residual torsion for an accuracy of 1 in 10,000. Suppose @ of torsion remain in the fibre and x be the coefficient of the torsion. 2H/ sin «a =n (@O—a), where « is the deflexion due to torsion. When the tube is rotated through 90° and the 458 Prof. W. A. Jenkins on : magnet system deflected 45°, 2 Hisin (45+2)=n (0—a)4+2 Flsin 45, H sin (45 +e)—H sin 2=F sin 45 sin 45 . ~~ sin (45 +2) —-sIn @ Le F ~ cosatsina~V2sina EF Ss GOS CE) sin a If sina is of the order i000 then cose isl—4 _ that cos « can be called 1. Therefore H=f (1—-41 sina), and in on that the torsion can be neglected sin a must be less than ~~ — The torsion which will give a deflexion of this order is 36°. The torsion was eliminated, as in the previous experiment, by substituting a brass bar equal in weight to the magnet and allowing the system to come to rest. “It is not probable that in such a case 36° of the torsion would remain in the fibre. The following additional errors are possible :— (a) KError in the determination of the angle between the mirrors attached to the magnet. (6) Error in the determination of the angle through which the solenoid is rotated. (c) Error in the determination of the coincidence of the zero of thescale with the cross-wires of the telescope. (a) The angle is an invariable quantity and can easily be measured to 10 seconds. Approximately H=Fcot45 An error of 10 seconds in 45 degrees gives an error of 1 in 10,000 in the calculated value of H. If desired the angle cone be measured with greater accuracy than that indicated. (6) Is similar to (2). (c) Phe distance of the scale from the mirror was 75 em. ; oe care the coincidence could be determined tol ofa division, 7.e.°0lem. This corresponds to an angle of rotation of the mirror of approximately 15 seconds. The actual angle through which the mirror is rotated is about 45°. Therefore we get H=F cot (45°+ 15"). This gives an accuracy of 1 in 6666, or say 1 in 7000. The accuracy of the experiment therefore reduces itself to the Determination of “f.? 459 the accuracy with which the angle of deflexion can be measured. A similar measurement is involved in the Kew Magnetometer method, and the accuracy of the two methods may therefore be said to be the same. Seconp Mrernop.—The principle of this method is ver similar to that described in the previous paper. An artificial field is produced in the solenoid opposite in direction to, and in intensity twice that of the Harth’s field. The equality of the two fields is tested by means of the deflexions caused by a subsidiary magnet placed in the Tangent B position of Gauss. Details of the Experiment.—The tube was first set accu- rately parallel to the Harth’s field. The octagonal shaped framework suspended underneath the magnet had mirrors attached to all its faces and the aljustments were made so that the scale was reflected from face D (diagram 2 a). The external magnet sliding in a groove at right angles to Diagram 2, x x a b Cc . Position under influence of Earth’s field alone. 2 Position under influence of Earth’s field + that of external magnet. . Position under influence of Earth’s field + external magnet + solenoid field. the solenoid was then adjusted until the seale was reflected from face C, diagram 26, into the telescope. Without moving the external magnet the solenoid field was switched on and adjusted in strength until the position of the suspended magnet was as in diagram 2c. Then the image of the scale was reflected from face B into the telescope. If the octagonal framework is accurately made and the angles between the adjacent faces are 135° exactly, then obviously the solenoid field is exactly twice the Harth’s field, and we get 2H=47nC cos a. The measurement of H is now simply a measurement of C. In practice, however, the angles are not exactly 139°, and 460 Prof. W. A. Jenkins on it is necessary to makean allowance for this in the calculations. The angles between the mirror faces can be accurately measured and, from the values found, the angles of deflexion 6, and @, in the two paris of the experiment calculated. Then the following equations will give the value of H :— F cos 6,;=H sin 6), F cos 0,=(X—H) sin 6, where X is the field of the solenoid. xX ~ tan 6, cot 6,41. In the experiment actually carried out, Gj — 142 Div 30r Dy 2 A i—=13 doz cos 2='99986 C=:037547 ampere 62955) fl ==3245 The experiment was carried out ten days after the former one, and hence coincidence of results could not be expected. EHrrors.—Possibilities of errors not previously described arise in this experiment owing to the following causes :— (1) A lateral displacement of the subsidiary magnet from its true position opposite the centre of the suspended one. (2) An angular displacement of the subsidiary magnet from its true position perpendicular to the solenoid. (3) Alteration of magnetic moment of the subsidiary magnet. ; (1) The following calculation shows the error likely to arise from a lateral displacement of the subsidiary magnet. Suppose the magnet is displaced 1 cm. from its correct position. Then a simple calculation shows that an error of 1 part in 3000 is introduced if, as was the case in the actual experi- ment, dis about 50 cm. and / about 18 em. If the displace- ment isnot morethan 3 mm. the error is not more than 1 part in 10,000. The subsiciary magnet can easily be placed to within 3 mm. of its correct position. (2) An angular displacement of 15° would introduce an error of 1 in 10,000. The magnet can be placed in its correct position to a greater degree of accuracy than this. (3) The only factor likely to cause a sudden change of the Determination of “H.” A6L magnetic moment of the subsidiary magnet in the interval elapxing between the two parts of the experiment is a tem- perature change. The temperature coefficient of magnetic moment is ‘001. Thus in order to obtain the required accuracy the temperature must be kept constant to within L° C. Suitable precautions can easily be devised to ensure this. In both methods described in the present paper the magnet when in a deflected position lies in a field of strength H 4/2 approximately, and hence owing to induced magnetism its magnetic moment will be slightly greater than when in a field H. This, however, introduces no error, for the magnetic moment of the suspended magnet is a factor which does not enter into the problem except in so far as it alters the angle of deflexion due to residual torsion. Atthe most the change introduced into « is a small fraction, and hence as @ 1s negligible the change owing to altered magnetic moment is LD) ee) oO also negligible. Comparison of the two methods.—Both methods give an accurate result, and both are easier to carry out than the Kew Magnetometer method. Once the constants n, cos a, the angles between the mirrors, and the direction of H are determined the only determination in the first method is that of the current required to give the necessary deflexion. When the solenoid is rotated through 90° the optical apparatus must also be rotated through the same angle, but in the apparatus which is being designed for permanent installation the optical system, consisting of eyepiece-scale and reflecting glass, is to be fitted into the end of the tube and will therefore rotate with it. ; In the second method no rotation of the tube is necessary, but it involves the use of a subsidiary magnet. Moreover, two adjustments are necessary. (1) The adjusiment of the position of the magnet to give the requisite deflexion under the influence of the Warth’s field. (2) The adjustment of the current to create an oppositely directed field which gives the same deflexion. Serevited that the solenoid is mounted upona large circular table whose rotation can be measured, as that of a spectro- scope table 18, the first method is extremely simple and accurate, and in the author’s opinion provides the best method of determining H with accuracy. The fact that the deter- mination of H by both methods gives substantially the same 452 Mr. J. D. Morgan on Impulsive result when allowance is made for the variation of H from ; day to day, is an indication of the relia ibility of the method. Check experiments with a not very reliable Kew Magneto- meter instrument gave the same value of H. In conclusion ] again wish to express my thanks to Dr. Hicks, who first of all suggested the use of the principle involved in the Solenoid method. Dacca College. 7th August, 1920, XL. Impulsive Sparking Voltages in small gaps. By J. D. Morean, B.Sc. * ie a paper entitled ‘‘ Time Lag in the Spark Discharge (Phil. Mag. vol: xxxvii., August Ee Dr. Norman Campbell discusses the fact that the im; pulsive sparking voltage isin certain gaps greater than the static sparking voltage. A's explained in that paper, when the voltage upplied to a gap is caused to rise rapidly from zero, the value it will attain before Sparkie occurs Is oreater that that reached by a voltage which is applied gradually. The ratio of these two values has been termed by Peck the “impulse ratio,” and this expression is now generally adopted. It is commonly believed that two conditions other than voltage are involved in the process of spark production, namely, time and initial ionization. Campbell recognizes these two conditions, and shows how they can be used to explain why the impulsive sparking voltage of a gap is 33- oreater than the static voltage, or in other words, why the gap has an impulse ratio greater than unity. Arguing from his own and Peek’s ‘investigations, the conclusion reached by Campbell is that there are two kinds of lag, a regular and an irregular one, in the process of spark production, though it is apparent that he sees a probable connexion between them. Taking Campbell’s mode of presenting the fetes tliere- would appear to be justification for distinguishing between regular and irregular lags, but it is questionable whether this manner of regarding the subject is likely to lead to the most usetul practical results. In the writer’ S opinion it is more convenient to assume only one lag, and to regard the sparking voltage of a given gap as dependent jointly on the three variable conditions which have hitherto been recog- - nized. These conditions are (1) rate of rise of voltage, * Communicated by the Author. Sparking Voltages in small gaps. 463 (2) the time for which the voltage must continue to act after it reaches a certain value, and (3) the initial ionization of the gap. When the voltage rises rapidly, variation of any one of these will affect the sparking voltage. By con- sidering these conditions together an explanation of the variations found in impulsive sparking voltages in a constant gap can be given in a direct and simple manner. Let it be supposed that the sparks are to be produced in a small gap between, say, small spherical electrodes, by means of an induction-coil. Assuming first that the gap is removed from connexion with the coil, and that the high tension terminals of the coil are so far apart that no spark ean pass when the primary current is interrupted, then the maximum voltage which can be attained in the secondary of the coil is variable in a regular manner by varying the primary current (7) at hen. Disregarding, for the sake of pumppcity , the double oscillation which occurs in the secon- dary, the rise of voltage can be represented by a sine curve, its period being constant, and its amplitude proportional to the primary current at break. The rate of rise of voltage from zero will vary with 7. In figure 1 a number of sine curves are drawn corresponding to the arbitrary current values noted in the graph. The ordinates represent Volts and the abscissee Time. A small gap is now connected to the coil and adjusted so that with 7)=1 in the primary, sparking just occurs. It will be supposed that the spark appears when V reaches the crest of the curve 7,=1. It will also be supposed that the static sparking voltage for the same gap is a slightly smaller value Vo such as is represented by the horizontal line. Assuming that there is ample initial lonization in the gap between the electrodes, then the time interval after the intersection of the curve ?;=1 and the line V, is that required for the voltage to act before sparking can occur. During that time the voltage rises to V,. As it is supposed that7,=1 is the smallest | primary current by which sparking can ‘be o! ‘tained, the effect of variation in the initial ionization is to alter the regularity of occurrence of the spark. From the crest of ij=1, draw any straight line as shown at an inclination to the horizontal. The inclination of this line will be different for different gaps. Thus for a gap between large spheres it will be small, but for a gap between points it al be relatively large. The horizontal distance or time between the intersection of this line on any curve and that of the line V, on the same curve decreases regu- larly from curve to curve. This may be taken to represent 464 Mr. J. D. Morgan on Impulsive in a simple manner the usual assumption (shared by Camp- bell) that the time during which the voltage acts on the gap after passing the line V, decreases as the voltage at which the spark appears increases. Considering the curve tg=2, then with sufficient initial ionization the sparking voltage is given by Vo. Ifthe initial ionization is diminished the voltage must be maintained for a longer time and the sparking voltage will reach some value higher than Vo. Likewise with the other values of ¢) the sparking voltages are represented by V3..... Ve. Volts ' Time The sparking voltage conditions for a given gap as indicated by figure 1 are purely hypothetical. What is there attempted is to associate definitely the rate of rise of voltage with time, and it has been mentioned that any of the bas ie Vo/ts Sparking Voltages in small gaps. 465. sparking values found (except V,) will vary with changes in the initial ionization of the gap. It is necessary now to test the matter by experiment. Two obvious checks present themselves. In the first place, if the argument is true, the sparking voltage for a given gap should vary with the primary current at break in a manner shown at figure 2 ig. 2: Current in Primary which is obtained from figure 1. In the second place the instant of occurrence of the spark should advance as iy is increased, and the amount of advance should progressively diminish as zp increases. This latter is easily tested. Starting with 7) as a minimum and focussing the cross wires of a tele- scope on the image of the spark in a rapidly rotating mirror, it was found that the image shifted relatively to the cross- wires in the manner anticipated. Regarding the variation of V with %, this was tested by an experiment on a small gap formed between polished brass balls of °5 inch diameter, both electrodes being insulated from earth. First the static sparking voltages for different gap settings were found. The results are shown by figure 3. Then using impulsive voltages given by a 10 inch induction-coil, the sparking voltage was found with the least current in the primary that Ki/o vel, ts A66 Mr. J. D. Morgan on Impulsive would produce a spark at a number of different gap settings. For measuring the voltages a thermionic valve was used in conjunction with an electrostatic voltmeter. The results are also shown by figure 3. Finally, with the same measuring | | | : | Impulsive aS eee eee ee | | | | | eros ot | | | | “O4 ‘G2 VUiicth eof Gap apparatus, the variations of V with 2, were found for each setting. These are given in figure 4. It will be observed that the curves obtained all have the same general character as the hypothetical curve given in figure 2. From the Sparking Voltages in small gaps. 467 experimental results there would appear to be good reason for believing that the assumptions on whichyfigure 1 is based are substantially true. Bo] Vand bint hilo volts. 2 3 4 Amps. Current in Primary. Before the curves shown in figure 4 can be fully appre- ciated it is necessary to mention certain supplementary details observed in the experiments. Taking any one of the curves, the first point (¢. e. the one at which sparking just occurred with the least current in the primary) was 468 Mr. J. D. Morgan on Impulsive always the most definite. The same experiments were repeated many times at intervals extending over several weeks, and always the same voltage value was found for the first point. The contact-breaker in the primary circuit of the induction-coil was operating at a speed of about four breaks per second, and with the primary current adjusted until sparking at the test gap was Just possible, the irregu- larities noticed at different times were not in the spark voltage but in the frequency with which the sparks occurred. The lag at the first point appears to‘ correspond with what Campbell terms the regular Jag, and the lag observed by Peek in his experiments. Proceeding now along any one of the curves, two changes were observed. In the first place, the sp arking became more regular until eventually a spark followed each break of the primary circuit, and in the second place the sparking voltage gradually became less regular. But in none of the experiments recorded by the curves was the voltage so irregular as to make the drawing of a curve unjustifiable (as is the case with some waps). These irregularities appear to correspond with Campbell’s irregular lag, and are mainly if not entirely due to vari- ations in the ionization of the gap immediately prior to the passage of the spark. Judging from his book (‘ Dielectric Phenomena in High Voltage Engineering, New York, 1915), Peek does not appear to have investigated sparking voltages for a given gap higher than those corresponding to the first point above mentioned, and this accounts in the writer’s opinion for the absence ‘(commented on by Camp- bell) of any reference by Peek to irregular sparking voltages. (n considering the above-mentioned irregularities it is “not overlooked that they may be due, at least in part, to irregu- larities in the operation of the contact-breaker causing the rate of rise of voltage to vary. Such a condition does undoubtedly account for some irregularity (a condition which would become more accentuated as the primary current is increased), but it does not account for all. dt is more probable that the irregularities observed were due mainly to variations in the initial ionization of the gap. As bearing on this point it is necessary to remark that on different days under different atmospheric conditions the same curves could not always be repeated. The dotted line curve in figure 4 shows for example one variation which was found. Here the first point was very detinite and the same as before, but with increase of primary current the voltage readings became not only rather irregular as before but also quite different. Subsequently the tests were extended to a variety of gaps Sparking Voltages in small gaps. 469 and in all eases the general relationship shown in figures 2 and 4 was found, the differences always being one of degree and not of kind. Where the gap is formed b electrodes of relatively large diameter or flat plates, the rise of sparking voltage with increase of primary current is usually small, and, when artificial means are employed to produce copious initial ionization of the gap, may be very small. But when pointed electrodes are used the variations are usually large, and witi the higher current values are very irregular. ‘The use with pointed gaps of a third insu- lated point reduces the sparking voltage and regularizes the action of the gap. It also diminishes variation of sparking voltage with primary current. Instances were noted with a flat-plate, an annular, and a three-point gap, in which the sparking voltage was apparently constant and independent of the primary current, but these results were occasional and could not be reproduced at will. No explanation of this exceptional behaviour can be given. The gaps in which the effect was observed behaved normally (2. e. showed an increase of voltage with primary current) at other times. It follows from such considerations as those outlined in this paper that the impulsive sparking voltage for a given gap is a quantity depending on three varia tble conditions. The rate of rise of voltage varies with the kind of apparatus used for producing the spark. With the same apparatus, keeping the circuit conditions constant, it can be varied by varying the primary current. The initial ionization of the gap varies with atmospheric conditions, but can be made more or less constant by artificial means. The element of time varies with the other two conditions and is different for different gags, as Campbell rightiy points out. When the potential gradient in the gap is uniform, the time for which a certain minimum potential must be imposed on the gap may be very small, but it may become relatively large when the potential gradient vari ies greatly (as in a point-gap). Due to the latter condition two gaps of different form but having the same static sparking potential and the same initial ionization will have different impulsive sparking voltages, as is well known. The subject is one of considerable importance, and has direct practical application in connexion with e. y, research on the ignition of gases by induction-coil sparks, ignition apparatus for internal combustion engines, protective gaps for certain kinds of electrical machinery, and overhead transmission systems subject to rapid electrical impulses. Phil. Mag. 8. 6. Vol. 41. No. 243. March 1921. 21 PATO | XLI. On the Design of Soft Thernione Valves. By G.Sreap, 1.A.* [Plate VILL] Pretiurwary Norr.—tThe experiments described in this paper were undertaken on behalf of H.M. Signal School, Portsmouth, and were carried out in the Cavendish Labor- atory, Cambridge, at various times during 1916-17. They are now published with the consent of the Admiralty. The writer wishes to acknowledge his great indebtedness to the members of the W/T staff of H.M. Signal School, and in particular to Professor C. L. Fortescue and Mr. B. 8. Gossling, with whom he was in close touch throughont the work. I. Introduction.—A general account of the development of thermionic valves for Naval purposes has been given in a recent paper by B.S. Gossling , and in section (7) of the paper the evolution of the “soft”’ or gas-filled valve is dis- cussed. It is thought that a more detailed account of some | of those portions of the work with which the present writer has been directly associated may be of general interest. The experiments described deal in the main with the effect on the valve characteristics of the pressure and nature of the con- tained gas, and of the position and form of the electrodes. Chronological order has been followed as far as possible, but has not been adhered to where rearrangement seemed likely to render the account more logical and connected. II. Position and Form of Electrodes.—In 1916, when the work described in this section was undertaken, little was known about the effect on valve characteristics of changing the position and form of the three electrodes, and experi- mental work in this direction seemed urgently required. Two kinds of experiment were carried out, one to determine the effect of altering the distance between the grid and the filament, and the other to ascertain the effect of altering the spacing of the grid wires. Gi.) Hxperiments with a movable grid—A special valve was designed in which it was possible to move the grid whilst the valve was exhausted. The arrangement is shown diagrammatically in fig. 1. The anode was an aluminium disk A, and the filament was in the form of a V with its * Communicated by Professor Sir J. J. Thomson, F.R.S. t Journai 1. l. Ii. 1vill. p. 670 (1920). ° On the Design of Soft Thermionic Valves. ATI plane parallel to that of the anode. Between them was a grid B, made of copper wire and mounted on a thin glass tube, whose lower end was sealed to a float C. The grid was connected to the source of grid potential by means of the flexible lead L sealed into the glass at D. ‘The float rested on mercury contained in the U-tube E, and the height of the float, and therefore the position of the grid, could be varied by altering the pressure of the air in the other limb of the U-tube. The distance between the planes of the anode and filament was approximately 2 cm., and the grid could move freely in this space, except that it was prevented by a stop from coming nearer to the filament than about 1mm. A valve of this type necessarily con- tained mereury vapour at the saturation pressure corre- sponding to room temperature (about ‘001 mm.) and was thus ie. fairly soft. The valve was exhausted as thoroughly as possible by a rotary Gaede pump followed by charcoal cooled by liquid air, and was baked during the process of exhausting so far as the presence of the mercury permitted, ?.e. to about 350° C. Characteristics were plotted for various positions of the grid, the filament current and the temperature of the mercury being kept the same throughout. Some specimen characteristics are shown in fig. 2 (Pl. VIII.), for three different distances between grid and filament. The general effect of increasing this distance is to diminish the slope of the anode current-grid voltage characteristic, and to diminish the Bel 2 472 Mr. G. Stead on the ratio of the grid current to theanode current. This is shown by the numbers in the following table, which were obtained by measurements on the original characteristics. | Distance from grid = = Maximum slope. | Grid current. | | to filament. | (Vi==40): Anode eurrent. | eM | 3mm. 0-6 miiliamp. per volt. 0742 | | der. Rue eee ote 1-05 7 | | 10 | 0:20 # si | 0-89 The second column gives the maximum slope of the anode current-grid voltage curve for an anode voltage of 40. The third column gives the ratio of the grid and anode currents for the case when the anode voltage is 40 and the grid voltage 15. (ii.) Experiments wich a grid of variable spacing.—A dia- gram of the special valve used for these experiments 1s shown in fig. 3. It was of cylindrical type and the grid Fig. 3. se 2 qe als AMO ADF saeeeees eS ee) — consisted of a steel spring, of helical form, which could be stretched by means of a thread attached to a small winch W operated from the outside through the ground glass joint G. The anode was a brass cylinder about 3°5 cm. in length and 2 em. in diameter. Small holes were drilled through the anode cylinder so that the grid and filament were visible from the outside. The filament was a straight tungsten wire about 2 cm. long and ‘0082 cm. in diameter. ‘The tube connecting the winch with the valve was made long, so that the ground glass joint could be outside the oven during the baking process, and the conduction of heat from the oven to the ground glass joint was prevented by coiling round the glass tube a thin lead pipe through which cold water was allowed to flow. , me. . Design of Soft Thermionic Valves. A473 The valve was exhausted in the usual way and was rendered as hard as possible by means of charcoal and liquid air, mercury vapour being also frozen out by liquid air. Hydrogen was then admitted. The hydrogen was prepared by electroly sis of a solution of barium hydroxide, and was dried by phosphorus pentoxide, followed by the freezing action of liquid air. Characteristics were then plotted for a series of grid spacings. Specimens of these characteristics are shown in fig. 4 (Pl. VILL ) for grid spacings of 1, 2, 3, and 4 mm. (By grid spacing is meant the width of the openings between the grid wires, not the pitch of the helix.) It will be noticed that for a given anode and grid voltage, the anode current first increases and then diminishes as the grid helix is pulled out, and the characteristics attain maximum steepness for a particular value of the grid spacing. The manner in which the anode and grid currents alter as the grid spacing is increased is shown in fig. 5 (PI. VIII.). The curves reproduced represent anode current (or grid current) aed against grid spacing for a fixed anode potential, and for several different orid potentials. In this particular valve the diaméter of the erid helix was about 0-6 em. and the diameter of the grid wire 0°5 mm. The grid ean which ee the maximum auode current increases from 2°3 to 2°85 mm.asthe grid potential increases from 1°6 to 15 volts. lil. Preliminary experiments in Mercury Vapour, Residual Air, and Argon.—The effect of the pressure and nature of the contained gas on the characteristics of a valve of audion type (with plane anodes and zig-zag grids) was examined in some detail, the atmospheres employed being mercury vapour, residual air, and argon. For the experiments on mercury vapour a small tube containing a drop of mercury was attached to the valve, and this tube was placed in a vacuum fiask containing water, or methylated spirit, at a known temperature. In this way any desired vapour pressure could be obtained. The range of temperature employed was from —13°C. to +56°C., corresponding toa range of vapour pressure from ‘00004 to ‘019 mm. «© Residual air’? was obtained by simply pumping down the valve to the required oo. und doubtless consisted chiefly of electrode gas, , hydrogen and oxides of carbon. Pressures between “008 mm. and ‘00025 mm. were employed. The argon was made from liquid air residues by prolonged sparking over potash with excess of oxygen, the latter being finally removed by phosphorus. The purity of the argon was ATA Mr. G. Stead on the tested spectroscopically. Characteristics were obtained for a range of pressures from ‘032 mm. to ‘00005 mm. These preliminary experiments showed that the general effect on the characteristics of variation of gas pressure was the same for all the gases studied, but that the pressure at which an audion works most satisfactorily is at least five times as great in residual air and argon as in mercury vapour. IV. General remarks on Soft Valves.—The chief difficulty to be overcome in the design of a satisfactory soft valve is that caused by variations “of gas pressure whilst the valve is in operation. It seemed likely, at first sight, that changes of pressure would be less marked if larger amounts of gas could be employed, but in valves of the audion type, with plane anodes, a general glow readily occurs throughout the bulb if the anode voltage is too high, and the valve is at all soft This glow, which is accompanied by rapid disinte- eration of the tungsten filament, is probably associated with ionization along the tracks of electrons which travel from the filament to the anode by long paths outside the electrodes. The probability of such paths is greatly diminished by the adoption of a cylindrical construction for the electrodes. Hence a higher gas pressure should be possible in cylindrical valves than in those of the audion type. Experiment showed this to be the case. The cylindrical construction has the further advantage that all four leads can be carried by the same stem; the electrodes can therefore be completely assembled outside the bulb, and then sealed into the bulb at one operation. In order to test the theory of the general glow given above, a special cylindrical valve was constructed in which one end of the anode was permanently closed, and the other end could be closed when required by means of a small brass cup which was carried on a stem attached to a glass bulb floating on mercury. It was found that when the end of the cylinder was closed by the cup the general glow was entirely eliminated, and “ kinks ” in the doi vanaricnrs could be avoided at pressures high enough to produce them if the anode were open. V. Soft Valves as Oscillators.—A detailed investigation was made in order to determine the pressures at which various types of valve would generate oscillations in two different circuits. A diagram of one of these circuits is shown in fig. 6. Nitrogen was the gas first selected for these experi- ments, and five different valves were employed. Two of lc Design of Soft Thermionie Valves. 475 them were of audion type, and the other three were cylin- drical_valves closely resembling the French type. The grid Fig. 6. A =anode. B,=anode battery. Gord: T =telephones. F =filament. L =inductance. B,=filament battery. Cj, Coy C3, Cx, ANd C;=cCondensers, wires of the three cylindrical valves were of different dia- meters, and so possessed different projected areas, but the three grids were of the same effective degree of closeness, 2.é., possessed the same value of the quantity m in the formula tN! log (d/d’) log[1/(aNd,) | where d and d' are the anode and grid diameters, respec- tively, d is the diameter of the grid wires, and N is the number of turns of grid helix per centimetre. This formula was communicated to H.M. Signal School by Professor Sir J. J. Thomson and is an adaptation to the cylindrical case of a formula given by Maxwell * for a plane anode, grid and cathode. (See also the paper by Gossling, already cited.) Of the two audion valves employed in this investigation, one had a normal type of grid and the other a very open grid, Experiment showed that under given conditions a soft valve would not generate oscillations unless the anode potential lay between certain limits. Both limits rose as the pressure of the gas was diminished, but the upper limit rose much more rapidly than the lower limit, so that the oscillating range increased as the gas was pumped out. For high gas pressure the range of anode voltage over which oscillations would take place was very limited, e. g., less than one volt, so n= * ‘Klectricity and Magnetism,’ i. p. 312 (1892). A476 Mr. G. Stead on the that the adjustment of the anode potential to obtain oscilla- tions was very critical. The upper limit of pressure, however, depends very much on the nature of the gas and also onthe design of the valve. Thus the amount of helium which can be introduced into a valve is about ten times the amount of nitrogen which is practicable, and this again several times «.s great as the amount of mercury vapour which is pessible. In these experiments, in order to find the lower limit of the oscillation range the anode potential was gradually raised until the anode current showed a sudden drop. This in- dicated that oscillations had started. The upper limit was determined in a similar manner by starting with an anode potential so large that no oscillations were generated and gradually diminishing this until a sudden drop of current occurred. As regards the determination of the potential of the grid itis to be observed that the circuit shown in fig. 6 has an insulated grid, and for this reason the method adopted was to observe the anode current and then connect the grid to a potentiometer and adjust this until the anode current was the same as before. The range of anode voltage over which oscillations could be produced under the given conditions was determined for each of the five valves for a series of pressures lying between the limits 0°6 mm. and ‘0074 mm. The oscillations were usually produced uncer somewhat difficult conditions, with an inductance of 1000 microhenries and a capacity of ‘005 microfarad, and an additional resist- ance of 2 ohms in the oscillating circuit. Some observations were also made under different conditions. In each case small portions of the valve characteristics were drawn for the region gonial nin the oscillation range. These experiments showed clear] ly that considerably higher gas pressures are practicable in cylindrical valves than in those of the audion type. Observations for two different valves are shown in the following table :— } Cylindrical type valve. Audion type valve. | Pressure. | | Minimum | Maximum Ranee: Minimum | Maximum ae voltage. voltage. e!| voltage. | voltage. JM Eo oe seats Joys oie a 06 mm. 196 | 19°8 0-2 = — = | 02 te ai 20 21-2 11] 200, |, 920 2)s mee | 05. eet? 26-1 19 | 21-9 23-9 | 20 "O1D - te Abe oe 375 54 28:5 2 isa 4:7 | 0074 ,, ree 65:6 | 20°5 a | Design of Soft Thermionic Valves. ATT One of the five valves was finally selected as giving the most satisfactory results in the circuits employed. This was a cylindrical valve having a grid made of wire of diameter 0-4 mm. coiled into a helix of mean diameter 4-5 mm. and having 5°5 turns percem. Full characteristics were drawn for this valve for a range of pressures from 0-5 mm. to 0045 mm. Finally a pressure of about ‘(06 mm. was selected as giving the most satisfactory characteristics. This valve was the Naval valve designated by the number R2. For reasons which will appear later (see Section VIII. below) nitrogen did net prove a satisfactory gas for valve purposes, and the next gas tried was argon, which was chssen as being the most easily obtained inert gas. Experi- ments on oscillation ranges In argon were carried out for pressures between 0°25 and ‘042 mm. The results given by argon were very similar to those obtained when nitrogen was employed, but it was soon found that there was a fatal objectien to the use of argon in valves. When the filament of an argon-filled valve was heated up there was no indication of a change of pressure, or of filament resistance, after di heurs, but as soon as a potential of about 20 volts was applied to the anode the filament resistance began to rise eely: and after 30 hours had increased by nearly 50 per eent., although there was still no alteration in pressure. It appears, therefore, that tungsten filaments disintegrate rapidly in argon under the conditions which occur in a valve during operation, and it seems likely that this disintegration is due to the bombardment of the filament by the very heavy positive ions of argon. Argon having proved impracticable helium was next tried. This gas should share with argon the advantage of not being © absorbed by the hot filament, without possessing the disadvan- tage of causing disintegration ofthe filament. Experiments on oscillation ranges in helium were carried out for pressures between 4 mm. and :042 mm. The results are shown for four different pressures in fig. 7 (PI. VIII.) Full charac- teristics for helium were also drawn, and these are reproduced in fig. 8 for two different pressures. These characteristics, instead of being drawn in the usual manner, represent the results in a very convenient way suze gested by B.S. Gossling. In this method anode potentials are plotted as ordinates and grid potentials as abscissee, and each curve represents a line of constant anode (or grid) current. The curves are therefore of the nature of contours. A general discussion of these contour characteristics has been given by Gossling *. * Journal I.E. F. Joe. et. 478 Mr. G. Stead on the From a study of the characteristics of helium it has been concluded that the highest pressure of helium which is practicable in valves of the design used is about 1 mm. Valve No. R2A was designed to contain helium at a pressure of about 0°6 mm. This pressure is considerably higher than would be practicable with any other gas except hydrogen, and for this reason it was thought at first that contamination by electrode gas would be less noticeable than in nitrogen. This did not prove to be the case, owing to the fact that positive ions appear in a valve containing helium when the difference of potential between the anode and the negative end of the filament amounts to about 21 volts, whereas the corresponding difference of potential required to produce positive jons in electrode gas is about 15 yolts*, so that a small admixture of electrode gas is able to cause a large alteration in the characteristics of a helium valve. This is brought out clearly in the contour characteristics, parti- cularly in the case of the grid current contours. It will be seen from fig. 8 (Pl. VIII.) that the grid curves rise nearly vertically until a potential of about 21 volts is reached, when they suddenly bend over and become nearly’ horizontal. This is due to the formation of positive ions, whereby the negative space charge is partially neutralized, so that a given grid voltage is able to produce a much larger current than previously, 2. e., a vertical line corresponding to any particular grid potential would at this point begin to cut across the grid contours in rapid succession. Now if the helium con- tains a small percentage of electrode gas, these horizontal portions of the grid contours occur at about 15 volts instead of 21 volts, and, in consequence, the characteristics of the valve are very much modified. Many attempts were made to produce satisfactory helium valves without bombarding the electrodes, the process adopted being to heat the electrodes as much as possible by radiation from the hot filament. It was found that, even when the filament was run at such a high temperature as to cause serious damage to itself, the removal of electrode gas from the anode was very incomplete, and the form of the valve characteristics was controlled mainly by the electrode gas and very little by the helium. Ultimately it was found best to subject the electrodes to heavy electron bombardment during the evacuation, just as in the case of hard valves, and then introduce the necessary quantity of helium, which had been purified by standing for some time in contact with charcoal cooled by liquid air. * Stead & Gossling, Phil. Mag. xl. p. 418 (1920). Design of Soft Thermione Valves. A479 VI. Methods of estimating Gas Pressure in Valves.—In the manufacture of soft valves it is very important to have a simple method of estimating the pressure of the gas before sealing off the valves from “the pump. A very convenient method, which is accurate enough for the purpose, is to measure the width of the Crookes’s dark space in a small vacuum-tube attached to the apparatus for the purpose. The vacuum-tube employed was provided with an aluminium disk as cathode, and a small aluminium rod in a side tube as anode. By experiment the relation between the pressure and the width of the dark space was determined for nitrogen, argon, helium, hydrogen, and carbon monoxide. The width of the cathode dark space depends on the current density through the tube *, but a pressure indication of sufficient accuracy can be obtained if the diameter of the cathode is specified, and an induction-coil of definite size is used. This method of estimating gas pressures is much simpler for manufacturing purposes “than the McLeod gauge method, and has the adv antage of not introducing mercury vapour into the apparatus. It is necessary that mercury vapour should be excluded since it ionizes at about 10°5 volts and a small trace of mercury vapour has a large influence on the characteristics. If evacuation is performed by means of a rotary oil pump and the dark space method of estimating pressures is employed, it is not necessary to introduce the complication of a liquid air-trap to keep back mercury vapour. The dark space method is, of course, no longer available after a valve has been sealed off from the pump, and it is very useful to have another pressure test which is applicable to sealed valves. For this purpose use was made of the phenomenon known to the staff of H.M. Signal School as **backlash.” When the difference of potential between the anode and the negative end of the filament becomes suffi- ciently great positive ions are produced by collision, and, if under these conditions the potential of the grid is raanlela little negative to the negative end of the filament, the grid will pick up positive ions and the grid current will be reversed. The magnitude of this backlash current depends on the anode and grid potentials and on the pressure and nature of the gas, and it forms a very simple and convenient method of estimating pressures in sealed valves. The pro- cedure usually adopted was to make the grid have a definite negative potential (usually —2 volts) and then increase the * Aston, Proc. Roy. Soc. 1907, 1911. 4380 Mr. G. Stead on the anode potential until the backlash current rose to a specified value (generally 1 microampere). The anode voltage required for this purpose increases as the pressure is diminished, and the: relation between these two quantities may be determined by experiment for any given gas. It is evident that the required anode voltage will be related in some way to the ionization potential of “the gas, but the relation between the two is by no means a simple one. Experiments have been carried out. for nitrogen, helium, argon, hydrogen, and carbon monoxide, and for grid potentials of —1:5, —2°0, and —2°5 volts. The results for a grid potential of —2 volts are shown in ho 9 (ei ye): ‘When VIL. Cooling of Filaments by surrounding Gases. a filament is surrounded by a gaseous atmosphere the power required to maintain it at a given temperature is greater than when the filament is 2 vacuo, owing to loss of heat by conduction and convection through the gas, as well as by radiation. It was necessary to know whether this effect would be appreciable at pressures likely to be used in valvex.. The method adopted was to measure the current required to maintain the filament at a given resistance, first 2 vacuo and then when surrounded by gases at various pressures. From these measurements it was possible to plot curves showing the number of watts taken by the filament at different gas pressures. Measurements of this kind were made for nitro- gen and helium, and the results are shown in fig. 10: (Pl. VILL). It was found that in nitrogen at a pressure of i mm of mercury the watts required were only 10 per cent. more than in a good vacuum, whiist at the pressure employed in the R2 valve (06 mm.) the additional watts amounted to. 1 percent. The effect is thus of little importance in this latter case. The cooling due to helium is appreciably larger, SMELLS, to about 13 per cent. at a pressure of 1 mm., and 20 per cent. at 4mm. In hydrogen it is still larger, but no. systematic observations were made. It will be observed that, both for helium and nitrogen, as the pressure increases the curves become gradually less steep up to a certain point, and then the steepness begins to increase again rather suddenly.. This effect was always observed and appears to be genuine and not due to accidental causes. It seems likely that the upward turn of the curve occurs when convection in the gas. begins to be appreciable. VIE Objections to Nitrogen atmospheres—As already mentioned, nitrogen had to be abandoned as an atmosphere Sere Oe 2 ee en a Design of Soft Lhermionic Valves. 481 for soft valves because the pressure did not remain constant under working conditions. Special experiments were made to investigate this effect. life tests carried out at Ports- mouth indicated that a fairly rapid absorption of nitrogen occurred when the filament was running, and a voltage of the order of 20, or more, was applied to the anode. Very little nitrogen was left after 50 hours of such treatment. The remaining gas began to ionize «at a potential of about 13 volts, and its spectrum showed lines of argon and the usual carbon monoxide bands due to electrode gas. Atmo- spheric nitrogen had been employed and this Lad apparently been absorbed, leaving a residue of argon, which woule ionize at about 12°5 volts, and at the same time the elee- trodes had given out carbon compounds. In order to investigate this effect more fully special experiments were carried om in the Cavendish Laboratory. A valve was attached to a ‘‘dark-space tube” (see para- graph VI. above) and was exhausted and filled with nitrogen until the dark space was 9 mm. in length. No change of pressure occurred as the result of sealing this apparatus off from the pump, nor was there any change alter the valve had been allowed to stand idle for a week. The filament current was then turned on, and it was found that after 50 hours running tle length of the cathode dark space was reduced to 85 mm. At the same time the dischar.e had become somewhat less pink i in colour. Thus a small quantity of gas must have been liberated from the electrodes, or from the walls of the tube. A potential of 20 volts was then applied to the anode, and after 7 hours the length of the dark space was found to have increased to 11 mm. At the end of 30 hours’ continuous running under these conditions the dark space was 14°5 mm. in length. At the same time the colour of the discharge had altered from pink to blue. It appeared, therefore, that nitrogen was being absorbed and that electrode gas was being given out. To test this point further a small spectrum tube, provided with external tin-foil electrodes, was attached fo another valve, and the arrangement was then exhausted and filled with nitrogen at a pressure of about 0°5 mm. Initially the spectrum consisted chiefly of nitrogen bands, but a few faint argon lines and carbon monoxide bands were also visible. The general colour of the discharge was pink. The apparatus was sealed off from the pump and allowed to stand idle for 10 days. This produced no change in the spectrum, and there was no great alteration after the filament had been run for 33 hours. As soon as a potential 48? On the Design of Soft Thermionic Valves. of 20 volts was applied to the anode the pink colour of the discharge began to alter, and after 24 hours there was no trace of pink colour left. The spectrum then showed carbon monoxide bands strongly, one or two faint nitrogen bands, and argon lines as before. It seems clear from these experiments that under the ordinary working conditions of a valve, nitrogen is absorbed and electrode gas given out. Furthermore, the evolution of electrode gas is at least in part a purely thermal effect whilst the absorption of nitrogen is apparently electro- chemical rather than chemical, since it does not take place unless there is a potential on the anode. There seems reason to believe that no absorption of nitrogen takes place unless the difference of potential between the ‘anodeand the negative end of the filament exceeds the ionization potential of nitrogen (i.e, about 17°5 voits) so that probably tungsten at a temperature of 2300° K does not combine with nitrogen moleculis, but is capable of uniting with nitrogen presented in the form of rapidly moving positive ions. ‘This is in agree- ment with the result previously obtained by Langmuir *. TX. Summary—An account is given of experiments which led up to the design of a satisfactory soft valve for Naval uses. The chief points considered are :— (1) The effect on the valve characteristics of the position of the grid with respect to the filament, and of the close- ness of the grid structure. (2) The effect of the pressure and nature of the gas on the valve characteristics and on the production of oscillations. Nitrogen, argon, and helinm were studied from this point of view, and it was found that there were serious objections to the first two gases, but that helium was very satisfactory. (3) A method of estimating the pressure of the gas ina sealed valve is given, and the effect of nitr ogen and helium in cooling the filament by conduction and convecticn is considered. In conclusion the writer wishes to express his sincere thanks to Professor Sir J. J. Thomson for his unfailing kindness, and his interest in the work. Cavendish Laboratory, Cambridge, August 1920, * Phys. Rev. ut. p. 450 (1913), and Jour. Amer. Chem. Soc. xxxv. p. 948 (1913). Paes XLII. A Method of finding the Scalar and Vector Potentials due to the Motion of KHlectric Charges. To the Editors of the Philosophical Magazine. GENTLEMEN,— ROFESSOR A. ANDERSON has published in No. 236 of the Philosophical Magazine (August 1920) ‘A Method of finding the Scalar and Vector Potentials due to. the Motion of Hlectric Charges.” He obtains divergent results from those obtained by myself in 1898 (Kelairage lectrique, tome xvi. p.5). I have no observation to present for the calculations of the beginning, but I do not agree with Professor Anderson when he states the following proposition :— > 2) erp A= iN herrea hs TA = 2-94 will vanish i=) at any point P where there is no electricity.” For that, it should be necessary that the elementary volume dx' ‘dy’ dz and the electric charge be carried by the same motion ; which is not the case. 132A In addition to “ans, it is easy to show that Vege ve will not vanish. In order to simplify the writing, I will suppose that w is constant as regards its magnitude and direction, in all points and at any time. In this case | 0’p . CE di! dy’ de’ ial - gat ea) ac p— Sala!) i Ors _ Ce eae Ou Oe? (292 one aN i: ' / ul = 3 a dy dz < nD : —- Yur(v—a') | c a A84 Messrs. W. E. Garner and Irvine Masson on the 2 2 ee and cs will be obtained by cyclic interchange. ‘Thus | rae ——— Lu(e—2') aie a: : : v4 — oA s 5 dx'dy' dz -and does not vanish. Iam, Gentlemen, Yours faithfully, A. LifNARD. XLII. The Activity of Water in Sucrose Solutions. By W. EH. Garner and Irvine Masson * Jones and Lewis (Trans. Chem. Soc. 1920, exvil. p. 1125), in investigating the catalysis of the inversion of cane-sugar by decinormal sulphuric acid, have obtained values for the activities of the hydrogen ion in solutions of varying sucrose content with the aid of the hydrogen electrode. These activities increase markedly with increasing concentration of sucrose ; thus, “‘ when the sucrose content is 70 per cent. the thermodynamic concentration [per litre of solution | of the hydrogen ion is 0°162 at 20°C. and 0°152 at 40°C although the maximum actual concentration cannot exceed Hai. It occurred to us that this increase in the activity of ‘the hydrozen goes pari passu with the abnormalities in the osmotic pressures of sucrose solutions observed by Berkeley and Hartley and by Morse and Frazer. Arrhenius (Zeit. phys. Chem. 1899, xxvii. p. 317) has already correlated velocities of inversion with earlier measurements of osmotic pressure. That such a connexion exists is best shown by comparing the ratios of the activity of the hydrogen ion in sucrose solution to that in pure water with the ratios -of the observed osmotic pressure to the osmotic pressure calculated according to van’t Hoff. The former set of ratios is denoted in the following table by (H° )su:r./(H* )aq., and -the latter by P.sps./Pea:. The ratios of the solubility of hydrogen gas in pure water ‘to those in sucrose solutions are also given in the last column, -as a further indication of the change in the activity of the * Communicated by Prof. F. G. Donnan, F.R.S. Activity of Water in Sucrose Solutions. A85 water. It has already been pointed out by Miiller (Zeit. phys. Chem. 1912, Ixxxi. pp. 483-503 ; see also J.C. Philip, Trans. Faraday Soc. 1907, iii. pp. 140-146) that other gases are quantitatively similar to hydrogen in this respect. Lewis § Jones. Berkeley & Hartley. Morse § Frazer. “TagiOeee | lal eG Sucrose. (H°)suer, Pp Pp tte mols./litre. (H*jaq. obs./ cal, H, sucr. 20°. AO?’ OS 20°. AO IO: -292 1:13 1-12 Uy oh 113 1-09 Lek 35 1:30 124 1:20 1-23 12 1-23 877 1:49 1:56 1:36 (1:38) (1°33) 1:39 1-169 1°75 1°82 1°56 dae Lee 161 1-469 Or 2°18 1-80 ae ue 1:87 1-755 2°32 2°60 2-11 8 Ade 217 2047 2°70 o'04 2:49 There is a very clear parallelism between these ratios, which is well shown when the data are plotted as curves. The general agreement shown by these figures indicates that the causes which are operative in increasing the activity of the hydrogen ion are substantially the same as those which _ eause the deviations of the osmotic pressure from van’t Hoff’s law and the decrease in the solubility of gases in sugar solutions. These causes would appear to be at work inde- pendently of the presence of hydrogen ions. Jones and Lewis have further shown, from their expe- riments on inversion velocities, that ‘the environmental catalytic influence of a molecule of sucrose is identical in magnitude with that exerted by one molecule of dextrose together with one molecule of levulose.”” Osmotic data are available for glucose (Berkeley and Hartley) as well as for sucrose ; and assuming that a molecule of leyulose is identical in its behaviour with a molecule of glucose, we can again tabulate the ratios of observed to calculated osmotic pressures. J. & L., Sucrose con- centration. "292 "$85 See elog | LOOK. 7a ZOkF : = p> Sucrose, B.&H., 11 120 136 156 180 211 249 eal, OL: Pp . . poi Glucose+levulose, 1:07 US) 1-34 1-52 re. C0)" es) cal. B.& lake 0° C, (H*)suer, SS aire‘ 9 ' 6 C aad ne oY att (H*)aq. ? JI.&L. 20° CG. TSS} 1:30 1:49 1 io 1:97 9:39 9 70 Phil. Mag. 8. 6. Vol. 41. No, 243. March 1921. 2K 486 Mr. C. G. Darwin on the Collisions of Similar ratios are obtained for the solubilities of oxygen in glucose solutions. It is quite clear in the first place, on osmotic grounds alone, that the activity of the hydrogen ion should be practically unafiected by the inversion of the sucrose to glacose and levulose ; and in the second place, that the velocity-constant of the inversion will not be influenced by the change in environment which occurs. The increase in the activity of the hydroeen ion may be ascribed to a virtual increase in the concentration of the sulphuric acid, due to the inactivating effect of sucrose molecules upon water molecules. This effect may consist in hydration of the sucrose, and in this case there would be an actual increase in the hydrion concentration in the free water. The water which is thus fixed by the sugar (changing from 10H,O to 5H,O per sucrose-molecule in the foregoing serles) is not ae ails as a solvent for the hydrogen ion or for gases, nor is it osmotically active. From the data of nee md Lewis it appears that this water is nevertheless chemically active, since the rate of inversion is proportional to the total water present and not merely to the free water—a fact which is difficult to explain on any supposition other than the above. We communicated these results to Professor Lewis, who very kindly suggested that they might be published. University College, London. XLIV. The Collisions of a-Particles with Hydrogen Nuclet. By ©. G. Darwiy, I.A., Fellow and Lecturer of Christ's College, Cambridge”. Ll. Introduction and Summary. N a series of four papers entitled “On the Collisions of a-particles with Light Atoms” in the Philosophical Magazine of June 1919, Sir Hrnest Rutherford treated of a long course of experiments he carried out on this subject. The first of these papers f dealt with hydrogen, and he drew several interesting conclusions, the most. striking of which was that in these collisions the a-particle and the hydrogen nucleus could not both be regarded as simple point charges. He roughly explained his observations by imagining that * Communicated by the Author. + Rutherford, Phil. Mag. xxxvii. p. 5388 (1919). a-Particles with Hydrogen Nuclei. 437 the #-particle is a plate of radius 3x107%cm. As long as the e-particle does not approach within this distance of “the hydrogen nucleus, the ordinary law of repulsion holds, but if it does a collision ensues, which sweeps the latter straight forwards. His experiments were intended to be of a preliminary nature, and so were not carried to any high degree ot accuracy. The object of the present paper was to submit these experiments to a more rigorous analysis, so as to try and discover more definite information about the structure of the nucleus. But I have been recently informed through the kindness of Sir E. Rutherford that more detailed experiments are in progress in the Cavendish Laboratory to redetermine the curves with greatly improved methods. In particular, the number of hydrogen particles projected at various angles to the primary beam of a-particles is under determination by direct observation. The results so far obtained indicate that some of his earlier curves require modification. It thus appeared desirable to omit the more speculative conclusions to which I had been led, until the new more accurate experiments are finished, and the paper only makes a study of the question of the reduction of the earlier experiments to their simplest terms and an examina- tion of what certain assumed patterns of nucleus would ‘give. We first discuss what is the complete information which can be obtained from experiments of any kind whatever on the subject, and show that these must lead to a certain relation, which we call the ¢ollision relation. ‘This relation is then found for Rutherford’s experiments, and it is seen to be quite different from that for the collision of two point charges. The complicated reduction was carried out in considerable detail before it was realised that the new methods were possible. In view of the extraordinary difficulty of experimenting in the lige and the probability that these new methods will not always be possible, it seems desirable to retain this detail, as showing how quite com- plicated experiments can be reduced to simple terms. The rest of the paper is an attempt to make models of nuclei which shall give the same collision relation as the ex- perimental. These models must all obey the inverse square law at great distances, and the attempt is of necessity limited to cases wheré the orbits are integrable, which makes them rather artificial. In all of them one particle ig still taken as a point charge, and the complexity is attributed to the other. In view of the trend of modern 21 2 488 Mr. C. G. Darwin on the Collisions of physics, it is natural to suppose that the «-particle is the complex one. From the first model we conclude with some probability that the experiments cannot be accounted for, if the «-particle is equally likely to be orientated in any direction—in fact, Rutherford’s suggestion of a plate is. supported. Two plate-like models are then tried, and are seen to give closer resemblance to the experimental curves. For the present, however, until the latter are confirmed it would appear undesirable to draw conclusions. 2. The Collision Relation. Throughout the paper we shall be dealing with the action of an a-particle in setting a hydrogen nucleus in motion. It will conduce to shortness to refer to the two simply as @ - and H respectively. We shall first consider what conclusions. can be drawn purely inductively from any type of experi- ment on the collisions of « with H. Neither particle may now be assumed to be a point charge, and the law of inter- action is supposed to be quite unknown. No experiments on any line that could be devised (at present, at any rate): could do more than give a statistical description of the numbers of H projected in various directions and with various velocities by a known beam of « We shall see that all such experiments can be made to lead to a certain relation between three variables, which we shall call the collision relation. In the first place, the velocity of H is determined by the angle between the initial line of motion of « and the line of final motion of H. This angle will be called the angle of projection. For, let H, M be the charge and mass of a, and let V be its initial velocity. Let e, m be the charge and mass of H, initially at rest, and let wu be its final velocity at angle @ to the initial line of motion of a, so that @ is the angle of projection. Then by a simple application of momentum and energy it is found that 2M 8 ; My | COS Oa NOR ere iw eee since M=4m. So it is a matter of indifference whether: the experiments observe u or 0. We shall use 0. In calculating the above it has been tacitly assumed that the collision involves no emission of radiation or change in the rotational motion of either body. Unless this is so, no progress can be made with the present method, as_ the. Spe a-Particles with Hydrogen Nuclet. 489 conservation of energy becomes inapplicable; but the absence of rotational change is made very probable by an argument based on the quantum theory, similar to one that occurs in the study of the specific heats of a gas. Let I be the moment of inertia of one of the particles, and suppose that v is a permissible frequency for rotation about the axis of I. Then the energy of rotation is $1(27v)?, and by quantum principles this must be equal to nhy, where n is zero or an integer. Solving, we have nh Vvp=— On]” ope lite elias totgtt Fol -, ee (2:2) and so the possible energies are of the form n*h? eye iio Wem aisle Hoven qiley etl ae (2°3) We conjecture the «-particle to be not larger than a sphere of radius about 2x10- em., while its mass is known to Pemororlyt+tseew . . . (1) fixes a limiting value of @ for every value of # Under conditions where / is a considerable fraction of 7, the equation may not give a solution at all for values of x near unity. This means that the «-particles, when they have got to e=1,. have lost so much velocity that even their most favourable encounters are incapable of producing a scintillating particle. We shall not need to consider this case here, but shall find that the case where / is only a small fraction of r is the important one. In the experiments it ranged between 10: and 20 per cent., and the former belongs to the most important experiments. Neglecting l/r, we “have cos? @> —-seew. - 2 0s) eee dy Substituting the value of 1, cos’ 0 > tlva-# sin? @), or in a form convenient for computing 4, —= =) 6 4 x< cosee OA | “(1 ( 1 sec 6) (3°3) The terminal values are 3 for z=0 cos Oyp= Ae Ar 4 fo — cos h= 4 / a. Ar - Thus 0;< 6, and so fewer particles near e=1 give scin- tillationsthan near v=0. Thisis not because of the reduction of velocity of the a-particles, but because those thrown off at a given angle strike the stopping foil more obliquely. 4 pe ie a-Particles with Hydrogen Nuclei. 495. If the area observed is A sq. cms., then the number seen [pa Pa v=r Beer i C34) 0) A Names te.) we (35) where yee eae Aqr|? . . e n e t s In this equation v is a known function of ine asa unknown function of 6, while a is a known function of @,. . t stipe . : depending on the parameter Ties This is an integral equation 9 of the ‘first kind’ of the type known as Volterra’s, with & do” discuss the general theory of such equations, but it appears that the following process gives quite a satisfactory solution. This is because the limits of integration @ and 6, are in all cases not very far apart. Fig. 3 illustrates an x, 8 curve, with the difference between ‘kernel’ The present writer is not competent to Fig. 3. | G Ny Sites ~° 0 ay O-—r 1 6 and §, somewhat exaggerated. The tangent at 0 is always horizontal. Assume that P isa function of @, that can be expanded in a Taylor series. Then in the field of integration oe d ) IE 2 d? 2.7 P(¢)= P(@,) +(A0—) nee (Ao) i 9 (9 — 4) 0,2 (0) (3 6) and so v Ay dP (°% Pa? ( % ? =P(6 n+ TN O-O)det 5 aa) (8—Ar)de. ( 0) da + a ( 0 )¢ Dae 5) d0.?. - ( ye The first integral is equal to unity, the second is the area of ABC taken negatively, and the third is twice the moment of this area about AB. The last two quantities are calculable 496 Mr. C. G. Darwin on the Collisions of by quadrature and depend only on ae We denote them by t AN Ai( =), A(z.) Gy dat aes es GA = P(6),—A,) a ape + amit e (377) The solution can then be completed by successive approxi- mations. First, neglecting second differentials we obtain a curve for P(@) from the experimental values of v. Then, by means of the method of finite differences, we calculate d?P curves analysed it was found unnecessary even at the point of greatest curvature to make any correction for second differentials. This implies that no discontinuities were found. It may here be said that a slight modification of our method would be capable of dealing with any of simple type that might occur. In obtaining the P, @ relation we neglected the loss of velocity of the «-particles in their passage through the gas. If this is to be allowed for, we have to use the accurate equation and so get a correction term for v. In the actual ‘= ( pe, V)de, LA yl taken between the limits given by (3:1) instead of (3:2). Assuming a Taylor expansion for P in V, we can write this as py ieee He yikes = { PO. V)- 3 i ae a ax. The process of approximation could be applied to this, though it would be very cumbrous. We should require a new, more accurate.set of curves like that in fig. 3, but calculated from (3°1) instead of (3°2}. There would now be two para- meters i, and = both influencing the calculations. We should then re-evaluate A, etc. We should also need from our previous solution to get relations between P and V, by taking together the experiments with «-particles of different velocities. In this way oe could be found, and so the second term evaluated, and applied as a correction to v. a-Particles with Hydrogen Nuclet. A9T A good approximation will be obtained by neglecting the change in the curves of fig. 3 and also taking ee = SP (b— LT) - for all the mh of @ in the rather short range of integra- tion. Then lat {P@, Das. "av P(O,—A,, Ve} ip | 1 SP = i P(Q—Ay- V) +5 3V My =P(@- A, V(1— -;)) ee ee pet PO (aay where V is the mean velocity of the a-particles in their passage through the gas. Thus we see that a complete determination is possible for the collision relation between P,@, and V. If at any time really accurate experiments should be carried out on the present principles, then it might be worth while to carry out in detail the processes sketched, but the labour would be very considerable, and it is to be hoped that it will be found possible to cast future experiments in a more tractable form. In view of the inaccuracy of the present ones I have contented myself with the simpler reduction previously described, together with the correction given in (3°9). 4. The Reduction of the Hxperiments. The reduction of Rutherford’s experiments then took the following course: (The pages mentioned in this section refer to the pages of his paper already cited.) Values of t/4r were taken at every unit decimal between 0 and 1. The values of 4) were calculated for these, and also the (x,@) curves, illustrated in fig. 3. By mechanical quadrature the areas and moments were found, so as to give A; and Ay. A table of the function @)— A, in terms of ¢/4r was thus drawn up, so that any value could be found by interpolation. The experimental curves of p. 550 were then measured up. They are drawn in arbitrary units, different for each curve, and require to be brought to the same scale by means of the table on p. 553. This process was carried out by reducing the latter numbers with the help of the curves of p. 550 to. a standard range of 7 cms. ‘The absolute size is determined 498 Mr. C. G. Darwin on the Collisions of from the experiment described on p. 554, which deals with the flat part of curve A. With our definition of p instead of p, the calculation on p. 555 stands good, and we thus have as the absolute scale p=2°4x10-® cm. for the value -of P on this flat part. In the curves of p. 550 the fastest H-particles at every range have very nearly four times the range of the exciting a-particles. This is the confirmation, referred to above, of -the law we have assumed for the loss of velocity of H-particles. For the reduction the longest range at which scintillations were seen was taken as 47. The abscissa of a point on the ‘curve is its t. Rutherford measured these quantities in their equivalents in cms. of air, not as we have done in hydrogen, but this will not affect the ratio t/4r. From the values of t/4r we get @)—A, by means of the table. These are to be -associated with values of P proportional to the square root ‘of vy. The value of the second differential was tested at the point of greatest curvature in A, and it was found that the resulting correction was negligible. 5 Se hb wae —_ a, ed — li See = Fig. 4. 1S, — > — oy a] > I AWS : 5 yeas a=" EXPERIMENTAL CoLLIsion RELATION (J, 6). p is measured in units of 107'* em. Each curve, as determined from experiment, ends at the point marked by a short cross line. H is the curve given by the collision of point charges, when a has the same velocity as in F’. The curves of fig. 4 are the results of this analysis. The curve H is the theoretical curve for point charges of the same velocity as . It would appear from the uneven way in which the curves are arranged, that there are some errors 10° 00° 30° WAGE a-Particles with Hydrogen Nuclet. 499 in the experiments of p. 593, which connected those curves together, especially in H and possibly B. From fig. 4 we deduce fig. 5, where P is plotted against (V/V)? for constant Pies of 0). Hlere Vgas 2 < 10% em. per sec., the velocity of the fastest «-particles. The reason for tiking this function ‘of V as abscissa is, that it is the form in which V naturally occurs, when the law of interaction is the inverse square. The irregularity of the lie of the curves of fig. 4 becomes very marked in the curves of fig. 5, and to obviate this some smoothing has been applied to them. NO ——> 3 EXPERIMENTAL COLLISION RELATION (p, V). The constant 6 curves have heen plotted with (V,/V)* as abscissa, where V,=2 x 10° cms. per sec. 5. The Reduction of the a-particle to Rest. We now proceed to study what special laws of force can roduce collision relations in any way resembling figs. 4 and 5. In doing so we are practically limited to those with integrable solutions, as the task of investigating orbits by the method of small arcs is too great to be undertaken, until some definite evidence is obtained as to the probable structure of the nuclei—especially when it is remembered that 16 would be necessary to treat of variations of P with both @and V. Before proceeding with these special cases it will be convenient to prove a simple general theorem applicable to the collision, of any two bodies. It is onlya slight extension 500 Mr. C. G. Darwin on the Collhcties of of the astronomical process of reducing one of two bodies te rest. Wesuppose them both incapable of rotation. Then if points fixed in the bodies are chosen as their ‘centres,’ and if these centres are at 21, 1, 21 3 2, Yo, 22, the mutual potential W will be a function of #,— 2s, ete. The equations of motion then are Mz,=— oW ete., oe (51) a sone is: e Oi Oe Suppose that « is approaching along the direction of the x-axis from the positive side. ‘The momentum integrals are Mi, +me,=—MV, My, + my=Q, M2,+mz,=0. (5-2) Also if &=2,—2z, ete. the relative motion is given by Mm pe oW ee. Mein. oe 0& ; Suppose (5°3) to be solved for the relative orbit. Let the angle between the initial and final asymptotes be 6, and suppose that the y-axis is so chosen that the final motion is in the plane a, y. From the conservation of energy the final velocity is V, and so the final values of E etc. are Mo = (5°3) F=2,—%,=V cos, 1=th—-jo=Vsind, Pp. - . - - (54) =%,—z,=0. Solving for # etc., we get = —MV(1+ cos6)/(M+m), Yo= —MYV sin 6/M +m, =O) and so the angle of projection is given by tan O=Ys/#2=tan 5 ; So we have @= : and we enunciate the result as follows. The angle of projection of H is half the angle between the asymptotes calculated for the problem of relative motion, in which one body is constrained to remain at rest at the initial position of H, while the other, endowed with mass a-Particles with Hydrogen Nuclei. 501 Mm/(M-+m), is projected along the same line and with the same velocity as « in the actual problem. It is indifferent whether it is H or a@ that is reduced to rest, and this shows that though it might be possible from experiments to attribute definite structures with certainty to H and 2, yet the experiments could give no information as to which had which. but there is strong evidence from other sources that «is more complex than H, so in considering our models we shall reduce & to rest and project H past it. In all the cases we consider, H is a simple point charge. 6. The Elastic Sphere. In our first model H is supposed to move according to the ordinary law of the inverse square, unless or until it approaches within a certain distance of the centre of a. If it gets within this distance it bounces off, as if from a hard elastic sphere. The calculation is quite straightforward, and need only be given briefly. If anauxillary angle } is taken, given by the relation [Dae HRCA AS ees ACH lan (OL) where, as in (2°5), ety eel aL ies a e + then the orbit under the inverse square law is the hyperbola Hsin A= cos (A) — cosd, aehnnvenl (Ore) and so the angle between the asymptotes is 2X. This is for the case where the apsidal distance p cot is greater than 6, the radius of the sphere. If p cot 6/4, there are no collisions at all. If w~1—2k. Then sech & can never exceed the value given by the third root of (9:3). The point where it reaches it is a turning point and may be called the apse. After the apse sech € must diminish Fig. 8. lees. » (9°3) A—seché E—>2-» =, again, while I. continues to increase ; and when sech é has reached zero I, will be double its value at the apse. In fig. 8 there is a sketch of the general character of I, plotted against a-Particles with Hydrogen Nuclez. D07 sech&; the exact shape will depend on the values of & and +. Now consider I,, the integral on the right. Here two cases arise which exhibit an important difference. If y>1 (that is, if p>a) the integrand has no turning points, as its é 3 5 GAR denominator never vanishes. If we plot ie against nal hed we have acurvé like HF in fig. 8. But if y<1 the de- f : a ae nominator vanishes as soon as sin G —7) =ry, and the curve bends back like [,. Zn then shrinks through zero to the value given by sin (5 = )= —y, where there is another turning point. So the curve is a sinuous line like GH in fig. 8. The three curves are all drawn with the same hori- zontal base line in the diagram ; k& and y will determine the size and shape of I,, and y of I, and also whether the latter is of the type EF or GH. To sce the form of any orbit it is only necessary to select the proper & and y curves and then draw a number of horizontal lines. These will give simui- taneous values of €and 7. In particular the line through C in fig. 8 will give the value of 7 on the second asymptcte. As 9 is the ordinary vectorial angle when & is large, the value of @ is simply (3 ~n). Notice that when y< 1, @ is almost as likely to be negative as positive. lt is unnecessary to describe in detail the rather tedious processes involved in the computation. I, has to be trans- formed to reduce it to a standard elliptic integral of the first kind, and two cases arise, differing only analytically, according as the second or fourth root of (9°3) is the larger. In both its cases I, needs little transformation, but the calculation is slightly complicated by the fact that the value of I, to which it is equated must always be diminished by some multiple of the ‘complete elliptic integral, (which determines the number of oscillations in the orbit) before recourse can be had to the tables. If we wish to obtain complete curves between p and @, we are limited by the fact that y? must be greater than 1—2h for all values of y, and therefore k must not be less than 1/2. Calculations were therefore made taking k=0°5, 0°6, 0°7, 0-8, 1:0, 2:0, and in each case the value of 0 was computed for a succession of values of p. The curve 0°5 has an infinite number of oscillations as 0 approaches zero. This is 508 Mr. C. G. Darwin on the Collisions of connected with the fact that if y=0, H has initially exactly such a motion that it would come to rest midway between the halves of a. For 0:6, 0:7 there is a single oscillation, but in the higher values # never becomes negative. After obtaining the p, @ curves we convert them into p, 0, a process _ only affecting those for which k<0-7. Fig. 9 shows the curves. The discontinuities in 0-5, 0°6, 0:7 are of the type Fig. 9. ‘ i ie : x i ae 20° 05. 5 7 Tans CoLuiIsIon RELATION FOR BIPOLE. The distance between the poles is 2 units. The numbers on the (7, @) curves refer to the value of k, which is proportional to1/V?. The inset gives the (p, V) curves in the same form as fig. 5. with one tangent vertical. Towards 90° all the curves rise very steeply, but it has been necessary to omit these angles in order that the important part, from 0° to 50°, might be drawn on a reasonable scale for comparison with fig. 4. From the p,@ curves with constant k, we deduce the p, V with constant @. These are shown in the inset of fig. 9, plotted with & (that is, 1/V?) as abscissa. This completes the description of the collision relation. 10. The Square Nucleus. The collision relation of §9 was calculated on the very arbitrary assumption that we can represent the action of a plate by means of a pair of poles, twisted round so that a-Particles with Hydrogen Nuclet. 509 they always lie in the plane in which the orbit is described, It is next necessary to see whether this is at ail a reasonable assumption. To do so we take a model nucleus and compare its equipotential surfaces with those of the bipole. The recent evidence indicates that the nucleus of helium is made up of two electrons and four protons, to adopt the name suggested by Rutherford for the unit positive nucleus, which in its isolated state is hydrogen. It was therefore natural to choose a model composed out of these parts, and for simplicity of calculation they were supposed arranged with the protons one at each corner of a square and the two electrons together at the centre. It was not intended that this should really represent the structure of the nucleus, (and indeed there is very great difliculty in-any model of this type in connexion with the theory of the relation of mass to energy,) ; it was merely adopted as giving’a convenient form Fig. 10a. Fig. 108. EQUIPOTENTIALS OF BIPoLE (a) AND SQUARE NUCLEUS (0). The potentials are measured with e/a as unit, where a, the unit of length in the figures, is the half distance between the poles of the bipole, or the radius of the square. ‘The dotted lines are the equipotentials in a plane inclined at 45° to the others. for examining the differences between the equipotentials of a real model and those of the bipole. Fig. 10a shows the equipotentials of the latter. In fig. 10} the firm lines are the equipotentials in a plane through the centre of the square containing two of the protons, and the dotted lines show those in a plane at 45° to the others; in the outer parts the two are indistinguishable. Now the inner parts of the figures are very different and some calculations of the 510 Sir J. J. Thomson on the Structure of position of the apses in the bipole orbits show that H does. penetrate into these regions. So it is really only possible to- put the argument in very vague form and to say that there appears to be no obvious characteristic of fig. 10, which would suggest that the bipole should have a more plate-like. effect than the square nucleus. LL. Conelusion. The comparison of any theoretical collision relation with experiment is a very exacting test, for it is a comparison of relations between three variables, and there is only oae- adjustable constant, the diameter of the nucleus. All the others, masses, charges, and velocity of approach are known from other work. Comparing fig. 9 with figs. 4 and 5, a rough measure of agreement was found with a nucleus- of diameter 2°7x 10-' cm., but in view of the doubt that has been thrown on the experiments not much confidence can be placed in this. It is better to leave the decision as to the shape and size of the nucleus, until accurate experi- ments are made. One point of very great interest may,. however, be noted as a possibility. Should future experi- ments confirm the flatness of curve A in fig. 4, it would be very strongly suggestive of the type of collision relation exhibited in fig. 7: that is to say, it would indicate that there was a discontinuity in the law of force between the nuclei. XLV. On the Structure of the Molecule and Chemical Combination. By Sir J. J. THomson, O.M., FBS. * ROM many points of view the structure of the molecule is even more important than that of the atom. For the structure of the molecule involves the method by which the atoms are linked together to form stable systems of different kinds, and is thus at the root of the enormously important question of chemical combination. In fact ‘the extension of the electronic theory of matter to chemistry depends upon the solution of the problem of the structure of the molecule. Regarding an atom as a stable arrangement of a mixture * Communicated by the Author. Many of the results in this paper were given by me in my lectures at the Royal Institution between 1914 and 1919, but owing to the pressure of other duties I have not hitherto- had leisure to prepare them for publication. the Molecule and Chemical Combination. Sle of a positive charge and, except for hydrogen, a considerably larger number of electrons, we have to consider what happens when two or more such atoms are brought close together ; what kind of rearrangement of the electrons takes place hen the atoms unite and condin 2 copluenent system, and what are the considerations which determine the properties and stability of such a system. I assume, as in my paper in the Phil. Mag. April 1919, that in the atom the électrons are in equilibrium under their mutual repulsions and the attractions exerted upon them by the positive charges. The repulsions between the electrons are supposed to follow the usual law and to vary inversely as the square of the distance. The force between the positive charge and the atom is supposed to be more complicated, and to vary with the distance 7 according to a law ex- G . ° ° pressed by ¢ (“) where ¢ is a length. When + is either very ry L cy) C A large or very small compared with c, o(;) reduces to 1/r”’ but when r is comparable with c the force is no longer of this simple type, but vanishes at certain distances, changing from attraction to repulsion or vice versd. My reasons for preferring this equilibrium arrangement to the more usual one of electrons describing orbits under forces varying rigorously as the inverse square, is that unless each electron is allowed to have a separate and isolated orbit the orbital arrangement is essentially unstable. Thus if the two electrons in a helium atom were to follow one another round the same circular orbit, the system would be quite unstable. Again, the scattering of light by hydrogen molecules is not that which would occur if the electrons in the molecule described orbits (J. J. Thomson, Phil. Mag. ser. 6, vol. xl. p. 393). When we have to consider molecules containing perhaps. dozens of electrons, the motion would be so hopelessly intricate and confused if these were all describing these large orbits that, apart from the fundamental difficulty about stability, the model would give us but little assistance in forming a mental picture of what goes on in chemical combination. Assuming that the positive charge exerts on an electron a central force which changes from attraction to repulsion and therefore vanishes at certain distances, we proceed to consider how the electrons would be arranged round a central char oe. If there is only one electron in the atom it must be at one of the places where the force exerted by the positive charge vanishes, and in order that the equilibrium may be stable the i Sir J. J. Thomson on the Structure of force must become attractive when the electron is displaced from this position away from the centre and repulsive when it is displaced towards it. If there are two electrons A and Bin the atom, these must be situated so that the centre of the positive charge is mid- way between A and B; the distance AB is determined by the condition that the repulsion between A and B is equal to the attraction exerted by the positive charge on either of these electrons ; three electrons will arrange themselves at the corners of an equilateral triangle, four at the corners of a regular tetrahedron and so on; the electrons are on the surface of a sphere concentric with the positive charge. When there are any number of electrons, the conditions for equilibrium are that the electrons should be so symmetrically placed that the force exerted on any electron by the other electrons should be along the radius, and that the magnitude of this radial force should be the same for all the electrons. It can be shown without difficulty (J. J. Thomson, Phil. Mag. ser. 6, vol. vii. p. 237) that the radial force on an electron P due to the other electrons Q, R.S ... is equal to e?$8,,/4ar?, where r=OP, O being the positive charge, e is the charge on an electron, and Sy ee ee * sin S POQ.” sin SPOR > ‘sin 4 POS ae or if we take @ to denote one of the angles $POQ... 1 Se sin 0’ where & denotes that the sum is to be taken for all the angles. Hence if we can find a distance so that the attractive force exerted by a positive charge on a electron at this distance is equal to e’S,/4a7r°, and if the electrons are so symmetrically arranged that 8, is the same for each electron, the electrons will be in equilibrium under the central force. This eqnilibrium, however, will be unstable unless another condition is satisfied, and it is the limitation imposed by this condition that in my opinion determines the structure both of the atom and the molecule. A simple illustration will show the stringency of this condition. Suppose that the electron P is displaced along OP from its position of equilibrium by a small distance 67, all the other electrons remaining fixed, then it can be shown that in consequence of this displacement of P the repulsion the Molecule and Chemical Combination. 513 exerted by the other electrons along a is increased by Ca ] au "87° 3(so9 Yin = If <6(*) is the attraction exerted by the central positive charge on the Siaeon at P, then when P is displaced the. attr nection is increased by th. AB eor > Ae (<). but unless this increase in the attraction is greater, or at any rate not less than that in the repulsion of the electrons, the. electron when displaced will not return to its original position and the equilibrium will be unstable: hence for any symmetrical distribution of electrons to be stable ©) > $3 (sag—aa Nvus As the number of electrons increases the angle between an electron and its nearest neighbour diminishes and the right hand side of this expression becomes greater. It becomes exceedingly large when the electrons are so numerous that 0 is a small quantity, and so the equilibrium will be unstable unless «$ @ is also very large. Now whatever be the law of force, we may assume that at the same distance from the centre both the magnitude of the force and its rate of increase with the distance will be pro- portional to the magnitude of the central positive charge ; this central charge is proportional to the number of electrons. We can see that if we disregard stability we ean find a position of equilibrium even though the number of electrons is very large, for as we increase this number we increase the repulsive force eS, 49" exerted by the electrons, but the increase in 8, is comparable with the increase in the number of the electrons and therefore comparable with the increase in the centra! attractive force Thus as the number of electrons increases, the attractive and repulsive forces can keep pace with each other, and it will be possible to find a position of equilibrium for a wide range in the number of electrons.. When, however, we come to consider the stability of the arrangement the case is very different. We see that the right-hand side of (1) contains the term }(1/sin*6;) and this term when 6; is small increases 514 Sir J. J. Thomson on the Structure of as 1/03. If the electrons were arranged round a ring 1/6,° would be proportional to n®, and if the electrons were on a sphere it would vary as n? where n is the number of the electrons, in either case it increases much more rapidly than n ; n however measures the rate of increase of the left-hand side of (1). Hence even if for small values of n, “8 (2) is greater than is necessary for stability by a considerable margin, as n increases the value of this quantity required for stability increases faster than the increase in the charge ; hence the margin of stability must ultimately disappear and the arrangement become unstable. I have worked out the ralue E of the central charge required to make the equi- librium stable for symmetrical distribution of varying numbers of electrons when the force due to the central charge E is expressed by = { = =F This law of force changes from attraction te seorleien only once, i. e. when r=c, and is thus about the simplest of its type. It will probably Tepresent with considerable accuracy the law of force in the neighbourhood of a position of equilibrium for a much more general type of force. I hope to give the analysis in a ‘separate paper. I will here quote two results which I have already given in lectures at the Royal Institution. The first table gives the value of the ratio of H/e to make the equi- librium of n electrons arranged at equal intervals round a circular ring stable when the motion is confined to the plane of the ring. if 10 42 A4SGRStsS Hije.... ‘70 -75 1°58 3:10 4:76 7732 142 24:48 35°9 oS "sous Thus to keep 10 electrons stable in a ring would require a central charge at least 25 times that on a single electron, while if Bere are more than 5 electrons in a ring the charge at the centre must be greater than the sum of the charges on the electrons. The next table gives the value of H/e when the electrons are arranged at the corners of a regular polyhedron instead of at equal intervals round a ring In one plane. n= 1 2 33 4 6 8 1220 Were re 10 Vio -158" 2-44 Se 7G 13 80 These results are independent of the value of ¢, the constant in the expression for the law of force. Jt may be pointed out that the cube is not the stable ae, the Molecule and Chemical Combination. 5 ey: arrangement of 8 electrons. This as Foppl (Proceedings of the International Congress of Mathematics, Cambridge 1912, vol. xi. p. 188) has shown is an arrangement when the 8 electrons are arranged in two sets of 4 in parallel planes, the electrons in each set being at the corners of a square, one square being twisted throngh an angle of 45° relative to the other so that the projection of the electrons on a plane parallel to either square is a regular octagon. This arrange- ment for 8 electrons shown in fig. 1 is analogous to that for 4 or 6, for the tetrahedron may be regarded as two Bigs: sets of 2 at right angles to each other and to the line joining their middle points; the projections of the electrons on a plane at right angles to this line are at the corners of a square, while the octahedron may be regarded as two sets of 3 points, each set forming an equilateral triangle; the planes of these triangles are parallel and the triangles are twisted relatively to each other so that the projection of the electrons on a plane parallel to either triangle is a regular hexagon. Whereas all the plane faces of a cube are four- sided, the twisted polyhedron has 8 triangular faces as well as 2 four-sided ones, thus two such polyhedra could be placed so as to have either 2, 3, or 4 corners in common. The stable arrangement for 5 electrons is when three are at the corners of an equilateral triangle with its centre at the central charge and the other two are at equal distances on opposite sides of this triangle. With 7 electrons five are atthe corners of a pentagon whose plane passes through the central charge and the other two are at equal distances ‘on opposite sides of this plane. The point in the tables to which I wish to direct special attention is that, whether the electrons are arranged in one plane or distributed over the surface of a sphere, whenever the number of electrons exceeds a limit, which may vary with the law of force, the positive charge required to keep them in stable equilibrium will exceed the sum of the negative charges on the electrons. Thus with the law of force Just assumed, if there are more than 5 electrons in the ring or 516 >Sir J. J. Thomson on the Structure of more than 8 on the surface of the sphere, the positive charge at the centre will have to be greater than the charge on the electrons. JI believe thatit is this fact which governs the structure both of the atom and the molecule and determines the qualities which the chemists group under the term valency. ; For the chemical atom is electrically neutral, the positive charge is equal to but not greater than the sem of the charges. on the negative electrons, ‘hence no ar ‘angement of electrons is possible which requires for its stability a central charge greater than the sum of the charges on the electrons. Very simple considerations wi rill show the consequences which follow from this fact. I will suppose that the electrons are arranged symmetrically on tle surface of a sphere whose centre is‘at the centre of the positive charges. This arrange- ment, as we have seen, is possible if there are not more than 8 electrons, and we can have atoms with from one to eight electrons on the surface of a sphere surrounding a positive charge equal to the sum of the charges on the electrons. Now let us consider the case i an atom containing 9 electrons. The symmetrical distribution of 9 electrons over _the surface of a sphere requires a central charge of more than 9 units to keep it in stable equilibrium, but when there are 9 electrons the central charge is 9 and hante is insuff- cient for this distribution. A new distribution will be required which will be of the following kind. The central charge 9 can hold 8 electrons in equilibrium on the surface of a conceniric sphere, so that 8 of the electrons will group themselves round the central charge and there will be one over, this will go outside the shell of'8 and find a position of stable equilibrium at a greater distance from the centre. Thus the external layer a this atom will contain only one electron and in this respect will resemble the atom with one electron and unit positive charge. Now suppose we have an atom with 10 electrons, eight of these will form the inner shell and two will be left over to form the outer ; thus we shall have an atom resembling as far as the ee ring is concerned the 2 electron atom with the double positive charge. If we increase the number of electrons still further we shall get outer rings with 3-4---8, but there will never be more than 8 in the outer shell. W het the electrons have increased so much that 8 on the outer shell is not sufficient to accommodate them, a third shell will be formed, and when this is filled up with 8 electrons a fourth will be formed, and when this is filled up a fifth and so on. Thus, if we confine our attention to the outer layer and arrange the elements in the Molecule and Chemical Combination. 517 the order of the number of electrons they contain, there will bea periodicity 1 in the number of electrons in the omen layer ; it will increase from one to eight, then drop again to one, increase again to eight, drop to. one, and soon. Thus as far as properties depending on the outer layer are concerned the element will exhibit a periodicity similar to that expressed by * Mendeleef’s periodic law in Chemistry. It may be pointed out, however, that the number of elemenis included within a period may possibly for the elements of large atomic weight be greater than eight. This might arise in the following way. Suppose that for a particular value of N a central charge N can hold n electrons in stable equilibrium, while N+1 can hold n+1. Then the atom of the element whose atomic number is N would have a layer of n electrons next the centre and other layers outside, the atom of the element next in order would have a central charge of N-+1 units and contain (N+1) electrons ; but as a charge N+1 can hold n+1 electrons in stable equilibrium the innermost layer might now contain n+1 electr ons, and thus the additional electron might be used up in the inner layer and nor affect the number im the surface layer. ‘Thus the atoms of the elements whose atomic numbersare N and N+1 will have different atomic weights and different central charges, while the number of electrons in the surface layer will be the same: hence, if there are any properties which depend exclusively on the number of electrons in the outer layer, these two elements will have these properties in common. If we are right in supposing that the valency of the element is a property of this kind, then the two elements will have the same valency. When there are a gre eat number of electrons in the atom arranged in many lay ers, 1t may require the addition of several electrons before a new electron finds its way to the outer layer, and thus there might be a considerable number of elements with different atomic weights but with very similar chemical properties. There are groups of elements such as the iron, nickel, and cobalt group, the rhodium group, and the erowd of elements known as the rare earths which fulfil this condition. Unless the atom is electrified the total charge inside the outer layer of electrons must always be equal to the char ge on those electrons. Thus, however many electrons there may be in the atom, the alec dvonne in the outer layer cannot be under the aiaemee of an effective charge ereater than the sum of the charges on these electrons, so that with the law of Phil. Mag. S. 6. Vol. 41. No. 243. March 1921. 2M 518 Sir J. J. Thomsen on the Structure of force we are considering no element could have more than 8 electrons in the outside layer. I shall suppose (as in my paper on the “ Forces between Atoms and Chemical Affinity,” (Phil. Mag. ser. 6, vol. xxvul. p- 757, 1914), that those atoms which have one electron in the outer layer form the first Mendeleefian group, those with two - in this layer the second group, and soon. Thus the hydrogen. and lithium atoms are supposed to have one electron in the outer layer, the beryllium atom 2, the boron atom 3, the carbon atom 4, the nitrogen atom 5, the oxygen atom a the fluorine atom ik and the neon atom 8. Let us now consider if any light is thrown on this view by the evidence afforded by positive rays. If eight is the maximum number of electrons which can exist in the outer layer, then the atom of neon already possesses that number and cannot accommodate another electron, and so cannot receive a negative charge. On the other hand, the atoms with a smaller number of electrons in the outer layer have, as a reference to the table 2 shows, a superfluity of stability, and so could accommodate another electron and thus acquire a negative charge ; the superfluity is however not great enough for them to accommodate two so that we should not expect to find any with a double negative charge. In experiments with positive rays neon has never been _ observed with a negative charge, while negative charges are common on hydrogen, chlorine, oxygeu, and car bon; we have no information about boron and beryllium : thus far the evidence is in favour of the view. On the other hand, no negative charge has as yet been observed on the atoie of nitrogen : : this is remarkable, as the atoms of the elements on either side of it, carbon and oxygen, readily acquire a negative charge. It must be remembered, however, that the number of atoms with negative charges varies very much with the conditions of the “discharge and the gases in the discharge- tube. To get a negative charge a neutral atom has to drag an electron from another atom or molecule, and unless an atom of nitrogen came in contact with an atom of some sub- stance which held its electrons more loosely than the nitrogen, it might not be able to aes the electron necessary to give it the negative charge. It will be seen that all the atoms except hydrogen, lithium, sodium, potassium, could lose more than one electron and thus have double or treble positive charges : this is in accdrdance with the evidence afforded by the positive rays. We have, however, to explain why anatom with 7 electrons the Molecule-and Chemical Combination. 519 in the outer ring, such as that of Fl or Cl, can have the same valency as an utom containing only one electron. yal ate ae B ' fA oS) “=. Suppose that A (fig. 2) is the centre of the 7 electron atom, B that of a 1 electron atom. Let « be an electron on the first atom, 8 one on the second, It phos electrons place themselves Sonora it as in the figure,/the attractions they exert on A and B may keep t these. elles in spite of the repulsion between the positive charges A and B, while the attractions A and B exert on « and 6 may keep these to- gether in spite of the repulsion between them. The addition of B to the seven electrons already round A will raise the number in the outer layer round A to eight. Now suppose we attempted to Mone another { electron «tom B’ to JV ‘This would introduce another electron in the layer round A, raising the number in this layer to 9. But we have seen ‘that 9 electrons in one layer cannot be kept in stable equi- librium by a positive charge of nine units, but 9 units of positive electricity is all we Thaw at our disposal, and two of these units are outside the layer: though they are outside they will make the arrangement of the electrons more stable than if they were absent; they will not, however, increase the stability more than if they were inside, nnd even in that ease they coald not make the arrangement stable. Thus A cannot hold a secorid atom of the type B, so that the com- pound AB, is impossible, while that represented by AB is saturated. Thus, if B be taken as the type of a monovalent atom, A with its seven electrons would act like a monovalent atom and would thus conform in this respect to the behaviour of Fl. The electrons in the molecule are thus arranged in a cell containing 8 electrons and surrounding the core of the fluorine atom. Next, suppose that A instead of having 7 electrons in the outer layer has only 6, and let an atom B get attached to it in the way we have supposed. This will raise the number of electrons in the layer round A to seven ; it can attach another atom before the number of electrons in the outer layer is raised to 8, and the limit of stability 2M 2 520 Sir J. J. Thomson on the Structure of reached. Thus, if A were a 6 electron atom it could form the unsaturated compound AB and the saturated one AB,, in which again there is a celi of 8 electrons round the more heavily charged atom; A could not combine with more than 2 of the B atoms. Thus, A with its 6 electrons would act like a divalent atom, and in this respect could represent oxygen. We see, too, that if A contained 5 electrons it could form the unsaturated compounds AB, AB,, and the saturated one AB;, but not any containing more than 3 B atoms. The 5 electron atom would thus act like a trivalent atom, and could thus in this respect represent nitrogen, the electrons in the molecule forming a cell of 8 round the nitrogen atom. We see too that since the attachment of a 1 one-electron atom like that of hydrogen requires two electrons on the layer round the atom with which it is combined, we cannot have more than four hydrogen atoms attached me a single atom; this is in accordance with experience. We can illustrate by means of the 5 electron atom the fact that, on the view of the constitution of the atom we have assumed, an atom may possess two valencies, the sum of these two being always equal to eight. Let us take for example the nitrogen atom, then, if it enters into combination with hydrogen, we have seen that 3 hydrogen atoms will saturate it as they will bring the number of electrons in the layer round the nitrogen nucleus up to 8, the limiting number. But suppose that, instead of combining with a “hy drogen atom, it combines with chlorine whose atom we assume to have "7 electrons in the outer layer. One of the electrons of the nitrogen atom may join the layer round the chlorine atom, bringing the number of electrons up to 8, the limiting ance and leaving 4 electrons in the layer round the nitrogen nucleus. ‘These 4 electrons can link up with four atoms of hydrogen, this will bring the number of electrons in the layer round the nitrogen nucleus up to 8, the limiting number, giving the compound NH,Cl, the arrangement of the electrons being as represented in fig, 3 oF where the ring denotes a cell of 8 electrons. Fig. 3. a As the chlorine nucleus has a positive charge of 7 and is surrounded by a layer containing 8 electrons, the system @) the Molecule and Chemical Combination. 527 has a unit negative charge. Since there is a positive charge of 5 on the nitrogen nucleus and one of 4 on the 4 hydrogen atoms, there are 9 positive charges on the system NH, while it is surrounded by only 8 electrons: thus on the balance there is a single positive charge on the system (Ht). Thus the compound might be represented by (NH,),Cl_,and would when electrolysed give NH, and Clas ions. In this case the molecule contains 2 cells of 8, one surrounding the nitrogen core and the other that of the chlorine. We have seen that the nitrogen atom could not take up more than three hydrogen ones, so that we could not have the compound NH;; we could, however, have that represented by NCl;. In this case each of the 5 electrons of the nitrogen nucleus would have gone off to complete the tale of 8 elec- trons round each of the 35 chlorine ions, and these cells would surround the nitrogen atom. Those electrons on the shells round the 5 chlorine atoms which are nearest to the nitrogen nucleus will form a layer round the nitrogen atom, and we should not expect stable equilibrium if the number of electrons exceeded 8. We should get this arrangement if three of the layers round the chlorine atoms presented an edge—each then would supply two elec- trons to the nitrogen layer, the other two layers would each present a corner and furnish one to the layer round the nitrogen ; thus two of the chlorine atoms would be more loosely attached to the nitrogen than the other three. The compound NCI; would also be possible on this scheme, for an electron round the nitrogen nucleus might link up with one from the chlorine and both form part of the layer round the chlorine as well as that round the nitrogen, as the number of electrons in the layer round the nitrogen nucleus cannot be greater than eight ; since there are 5 already round the nitrogen we can only get three chlorine atoms linked up in this way. Thus it is possible to have the two chlorides NCl;, NCI; ; though NCI; does not seem to have been pre- pared, phosphorus which, like nitrogen, is a pentavalent element, is known to have two chlorides, PCl; and PCI;. Ii, instead of an atom with five electrons in the outer layer, we had taken one like the atom of oxygen with six, it might attach to itself in the way just described six atoms of chlorine, and so be apparently hexavalent, as sulphur is in the remarkable compound SF, discovered by Moissan. In this way we are, as I pointed out in the paper on Chemical Combination already referred to, led to conclusions with 522 Sir J. J. Thomson on the Structure of regard to valency very similar to those advanced by Abegg and Bodlinder (Zeit. anorg. Chemx, 1899, xx. p. 453, 1904, xxxix. p. 330), who ascrixed to each element two valeneies according as it was combinet.wit4 a more electro-negative or more electro-positive element, the sum of these valencies always being eight. Cohen (Organic Chemistry, vole p. 3) says that the weak point of this scheme is that there are no compounds in which the alkali metals possess the valency 7. We should not expect, on the views given above, that these compounds could exist, for on this view the two valencies arise from saturation being arrived at in two different ways, one, e.g, when nitrogen was acting like a triad, through the lay er round the atom having acquired the maximum number of atoms consistent with stability ; the other, e. g., when nitrogen acts like a pentad, when all the electrons associated with the atom have been used up in binding other atoms to it. We should not expect those atoms which possess only a sma] number of electrons to show both kinds of vaiencies. For example, we should not expect. to find hydrogen acting like a septavalent element, for this would mean that a layer of eight electrons was in stable: equilibrium round the hydrogen nucleus which has only unit positive charge. Now a layer of eight electrons requires a very considerable positive charge to “keep it in stable equi-. librium, and it is improbable that a single positive charge inside, even though it were assisted by seven positive charges. outside, would be able to do so. The freedom of motion of the electrons in) an jetunmemeel importance in connexion with the attraction which the atom is likely to experience from other atoms. We can illustrate this point by consider ingjan atom containing one electron « which will be free to move in any direction provided its distance from the centre of the atom does not vary. Suppose that the atom is placed near a positively electrified body B,. then whatever may have been the initial position of the élec- tron «, it will swing round A until it gets as near as possible to B; the attraction of B on @ will then be greater than the: repulsion between B and the positive nucleus A, so that the atoms will always be attracted towards B. If « had not. been free to adjust itself under the force exerted by B, the atom would at as likely be repelled from B as attracted by it. We see from this that a very rigid disposition of the electrons in an atom will diminish the likelihoed of its being: attracted by or attracting other atoms. the Molecule and Chemical Combination. 523 It is worthy of notice that in a case like one when 8 elec- trons are distribnted over a single layer and are on the verge of instability, the readjustment of the electrons is prevented by another cause besides that of rigid arrangement. Tor any readjustment of their positions would be accompanied by a diminution in the minimum distance between two elec- trons, but in the undisturbed state this minimum distance has almost reached the limit consistent with s stability ; thus any rearrangement of the electrons will tend to be unstable and to break up. Thus the system comprising the atom and its electrons will not experience the attraction to which it would be subject if the rearrangement of the electrons could be maintained. The distance of the outer ring of electrons from the centre of the atom will vary with the valency of the atom, and can be caleulated from the condition that the attraction : the electron by the central charge balances its repulsion by the other electrons in the ring. We have assumed thai the Ren laGton between a positive ! a ) charge and an electron is of the form —~ ——: we must con- oe sider the expressions for a and 6 when there are negative electrons as well as positive charges inside the aie ring. Since the force between electrons varies rigorously as Lr, the existence of these |inside the ring will not affect the term 6/r? in the expression for the attraction: this term will be proportional to the as charge, and therefore to the atomic weight N. On the other hand, these electrons will affect. the mera) a/r?: they will make a proportional to the difference between the positive charge and the charge on the electrons inside the outer ring; but this difference in a neutral atom is equal to the charge on the electrons in the outer ring. Thus we mav write the attractive force on an electron in the outer layer in the form ne Ne.a 7 aT 78 cay where N is the atomic weight of the element and n the ry number of electrons in the tak ring, 2 isa constant. The repulsion exerted by the other electrons in the outer layer is etal b Ents equal to §,/47?: where Sa=>— a” where 2@ is the angle ; sin. subtended at the centre of the atom by the line joining 524 Sir J. J. Thomson on the Structure of 2 electrons in the outer ring. Since the repulsion must balance the attraction, we have we Ne. ies Sa, y? re > Ae Na or p= Op n—- — A. For hydrogen N=n=1, S,=0, | so. inal yieee ‘ere Ihnelavguna an IN asian ire = leas) ==) 5) eee forsberylliumin N95 —2, iS, —, > Teo oleae B I Se Sea a WO eee EOE, DOLOM Bao. 5. Neal for carbon ...... N=12, n=4, 8,=3°06, .,,” (7 § ones formiinogen ... N— In — 9), 3,2, ook Py = 3°18 a; for oxygen...... N=16, n=6, 5,;=6168, 4, "ope ees for fluorine N=19,n=7, Sx=8'08, 5, | 7a) eaoroee, (EOI TNO ea N=20,n=8, 5,=101, ,, 7, =d'bde; ory SOUIMM 42.22 NSS 235 1, Sn—=0); Pe i - Thus, taking the elements from lithium to neon, we see that the radius of the outer layer is greatest for the lightest element and diminishes rapidly at first and then very slowly to the end of the series; when we pass from neon, the last element in this period, to sodium the first in the next, there is a great increase in the radius: this increase is again followed by a diminution which continues until we reach argon ; when we pass to potassium, the first element in the next series, there is again a large increase. The radii of the layers in the same group like lithium, sodium, potassium, fluorine, chlorine, bromine, increase with their atomic weights. Thus the relation between the radii of the layers and the atomic weight is represented on this theory by a graph differing from the historic one for the atomic volumes and atomic weight given by Lothar Meyer and reproduced in almost every text-book of chemistry: here the minimum atomic volume comes in the middle of a period and not at the end. Recent investigations, however, have shown that the relation between atomic volume and atomic weight is not accurately represented by a curve of the type of that of the Molecule and Chemical Combination. 325 Lothar Meyer. Thus, Gervaise le Bas, ‘ Molecular Volume of Liquid Chemical C ompounds,’ p. 237, says :— “1. There is a periodic relation between the atomic volume and the atomic weight of the elements. ey Whreresisra tendency for the atomic volume to diminish in each series as the atoms increase in weight. The smallest occur in group 7. 3. There is a general increase in the atomic volume of the members of each group from series one on- wards, that is in the direction of increasing atomic weight.” This is in entire agreement with the results we have just found. The same thing is beautifully shown by the expe- moments or VW. i: Brago Cebil. Mag. xl. p. 169, 1920), which give a curve (fig. 4) for the atomic radii which for the period from lithium to neon agrees remarkably well with the figures given above. We should expect, since the law of force we have assumed is probably only an approximation holding near a position of equilibrium, that the relative values of the radii of the elements within one period would be more reliable than those for elements in different periods as the value of « would vary from one period to another. Work requred to separate one electron from the atom. The work required to separate an electron will depend upon whether the electron is ejected so quickly that the electrons left in the atom have not had time to alter their positions appreciably before the ejected electron has passed ‘out of the sphere of their influence, or whether the process takes place so slowly that the other electrons subside gradually into positions of equilibrium without acquiring any kinetic energy. If W,, We. represent the amounts of 526 Sir J. J. Thomson on the Structure of work for these processes respectiv ely, it is easy to show that if the attraction is mer as on page 523 by rece tc Se Naat W i= ON (» ale 18) z Sn 2 1— z Wie se { n(n") —(n-1) (n— a) } . W, is always greater than W,. From these expressions. we find the followit ng values for W, for different types of atom. Gas. W,. Wie Gus. W,. W.. ye le? 2 2 Hydrogen., = — a es le? eee yarog De Ne Nitrogen... 5 X99 Dg Ne ae 1 ¢? 12? Le ie w 29 a an | a =e € Lithium ... a4 333 on x'°333 | Oxygen Secs a A ae ilke2 if ieee . Beaty tae ee Pio : e é Beryllium .. 54 XO 5°03 |Fluorine... 5 -x:55 a, XA = “aa a ] e if v2 oe Boron) ea. dn x 62 ee 1 2? 1é a Ie2- a Carbon = Eo a - X48 Sodium. 3.07 = 5109 Date 2 @ 2a Thus the ionizing potential is least for lithium ; it then rises sharply for beryllium, and remains nearly constant for- the rest of the period ; there isa great drop in passing to sodium, and again a rapid rise, fhe ease with which the. atoms of the alkali metals are ionized in comparison with those of other elements of the same period is very marked,. and accounts for the sensitiveness of the heavier alae metals to light of long wave-length. The ionizing potentials in the preceding table relate to. atoms and not a molecules. formation of Molecules. The molecules we consider are in the gaseous state, where each molecule is separated so far frem its neighbours that it. ean be regarded as having an individual existence and not merely as forming a brick in a much larger structure. The term molecule when applied to the solid state is quite ambiguous without further definition: for example, from many points of view we can quite legitimately consider the the Molecule and Chemical Combination. BZ whole of a large crystal as forming a single molecule. We should naturally expect that when the atoms are crowded together as in a solid, when each atom may come under the influence of a large number of neighbours, the arrangement of the electrons relativ ely to the atom may differ subst: antially from that in a gaseous molecule. Let us now take the simplest type of molecule, that of an elementary gas when the two atoms forming the molecule are identical, For the union of two atoms each containing one electron we have the arrangement represented in fig. 5, where the positive charges AB and the electrons a 8 are at Fico. 5. (2) at ZF ‘ the corners of a parallelogram. The angle of the parallelo-- gram will depend upon the law of force hone een two positive charges when separated by molecular distances. If eé’ “4(< ‘) 5 is the attraction between unit positive charge and an electron. at a distance 7, oy(*) the repulsion between two positive: {fh charges at a distance r, then if Ae (fig. 5) =7 and the angle An8=6 we have for equilibrium 9 e~ C 2e*d| - | cos d= (*) Ar” cos? 6? 224°) sin @=e'p(27 sin @). Thus we may regard the electrons in the atoms as a kind of hook by which one atom gets coupled up with another ; this disposition of electrons may be regarded as forming what is alled the ‘“‘ bond” by the chemists. Inasmuch as each bond of this kind requires two electrons, the symbol, whether a line or a dot, used by chemists to denote such a bond represents two elect ons; if,as in my paper ‘ Forces between Atoms and Chemical Affinity,’ l.c., we use two of these symbols in place of one, the number of symbols will equal 228 Sir J. J. Thomson on the Structure of the number of electrons concerned in binding the atoms together ; as was shown in that paper and will be seen in this, there is much more involved than a mere change in notation. . The union of two bi-electron atoms may be either of the type fig. 6 or 6a. When only one pair of electrons is used Figs. 6 & 6a. ! oe —np B 6 A Bg ‘2 ae 3! up in uniting the atoms, this type would be represented by —A—A—; or both the electrons in each atom might be used for coupling the atoms, the four electrons being at the corners of a square in a plane bisecting AA at right angles, fig. 6a. nis system of four electrons between the positive charges may be regarded as a double bond between the atoms. Inasmuch as the equilibrium of four electrons in one plane when the displacements are not confined to the plane re- quires very strong restoring forces to make it stable, we should not expect the double bond to be permanent when the positive charges which exert these forces are as small as in this case when their sum is only equal to the charges on the four electrons. If there are two-trielectron atoms in the molecule, if all the electrons were coupling up the atoms there would bea hexagonal ring of electrons in a plane bisecting AA at right angles ; as this ring requires a central positive charge greater than 6 to keep it in stable equilibrium, it would be unstable, and the more prebable arrangement of electrons is B ayy ce the octahedral one shown in fig. 7, and represented sym- bolically by —A=A—. With two four-electron atoms in the molecule we have 8 electrons to dispose of, the maximum the Molecule and Chemical Combination. 529 number which can be in equilibrium on the surface of a single cell. A way in which these might be distributed with 4 electrons between the two atoms is shown in fig. 8, where there is a double bond between the atoms A, B. Fig, 3. DN B3 H B, €25) Ba 4 Ny ‘ When we proceed to consider the union of two five-electron atoms, new considerations come in, for in the diatomic molecule we have ten electrons, two more than can be accom- modated on a single layer. A simple distribution of these 10 electrons is shown in fig. 9 where we have an outer cell, Fig. 9. 1 hy a, a; 2, 8.8.85 of 8 electrons, inside this are the two positive charges and midway between these two electrons a; 8; which together with the double bond a, 8, B, a, help to hold the positive parts together. With two atoms each containing 6 electrons, there are 12 electrons to accom- modate in a diatomic molecule. These will be sufficient to surround each positive charge with a cell containing 8 elec- trons, provided the cells have 4 electrons in common ; these electrons coming between the two positive charges will form a double bond tending to bind them together. electrons, we have When each of the atoms contains 7 14 electrons at our disposal in the diatomic molecule ; these are sufficient to surround each of the atoms with a cell of 8 electrons, provided the cells have two electrons in common, these two electrons forming a single bond. We might expect as the two cells have only two electrons in common, while those for the six-electron atom had four, that the cells. 030 SirJ.J. Thomson on the Structure of of the seven-electron atom would ceteris paribus come apart more easily than those of six-electron atoms. Thus we should expect substances with a seven-electron atom like chlorine to be more easily dissociated and more ener getic in their action than those like oxygen, whose atom only contains 6 electrons. We can apply simuar considerations to the union of atoms of two dissimilar elements, and we shall take as our text the molecule of carbon monoxide. This substance has excited a good deal of attention, as though from the valency point of view it is highly unsaturated, its physical properties, for ‘example the difficulty with which it ‘is liquefied, indicate than its molecules exeréise oe less than the normal attrac- tion upon each other. We know from experiments on the Positive Rays that non-permanent molecules can be formed which violate all principles of valency. Thus, when these rays go through such a gas as COC], we get evidence of the existence of ail kinds of combinations of carbon and chlorine atoms. Where valency comes in is in connexion with the duration of-the compound after it has been formed. A molecule built up in conformity with valency principles is one which, as we have seen, is not likely to attract or be attracted by other inolecules as much as one that is not made up on these lines. ‘he principle of valency depends in fact upon the * survival of the unattractive ” : attractive molecules die young. If we can secure this utiaktrachiwenaee by other than valency conditions there is no reason why the compound should not be as permanent as the orthodox one. Now let us consider the case of CO. Here we have 10 electrons to dispose of, the same as for Ny, and we might expect that a Poenibat siniliar arrangement to that shou in fig. 9 would produce a stable nd permanent molecule. As the positive charge on the carbon atom is not the same as that on the oxygen, the cell will be distorted and will not be symmetrical about the plane through the middle point of the } line joining C CandQ. It is interesting to notice that some of the physical properties of CO and N, are very similar. Thus, if aand dare the constants for these gases in Van der Waals’ equation, b being what is called the ome volume anda@aa - constant which is connected with the magnitude of the force which one molecule exerts upon another, ,—then (Kaye and _Laby’s Tables) : a. b. 10) ao OO eee "00275 "00168 JSS aA 00259 “00165 the Molecule and Chemical Combination. 531 Thus we see that a combination of this type, not deter- mined in any way by considerations of valency, might be as ‘stable and as saturated as the combination of atoms in the molecule of an elementary gas. One condition for the existence of compounds of this type is that the sum of the electrons on the two atoms should be greater than 8. Thus we should not expect, what perhaps on the ordinary theory a -of valency we might expect, that the existence of CO implied the independent existence of CH,, for this would have only ‘6 disposable electrons, and as these are insufficient to pro- ‘duce a completely saturated layer, they could not form a completely saturated compound. It is important to point out that we distinguish between the molecule of carbon monoxide and that of the carbonyl radicle CO. In the latter we suppose that two out of the four electrons of the carbon atom have gone to unite it with the oxygen and to make up the 8 required to form the cell round the oxygen atom, while the other two are free to join up with other electrons, so that while the molecule of CO is represented by fig. 9 that of the carbonyl radicle as represented by Fig. 10. The constitution of CO, is I think best regarded as the union of an oxygen atom with the two electrons a; a, of the carbonyl radicle and having the configuration represented by fig. 11, where the 16 electrons are arranged in 3 cells of Fig. 11. ‘ er eet cerry 8 round each of the atoms, each cell having 4 electrons in common with its next neighbour, these forming a double bond between each pair of adjacent atoms. ‘To simplify the diagram the cells are represented as cubes. It must be acknowledged that there are some compounds 532 Sir J. J. Thomson on the Structure of which we might expect from this point of view and of which we have no evidence: for example, the molecule NF contains 12 electrons like the oxygen molecule, and we might. expect that a configuration of electrons resembling with some distortion that of O, might give a stable molecule with properties somewhat simmilan to QO,. - Sov targacuel! am aware, no such compound has ever been detected or suspected. It must be remembered that the molecule N, which forms the type for CO is singularly inert: if a ah stance formed on the model of O, were to have more energetic chemical qualities than those of its prototype, it might enter into combination so readily as to escape observa- tion in the free state. We have, however, in the compounds of nitrogen and oxygen probably other examples of this type of molecule. The molecule of NO contains eleven.electrons : if we take eight of these to form the outer cell we are left with three which, by taking up positions at the corners of a triangle ina plane between the atoms of nitrogen and oxygen, may help to keep these atoms together. The molecule of nitrogen monoxide is not the only form in which the combination NO occurs; besides this, there is the radicle NO, and just as we supposed the carbonyl radicle to have a different configura- tion from that of the molecule, so we suppose that the radicle NO has a configuration where 8 electrons form a esl or 8 round the oxygen, while there is a cell of 7 round the nitrogen, the two cells having 4 electrons in common. Thus the cell round the nitrogen is unsaturated and the combination will act like a seven-electron atom, 2. e. be univalent. The three electrons left over from the five nitrogen electrons after two have been used to saturate the cell round the oxygen atom might bind three hydrogen ions, and thus it is possible that Tae Oaiein Grenmenmess fos radicle might act as if its valency were three. The view we have taken of the structure of molecules is consistent with the existence of many compounds which violate the ordinary valency conditions. According to this view a molecule would be saturated, if the disposable electrons—z. e., those which before combination occupied the outer layers of the atoms which make up the molecule— can be arranged in a number of connected cells, each cell containing 8 “electrons. The cells must be numerous enou gh for the core of each atom which possesses four or more electrons to be placed inside a separate cell. The reason the Molecule and Chemical Combination. 533 that each cell must have a heavily charged nucleus inside it is to ensure stability. If the cell stood alone it would require a positive charge of about seven units to ensure stability ; when it forms one of a group and there are positive charges outside it, it will not require so many inside to make it stable: the assumption that four will be sufficient is an arbitrary one, which fits in with chemical facts. It is clear that there must be a lower limit, though its calculation would be long and tedious. A simple way of determining the number of disposal electrons is by the rule that it 1s equal to the sum of the valencies of the atoms in the molecule: we must be careful, however, when we apply this rule, to count the valency of the halogens as seven and not one, that of oxygen and sulphur as six and not two, and that of nitrogen and phosphorus as five and not three. Perhaps the clearest way of seeing the differences between the consequences of the view of combination we are discussing and those of the ordinary theory of valency, is to consider some special cases. Let us first take the case when there is only one cell: then, to get the eight electrons required to complete this, a four- electron atom must combine either with an atom with four electrons, or preferably with a number of less highly charged atoms, the sum of the charges on these atoms being equal to four. A five-electron atom must combine either with a three- electron atom, or singly or doubly charged atoms containing altogether three electrons; the six-electron atom must combine either with one two-electron atom or with two one-electron atoms; the seven-electron atom must combine with one one-electron atom ; while the eight-electron atom could not. combine at all. These results are identical with those we should arrive at from the ordinary theory of valency, provided we ascribed to the 8, 7, 6, 5, 4 electron atoms valencies 0, 1, 2, 3, 4 respectively. Let us now, however, consider the case when we have two of these 8-electron cells connected together: these will require 12 electrons if the two cells have four electrons in common, 13 electrons if they have three in common, and 14 if they have two in common. We do not consider the case when they have only one electron in common, as it seems probable that this connexion would be too slight to keep the two cells together. Thus if we have 12, 13, or 14 electrons at our disposal, and only two highly charged atoms, we can make up a molecule which will fulfil the conditions of saturation. Let us consider the combinations possible for a four-electron atom. We shall, Phil. Mag. 8. 6. Vol. 41. No. 243. March 1921. 2N 5384 Sir J. J. Thomson on the Structure of for clearness, always in what follows represent 4, 5, 6, 7, 8 electron atoms by the symbols C, N,O, Fl, Ne respectively, while we shall represent the one-electron atom by H. Con- sider the combinations in which the C, N, O... atoms form part of a molecule in which there are two cells together containing 12 electrons. The possible combinations are given in the following table :— Two cells—-quadruple connexion. C.Ne; C:F.H, *COH,, *CNH;,) * CL ae Ngee 3 = NOES UNE 0: When there are 13 electrons in the two cells we have the following results :— Two cell—triple connexion. C.Ne.H, G.F.H, COQH, CNH, C,H Ngne NG ONO, ONE.) OY OnE -When there are 14 electrons in the two cells we have the follewing results :— Two cells—-double connexion. CNeH,, *C.F.H,, *COH,, *CNH;, *C,H,, N.NeH, *NFH,, ~) *NOH:, *N,H, OF .H, *O,H,,. *B Ome Of these only those marked with an asterisk fulfil the conditions required for the ordinary theory of valency. The theory does not assert that all these compounds can be formed: for example, we have included compounds containing the ‘8-electron atoms, these are the atoms of the inert elements. Owing to the very small attraction which such atoms exert on other atoms, it is not at all likely that such compounds can be prepared ; or, again, some of the compounds might break up with great rapidity into still more stable forms. All that the theory involves is that such molecules would be saturated and would be stable if they were subject only to small disturbances. It will be noticed that practically all the compounds where the two cells have two electrons in common, 2. e. are united along an edge, fulfil the condition of being in accordance with valency principles ; that for those where the cells have four electrons in common a considerable number are in accordance: while none of those which have three electrons in common satisfy valency conditions. We can see without much difficulty the Molecule and Chemical Combination. 535 that contact along an edge of a cell is analogous to what is described on the valency theory as union by a single bond; while the fourfold connexion, 7. e. contact along two lines, is analogous to the state known as union by a double bond. If we use the modified notation previously described, the number of bonds between two atoms is equal to the number of electrons their two cells have in common. There is nothing in the ordinary valency theory analogous to the cells having three electrons in common, though there is on the modification I gave of that theory in the paper on chemical combination already referred to. The three-electron contact has to be invoked (if we suppose the one-electron contact to be too fragile) to account for the existence of compounds like ClO,, which are inexplicable on the theory of valency if chlorine has any odd valency. On the theory we are discussing there are 3 cells in this com- pound, and these have to be formed from 19 electrons: this could be done by making the cell surrounding one of the oxygen atoms have a double contact with the other oxygen cell, while one of these oxygen cells has a threefold contact with the chlorine cell. Another example where the threefold contact might come in is the well-worn one of the benzene ring. In benzene C,H, we have on our theory to make up 6 cells and we have | 30 electrons with which to do it. The simplest and most symmetrical way of doing this is to have the six cells in contact round a ring with threefold contact between each two. As two opposite triangular faces of the twisted cell represented in fig. 1 are inclined to each other, this could be done without introducing much strain in the system. With this arrangement we have complete symmetry, and it is analogous to the Armstrong and Baeyer or central theory of the benzene ring. The analogue on our theory to Kekulé’s conception of the constitution of the benzene ring would consist of three sets of pairs of cells, the cells in one pair having fourfold contact with each other, but only double contact witha cell in a neighbouring pair. It has often been pointed out that it is dificult to explain on this view why we do not get, when we replace two atoms of hydrogen by two atoms of chlorine, more isomers than have been observed. Let us consider from the point of view of this theory the changes which must take place in the disposition of the cells if one of the carbon atoms takes up another atom in addition to the hydrogen atom already attached to it. Let us suppose that this is another atom of hydrogen. This atom will introduce another electron in the system which will have to 2N 2 536 Sir J. J. Thomson on the Structure of be accommodated in the eight cells. To do this the contact between one or more of the cells must be altered; to find room for the new electron one of the three electron contacts must be reduced to a line contact, 2.e. two of the cells must have only an edge in common instead of the triangular face which formed their interface before the new electron was introduced. But if one cell moves so that it opens out the contact with the cell on its left from a three- to a two-electron contact, it will also alter its contact with the cell on the right, unless this moves also, and reduce this also to a two-electron contact ; hence the system will be unsaturated unless another electron is introduced to fill up the gap caused hy the loss of a contaet. To supply this electron another atom of hydrogen must be introduced. Hence we see that the addition of the hydrogen atoms must occur in pairs, and unless there is a movement of more than one cell throughout the chain these pairs must be on adjacent carbon atoms ; if one half the ring were to move as arigid body past the other half, the pairs would be on opposite carbon atoms. These conclusions are borne out by the study of the addi- tive compounds of benzene. When there are two rings, as in naphthaline C))Hs, there will be 10 cells and 48 electrons; this would correspond to an arrangement like that in fig. 12, where the two cells round the two central carbons have two-electron contact, while the ten contacts between the outer cells are three- electron ones. The view that the electrons in a molecule are arranged in sets of eight forms the basis of the very interesting papers on chemical combination recently published by Professor Lewis and Mr. Langmuir, their view as to the origin of these sets differs from that given in this paper. We shall now turn from the chemical side of the theory to the physical one, and consider how far its consequences are in accordance with our knowledge of the physical properties of atoms and molecules. As positive ray analysis is the most the Molecule and Chemical Combination. 537 powerful method for the study of individual atoms and mole- cules, we shall consider from the point of view of the theory the evidence afforded by this method. Let us begin by considering the positively electrified atom. Such an atom is one that has lost one or more of its electrons : but diminishing the number of the electrons round the central charge will increase the stability of those that are left; hence, if we could produce them, atoms which had lost one or more charges would be stable and might be expected to be in evidence in the positive rays. The magnitude of the positive charge cannot be greater than that pr oduced by removing all the electrons from the outer ring; hence we could not havea positive charge of more than 8 units, which is the maximum number of electrons in the outer ring: as a matter of fact, 8 is the greatest charge yet observed on any atom in the positive rays. Again, hydrogen has never been observed with more than a ‘single positive charge, which is in agree- ment with the view that its atom contains but one electron. Another way in which the theory might be tested would be to study the positive rays for the vapours of the alkali metals and the alkaline earths : these on our theory contain respec- tively one and two electrons in the outer ring ; hence the vapour of an alkali metal ought not to be able to acquire a double charge, nor that of an alkaline earth a triple one. It is difficult. to get these vapours in the positive rays in the ordinary way, but it is hoped that the application of the positive ray methods to anode rays will result in a thorough study of the properties of the atoms of metallic vapours. One result we may deduce already from our observations on the positive rays, and that is that though chlorine is regarded as a monovalent element it contains more than one electron in its outer layer, for chlorine atoms with a double positive charge occur whenever the positive rays pass through chlorine. As chlorine on our view has 7 electrons in its outer layer, 1t would not be surprising to find chlorine atoms with three or four positive charges. Let us now turn to the case of positively electrified molecules. These with one positive charge are to be found in almost every positive ray experiment, thou oh Tam not pre- pared to say that every molecule can survive » the removal of an electron and acquire a positive charge without dissociating into atoms. As an example of the way in which the removal of an electron might dissociate a molecule into atoms, we may take the case of a molecule of chlorine in which the two cells are supposed to have only two electrons in common ; if one 538 Sir J. J. Thomson on the Structure of of these two were removed, the two cells would be connected only by a single electron which might quite likely be insufficient to hold them together and the molecule would dissociate. This, however, would only happen if the electron removed were one of two special electrons ; if any other of the 14 electrons were removed the connexion between the cells would be unimpaired and the molecule, though un- saturated, would be stable: as a matter of fact, positively electrified chlorine molecules are found in the positive rays. It is when we consider systems with more than one posi- tive charge that the difference between atoms and molecules becomes most apparent; for while the parabolas corresponding to doubly charged atoms are to be found on nearly every positive ray photograph, those corresponding to doubly charged molecules though not unknown are rare. If the molecules were those of elementary gases, the parabola of the molecule with the double charge would coincide with that of the atom with a single charge, and so might escape notice ; this, however, would not apply to molecules of compound gases, and even in the case of elementary gases the existence of doubly charged molecules would modify the appearance of the parabola due to the singly charged molecule in a way that would lead to their detection. This very striking difference between atoms and molecules is, I think, due to the fact that it is the electrons which hold together the atoms in a molecule; they are, in fact, structural, and in general, when more than one of them is removed, the structure is weakened to such an extent that the molecule splits up and ceases to be a molecule. The mechanism by which the double charges are produced may be one which is much more likely to produce doubly charged atoms than doubly charged molecules. If the double charge was due to the system losing one electron by one collision and a second one by a subsequent collision, we should not expect to find the marked discrepancy between the numbers of atoms and molecules with double charges. There is, however, ample evidence from the positive rays that the atoms acquire their double charges by a single operation, and not in this way. This operation may well be the breaking up of a molecule containing the atoms. Thus, for example, if from a molecule of oxygen an cctet were to break away it would leave behind it an atom of oxygen and four electrons ; as the oxygen atom carrries 6 positive charges, the system would have a double charge. ‘Thus if the double charge, as seems probable, is due to the breaking up of molecules, we should expect to find the atoms rather than the molecules in possession of these charges. If one of the the Molecule and Chemical Combination. 539 products of dissociation of a complex molecule were «simpler molecule, we might get this with a double charge. Thus if a molecule of CO., for example, were to dissociate by an octet round one of the oxygen atoms breaking away from the system, the atoms C and O with 8 electrons would be left behind ; as the positive charges on the atoms amount to 10 units, the molecule of CO formed in this way would have a double charge. As a matter of fact, CO is one of the few doubly charged molecules I have found in the positive rays. Let us now turn to the negatively electrified atoms. These have had an extra electron added to the outer layer ; but if a cell of eight is the maximum number consistent with stability, © it is clear that an atom such as one of neon or argon, which already contains 8 in the outer layer, is not in a condition to receive another electron and so cannot be negatively charged. This is borne out by observations on the positive rays, for we never find the atoms of these elements occurring with a negative charge. Again, we have supposed thatin molecules, when the number of electrons is sufficient, the electrons are arranged in a series of cells of 8 ; since each cell is a satu- rated system, there is no place for an electron to find a resting-place and so no possibility of the molecule acquiring a negative charge. We find this borne out by the positive rays: a negatively electrified molecule is exceptional, though there are cases like those of oxygen and unsaturated hydro- carbons where the molecule can acquire a negative charge. The molecules which can receive a negative charge are, I think, those consisting of a pair or more of cells which have more than two electronsincommon. Let us take, for example, the molecule of oxygen: we have regarded this as consisting of two cells with four electrons in common; as long as this contact is intact, neither cell can receive an electron without losing its stability by containing more than 8 electrons. Suppose, now, that the contact were to open out so that the cells had only two electrons in common ; there would now be room on each cell for an electron without the number on either cell exceeding 8, so that this molecule might receive one or even two negative charges. Compare this case with that of a molecule of chlorine, where the two cells have only two electrons in common : it cannot receive an electron while the contact is intact, and any loosening of the contact would lead to a separation of the atoms; hence we should anticipate that while a molecule of oxygen could acquife a negative charge, one of chlorine could not. A molecule in which none of the cells have more than two electrons in common is one DAO Sir J. J. Thomson on the Structure of which, if its structure were interpreted in terms of the ordinary theory of valency, would contain no double bonds, z.e. it would be a saturated compound; such a molecule would on this theory be incapable of receiving a negative charge, whereas if it contain double or triple bonds some of the cells would have more than two electrons in common, and by opening out this contact might be able to accommodate more electrons and thus receive a negative charge. The negative charge on the molecule indicates that the contact or, in the usual terminology, the double bond has been loosened ; so that it would seem possible that an effective way of loosening this bond, z.e. turning unsaturated into saturated compounds, would be to give the molecules a negative charge by exposing them to a stream of electrons. This may play a part in the Sabatier-Senderens method of reducing unsaturated compounds by passing them over finely divided metals at a high temperature. The hot metal is well known to be a source of electrons. Let us consider from this point of view the formation of negative ions in another case—that of a gas ionized by Roéntgen rays. Wemay suppose that the first effect of these rays is to eject electrons from the molecules, so that initially the negative ions are electrons. They will remain electrons, and so have much greater mobility than the positive ions, unless they can attach themselves to atoms or molecules. But if the gas, like neon or argon, consists of atoms with 8 electrons in the outer layer there is no room for the electron on the atom, and if it collides with the atom it will rebound and remain a free electror—the mobility of the carriers of negative electricity will be that of an electron, and will far transcend that of the positive ion. Franck and Hertz long ago called attention to the great mobility of the negative ion in argon. Next suppose that the gas is not monatomic, but that in the molecule there is only a single octet of electrons, as in N,and CO; in this case, again, there is no room for anelectron, and we should expect the electron to remain free and havea high mobility. Franck and Hertz have shown that this is true for nitrogen. Suppose, however, that the molecule, like those of oxygen or chlorine, contains two octets with some electrons in common; then, by opening up the contacts the molecule could accommodate more elec- trons, so that in this case the electron could attach itself to the molecule and thereby make its mobility comparable with that of thegpositive ion. With oxygen, where the octets have four electrons in common, the cpening of the contacts might occur without the separation of the atoms. In chlorine, however, where the octets have only two electrons in common, . the Molecule and Chemical Combination. 541 the opening of the contact would probably result in the dis- ruption of the molecule into a negatively electrified chlorine atom and a neutral one, so that the electron would be attached to a chlorine atom and not to a chlorine molecule. Similar considerations will apply to more complicated molecules, and we are led to the conclusion that when the molecule of the gas contains two or more octets having electrons in comnion, the electron may be caught by the gas and its mobility reduced. If the molecule contains the ‘double bonds” of the chemists, 7.e. if two or more octets have 4 electrons in common, the electron will attach itself to the molecule of the gas. If there are only single bonds in the molecule the molecule will be dissociated by the electron and the electron will be attached to one of the products of dissociation. If, as in the water molecule, though there is only one octet there are a number of positively charged atoms outside it, the electron might attach itself to one of these atoms without destroying the equilibrium of the cell. It is interesting to find that, as shown by observations on po- sitive rays, radicles such as OH, CH,, CH;, which are highly unsaturated molecules, readily acquire negative charges. The ability of an atom to receive a negative charge depends on its positive core being able to hold in stable equilibrium one more electron than is found in the outer layer of the neutral atom: thus the existence of the negative hydrogen atom shows that a single positive charge can hold two electrons in stable equilibrium; the negative carbon atom shows that a positive charge 4 can hold 5 electrons in stable equilibrium; the negative oxygen atom that 7 electrons can be held by a charge 6; and the negative chlorine atom that 8 can be held by a charge 7. The absence of a nega- tively electrified nitrogen atom is remarkable, since the atoms of the elements on either side of it readily acquire a negative charge. ‘The reason may be that, though the nitrogen atom could hold a negative charge if it could get it, it is not able to snatch one from the molecules of the gas through which it is passing. When the number of electrons in a molecule is less than eight, as it would be in such compounds as NaH, BeH,, BHs, the molecules might be expected, if they could be obtained in the gaseous state, to behave something like an atom having on its outer layer the same number of electrons as are present in the molecule. These molecules might therefore be expected to be able to acquire a positive charge. It is worthy of notice that, with the exception of hydrogen and helium, no elements whose diatomic molecule contains less than 8 electrons is gaseous at ordinary temperatures. We 542 Sir J. J. Thomson on the Structure of have therefore no data by which we can compare the values of the physical constants for molecules of this type with molecules where the electrons are numerous enough to make up a series of saturated cells. We can, however, compare the properties of a gas like nitrogen, whose molecule contains only one cell, with that of the molecule of oxygen which is made up of two cells. It is, perhaps, a little surprising that what is called in the Kinetic Theory of Gases the radius of the molecule is actually less for the two-celled oxygen molecule than for the single-celled nitrogen one. We must remember, however, that unless we reoard the atoms and molecules as hard impenetrable solids, the size of the atom depends upon other than geometrical considerations ; it depends essentially upon the range of the forces exerted by the electrons and positive charges in the molecule. Thusasystem consisting of a uniform distribution of negative electricity over the surface of a sphere and an equal positive charge at the centre would produce no effect outside the sphere ; an atom of this character would have a long free path and behave like a small atom ; while an atom consisting of two equal point-charges, one ‘positive and the other negative, would have a large stray field and might behave like a much larger atom than the spherical one, even though the distance between the point-charges were less than the radius of the sphere. The nitrogen molecule, con- taining 10 electrons, has two of these inside the cell ; this will introduce a want of symmetry and increase the stray field, and thereby the apparent size of the molecule. But if there is not a break in the size of the molecule as we pass from nitrogen to oxygen, there is one in another property, which, as it depends essentially on the configuration of the electrons, is, I think, very suggestive. I allude to the effect of the molecule on the scattering of polarized light investigated by Lord Rayleigh (Proc. Roy. Soe. A. xeviii. p.57). The light scattered by a single electron in a direction at right angles to the incident beam is completely polarized, and can be extinguished by a Nicol prism. The same thing is true when the light is scattered by a perfectly symmetrical body such as a sphere; if the scattering body is not perfectly symmetrical—if, for example, it is ellipsoidal instead of spherical—the scattered light is never completely polarized, and therefore cannot be completely quenched by a nicol. The ratio of the minimum to the maximum intensity of the light as seen through a nicol, which would be zero for a sphere, would increase with the ellipticity of the ellipsoid, and may be taken as an indication of the deviation of the scattering body from sphericity. PUCROIEC Mie endl Chemical Combination. 543 For the scattering of light by the electrons in the molecules of different gases, we should expect the arrange- ment of the electrons in one shell to approach nearer to the spherical symmetrical form than an arrangement in two, and that two cells would approximate to the “symmetrical more closely than three; so that the ratio of ‘the minimum to the maximum intensity of the light would increase with the number of cells. The following are Lord Rayleigh’s deter- minations of this ratio for several gases :— Gas. Ratio expressed as a percentage. Argon eh ab an eA BivdroGenyye rs cin or Oo INitrosen ya ee lan!) 4206 Oxyvoomi agri iy ore Carbom dioxides 2) lile7 Nitrous oxide...) 1974 We see how small this ratio is in the case of the sym- metrical distribution of electrons round the argon atom, but the most striking feature of the table is the great Jump between nitrogen Pand oxygen ; this is just what we should expect on our view of the constitution of these molecules, as the oxygen molecule contains two cells while the nitrogen only has one. The carbon dioxide and nitrous oxide mole- cules are 3-celled molecules, and we shouid expect them to have a higher value than the 2-celled molecule oxygen. The measurement of this ratio promises to be a very valuable aid in determining the configuration of the molecule : it would, for example, be very interesting to know what is the effect of replacing one or more of the hydrogen atoms in CH, by halogens, and to compare its value for CH,, CoH., (,H,, and C,H, ; the first and second of these are one-celled molecules, while the third and fourth have two cells. In one sense even the octet is not fully saturated, for though it cannot receive any more electrons its own electrons may serve, as it were, as party walls against which the electrons round other atoms may fill up their gaps and become octets ; thus, for example, one of the electrons in an octet might complete the tale of 8 for a 7-electron chlorine atom. This might come about as follows :—We know from experi- ments on positive rays that the chlorine atom when moving through a gas seizes and retains an electron torn from one of the molecules of the gas. In the positive rays the chlorine atoms are moving at a high speed, and have therefore great energy ; this enables them to drag the electrons they erip out of ‘ine molecules. If, hemeven they have less than a critical amount of ener gy they will not be able to tear off the electron; this will grip them, and the chlorine atom will become the prisoner of the molecule. 544 Mr. A. Gilmour on the Resistance of Any of the electrons in any of the octets might act in this way to fill the gaps in the layer of electrons round an atom. Thus any octet might act like a nucleus from which chains and side-chains of atoms ramified in every direction. As far as geometrical considerations are concerned, there is nothing to limit the number of certain kinds of atoms which could be linked together in this way. It is probable that systems built up in this way are too weak to have any but the most transient existence; but the possibility of their formation to a limited extent in certain cases ought not to be lost sight of. XLVI. On the Resistance, of Solutions of Copper Sulphate in Glycerine. By A. Gitmour, M.Sc., 1851 Hahibition Scholar, Queen's University, Belfast *. la many laboratories a high resistance of very small induc- tance and capacity is becoming very desirable. In the physical laboratory of Queen’s University, Belfast, a mixture of glycerine and copper sulphate has been used for some years for high-resistance potentiometer and other high- resistance work. During the past year variable resistances of very large value and negligible inductance and capacity were required in connexion with some work undertaken on valves t, and, on investigation, glycerine-copper sulphate solution proved very suitable. Though glycerine has been known for a long time asa solvent of very wide range—at least as extensive as that of water—very little work has been done on the resistance of glycerine solutions prior to that of Jones and his colla- boratorst. They investigated the molecular conductivity of a large number of salts dissolved in glycerine and in mixtures of glycerine and water, but make no mention of glycerine- copper sulphate mixtures. In this paper the specific re- sistances of various mixtures of these substances are dealt with for convenience in laboratory work. Method of Measurement. The cell used to measure the resistance was an inverted U-tube a (fig. 1) into the horizontal part of which a glass tube with a bulb 6 was sealed ; the latter acted as a reservoir for any extra solution drawn up by the rubber bulb c, by compressing and releasing which the U-tube was filled. The ends of the U-tube were encased by spirals of copper wire ee, which acted as electrodes and dipped well under * Communicated by the Author. + Beatty and Gilmour, Phil. Mag. Sept. 1920. £ Jones and collaborators, Carnegie Inst. of Washington, No. 180, 1913. Solutions of Copper Sulphate in Glycerine. 545 the surface of the liquid in the test-tubes dd. The diameter of these test-tubes was large compared with that of the Figg. HEATER Lf ma tube a, so that any casual displacement of the ends of the tube a had no effect on the resistance measured. This was verified experimentally as was also the fact that the height of the liquid in d had no effect on the resistance. For the higher values of the resistance the cell was con- nected in series with a battery and a Broca galvanometer and the resistanee calculated from the deflexion and the voltage applied. For a few of the lower values the galvano- meter was too sensitive. A resistance of 10 ohms was put 10,000w 1lOw . . e ® in parallel with the unknown resistance R and the galvano- meter. When a resistance of 10,000 ohms was put in series in the cirenit (fig. 2) the galvanometer deflexions were 546 Mr, A. Gilmour on the Resistance of within the scale of the instrument, and the resistance R could again be calculated, knowing the deflexion and voltage applied. The sensitivity of the galvanometer was about 1:°3x1078 amp. per mm. scale-division, and was tested at the beginning and end of each set of readings to ensure that it remained constant during the experiment. To get the specific resistance factor of the cell, the latter was filled with clean mercury and the resistance measured. The specific resistance of mercury divided by this measured resistance gives the factor of multiplication which transforms the actual resistance of any liquid in the cell into the specific resistance in ohms per c.c. Solutions. Owing to the difficulty of obtaining drainage of the glycerine from burettes and measuring-vessels, the com- position of the solution is expressed in grams of copper sulphate per gram of glvcerine. Biue crystallized copper sulphate CuSO,, 5H,O was used. The required weight of it was added to a known weight of glycerine, and the whole heated slightly until the copper sulphate dissolved. The resistances were measured while the solutions were fresh. Results. The results are given in the following table and are represented in Curves I, II, Il]. It will be seen that when the concentration of the copper sulphate (as measured by the scale of numbers used as ahscisse) lies between ‘01 and | | Gms. CuSO, Sp. Res. | Gms. CuSO, Sp. Res. Gms, CuSO, Sp. Res. added to 104 | added to 10! added to 104 | 1 gm. in (hi lsenia' in 1 gm. in | glycerine. ohms/ce. | glycerine. ohms/ce. | glycerine. ohms/ce. Ee 441 | -0102 204 | -9gge 0) ae 0010 245) 1) 0110 41:0 -0400 ge pee 0022 £537 0 | eee OLA 344 | 0474 5:52 | 0033 1190 | 0140 20:55 lle Ono 4-70 | 0040 919 | 0170 212 ae eood 3-40 0060 648 | -0200 2370" ree tom 2°50 | 0070 525 | . 0280 162 | +1683 1:36 0086 401 | -0254 878 | -2008 1-20 0095 315 | 0285 9:05 2584 1-08 03, maxima and minima occur in the curve. From the positions of the points in the vicinity of the first maximum it seems possible that subsidiary peaks are there present, hut no close investigation has been made of these, as this Solutions of Copper Sulphate in Glycerine. 547 Curve I. iad I ft) (@) 12) ohms onms— CC. Specific Resistance 100 com) 002" 003 00%) G05), 006 | -OO7 | O08 000 a Gms. Copper Sulphate added per gram. glycerine Curve II. eee Specific Resistance ohms fo# TS es 1) | ‘02 Gms, Copper Sulphate added to / gm. glycerine, onic Resistance chm Spe CL. 10 545 Resistance of Copper Sulphate in Glycerine. portion of the curve is evidently one to be avoided by any person desiring to make up a solution to give a definite resistance. ‘The peaks correspond, in a general way, to the discontinuities in the curves for solutions of other salts got by Curve III. “02 “04 iy ‘06 i O08 1D ECE iy “2 cian {4 16 18 Gms. Copper Sulshate edded to 19m. glycerine, Jones and his collaborators *, and attributed by them to the differences in association and dissociation of the molecules when the amount of the salts and water added to the glycerine is varied. Glycerine solutions are hygroscopic and their resistances, especially the higher values, vary slightly as they absorb water. Accordingly, it is desirable to seal in the electrodes when a fixed resistance is required ; while in the case of variable resistances, the rod carrying the movable electrode may be made to slide in a hole in a rubber cork which forms a fairly air-tight joint. The solutions have been found to be verv free from polarization, and the curves show that a very wide range of resistance is obtainable. ~ | In conclusion, the writer wishes to thank Dr. Beatty, who suggested this research, for his continued interest and advice during its progress. * Toe, ctt, 2c THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE. AND Nek OF SCIENCE. Le e ei eS (SIXTH SERIES. ] wee — — A ea? |. XLVIT. On the Supposed Weight and Ulumate Fate of Radiation. By Sir OLIvER Lovee N regions where our ignorance is great, occasional guesses are permissible. Some guesses occur in this paper : let an apology for them be understood. If light is subject to gravity, if in any real sense light has weight, it is natural to trace the consequences of such a fact. One of these consequences would be that a sufficiently mas- sive and concentrated body would be able to retain light and prevent its escaping. And the body need not be a single mass or sun, it might be a stellar system of exceedingly porous character so that light could penetrate freely into the interior and be subject to the combined gravitative attraction of all the constituent masses. Given a material universe of any shape, bounded by a surface 8, with an aggregate mass M distributed anyhow inside it, the average intensity of gravity, F, at the surface is given. by Green’s theorem as —FS=47M. A large enough mass, not spinning too rapidly, tends to be spherical ; so, if the average density of the distributed matter is p and the radius is R, L'= —4mpR. If the distribution is taken as anton m, for the sake of an example, that is with equably distributed masses and great * Communicated by the Author; being the substance of an Address to the Students’ Math. & Phys. Soc. of the Univ ersity of Birmingham, Phil. Mag. 8S. 6. Vol. 41. No. 244. April 1921. 20 550 Sir Oliver Lodge on the interspaces so as to be extremely porous,—for a hetero- geneous mass is more not less effective than a homogeneous one,—the average force at any point in the interior of the sy stem at a distance » from its centre of gravity, introducing the Newtonian constant to make the specification complete, is f=—4mpGr. And the potential there is V = 27 p(R?—47°)G. The speed of anything amenable to gravitation would be a maximum near the centre of such a system and a minimum near or outside the periphery ; consequently light unable to escape would accumulate near the boundary, and if liberated by an expansion or other catastrophe happening to the system, such as might occur through a gradual growth of instability, would burst forth in a blaze. Such a blaze, rapidly rising in intensity, would die down gradually during the time that the deeper seated portions of the luminous shell took to rise to the surface. The speed which a heavy body could acquire by falling from periphery to centre is ,/(gR), by falling from infinity to surface ,/(2qR), ae by fallin g from infinity to centre ,/(39R), g being the maximum gravitational intensity 4moRG, where G is the Newtonian constant whose value in c.g.s. units is 666 x 107?° and whose dimensicns ae the sla ofa velocity divided by a linear density, or M7! Lt? T-? This means that a body able to prevent light emanating from centre from escaping altogether would have , rT = ra » e > ? . - y To prevent light from centre from reaching the surface would need while to control light from surface into an orbit, the inter- mediate value Boroce 87G Po would suffice. : The estimates are all of the same order; so, taking the Weight and Fate of Light. Dd1 intermediate value, we find that a system able to control and retain its light must have a density and size com- parable to na rad Stal Oo 25 x 6°66 x 107° — ib aa For a ray travelling along a radius vector, rd@ &e being then zero, this gives dr ——— — or ie while for a ray inclined at angle e to a radius vector, 1 eee alin ones ~ (v cos €)?+(v sine)? =ye?, iy, v ry sec € ve — J (1+ vy tar 2 Fn ork Be . ° e ° 2 (3) ¢ (l-+y¥ tan’ e) The reciprocal of this expression is the refractive index in a gravitational field, which, to a first approximation, is pap radially, and ae tangentially. or Incidentally we can see that my elaborate experiment on the influence of matter on ether in its neighbourhood (Phil. Srans. A. 1898 and 1897), which gave a negative result when tested between a pair of whirling disks, would have failed to give a positive result even if the whirling disks had heen so absurdly massive as to cause a gravitational potential comparable with that existing at the surface of the sun. For, in accordance with Fizeau’s experiment, light is carried forward with the extra speed v(1—1/°), which, since the peripheral speed v was usually 150 metres a second in that experiment, gives a ratio to the velocity of light ‘\ . ou — AX 10= 22. Re> ¢ The smallest ratio that could have been observed in that apparatus was the hundredth of a wave-length divided by the total effective journey of the light, say 22 metres,—a ratio which is 3x107~", or seventy-fold lacking in what would be necessary. Weight and Fate of Light. D990 Similarly the Michelsun-Morley experiment, even if performed on the surface of a mass as big as the sun, would be likely still to give a neyvative result: hence it is useless to try to repeat the experiment on some mountain elevation or in the interior of the earth. So, by (3), it is possible for the speed of light to be zero in a region where y=0, that is in the neighbourhood of a mass so great that 2a RG? and in that case light cannot altogether escape from the body. It would have a better chance of circling round and round the body, with e=90°, because rd@/dt is controlled by Vy instead of y; but strictly speaking it could not do this either, but would be stopped in its tracks. Hinstein’s method thus makes the speed of light a minimum where ordinary gravitational considerations would make it a maximum. Instead of increasing in speed as it approaches a massive body, light lessens in speed, as if it were repelled, not attracted. But the striking discordance between the two systems is that, whereas the speed of light fully subject to gravity would depend on the distance it had travelled against a retarding force, Hinstein makes it assume a velocity characteristic of the place where it is at each moment, without reference to past history. e So taking y as equal to 1— a , and the velocity of light along a radius as yc, the velocity of light anywhere inside a stellar system such as has been considered above, pp. 549 and 551, at a distance rv from its centre, is v=c0— val (R? — dy?) G. An aggregate mass whose plz — eA Goh x 107 Cel ors, would therefore reduce light at its centre to relative rest. The question has often been asked, What becomes of all the radiation poured into space by innumerable suns through incalculable ages? Is it possible that some of it is trapped, without absorption, by reservoirs of matter lurking in the depths of space, and held until they burst into new stars ? And a further more important question begins to obtrude itself :--What happens to light when, in free though modified ether, it is stopped relatively to a gravitational mass? Does it retain its energy, mainly in rotational form, tie itself into electrons, and add to the mass of the body ? 556 Sir Oliver Lodge on the fo attempt an answer would involve looking into the manner in which light is retarded by a denser medium, the front being thrown back upon itself (see a detailed theory of refraction by Sir J. J. Thomson, Phil. Mag. June 1920, p. 687, and Dec. 1920, p. 715) ; also into the suspected tendency of a wave front to break up laterally and concentrate into guunta-like units akin to those whence it arose. It may suffice for the present to remark that Oa wtL or what the familiar expression for reflected amplitude, represented in the gravitational case by Vae 0 fs ll ? F 1 . —, OY practlica —,1f, as usua 1s 1% + mM ? Pp vi r z) 9 y) 2 “ 9 written 1+ ee Also that if the momentum conveyed per would be written as second through any area is equal to the energy per unit length even ina dense medium,—which was Prof. Poynting’s assumption and is now being made the subject of careful experimentation by Dr. Barlow,—the momentum of that part which is transmitted increases, because of the longitudinal compression, the increase being accompanied by a pull. The resultant force acting on light when suddenly entering a denser medium perpendicularly comes out 2.8 times iv the incident energy-density ; the transmitted energy being Aw (u+1)? Bac OtL FNS ‘ Bes nae second a) ; which momentum, instead of remaining uniformly distributed, may become concentrated and localised in specks in the immediate neighbourhoed of matter or an intense electric field. For the potential close to an electron is one third of a million volts, and the gradient is enormous. The theory of refraction above referred to shows that, inside matter, lateral elements of a wave must contribute to the formation of the new and retarded wave-front ; hence lateral as well as longitudinal concentration is to be expected. We may also observe that the density of ordinary sunshine near the earth, being 3p/c’, is of the order 10~” gramme per c.c.; so if 10 cubic millimetres of earth sunshine, or the equivalent of 1/4600 c.mm. of solar emission, could be checked and condensed till its density was 10’? it might be converted into an electron of mass 107?’ gramme. Its -original momentum, 3 x 1077’, will presumably be communi- cated or restored to whatever stopped its translation ; while of the original, and the transmiited momentum per Weight and Fate of Light. 5a7 the energy of the light, which is ¢? times its mass, will become the constitutional or vortical energy of the electron, whose intrinsic circulation must therefore be of the same order as c¢. | Hinstein’s escnulion of the Une ont oy of matter is harmonious, for it takes the form mc?+4mv’, the first term being apparently the intrinsic or constitutional energy of each ultimate particle. On this view the interior of an enormous stellar system could be the seat of the geveration of matter, on rather different lines from those sugested by Prof. Eddington in his Address to Section A at Cardiff last year. “If the concentrated and controlled luminosity were locked up satisfactorily as an electron, its existence would be per- manent ; but is it possible that under exceptional conditions a por table collection of half-formed, either half-materialised or half-shattered, units could go abort together in an un- stable state, as globe-lightning, and be liable to explode back into ight again ? Dr. Barlow questions whether the Cee of light here considered need be very different from ordinary stoppage by absorption, and asks whether the @ particle ejected by an impinging X or y ray is liberated merely, or manufactured ! I do not attempt to answer any of these questions at present, but I may refer to page 466 of the Phil. Mag. for May 1919 (where there is a misprint in the footnote of 10~¥ for 1072), and also to the Phil. Mag. for February 1920, pp. 172 & 173, especially to these words :— “A wave-front is an evanescent kind of matter—a sort of attempt of an accelerated electron to reproduce itself; the question is how such a peculiarity, when generated, can be made permanent and its violent locomotion checked. We must find out how to disturb the ether in such a way that the modification shall remain concentrated, and not instantly rush away and disperse itself with the speed of- light. The electric and the magnetic components must be separated, the one kept and the other annulled.” Or, at least, they must be put out of phase. The simul- taneous electric and magnetic displacements which constitute a light-wave start from a Hertz vibrator with 90° difference in phase, though the electric rapidly overtakes the magnetic, and the two get into step. Retardation should reverse the process set forth by Hertz in his Hlectric Waves, Chapter IX, (especially pp. 142, 146), and cause them to finish in a modi- fication of their original predicament—absorption corre- ‘sponding to emission,—thus reproducing the kind of etherial ‘singularity whence they arose. [ 558 | XLVI. The Spectrum of Hydrogen Positive Rays. By L. Vecarp, Dr. Philos. and Professor of Physics at the University of Christiana *. N the number of the Philosophical Magazine for August 1920, Mr. G. P. Thomson has published some results of investigations with regard to the light produced by positive rays in hydrogen. He has studied the light emission of positive rays of a different composition, and according to his interpretation of the results the positive rays themselves in the molecular form should emit the so-called second or many-lined spectrum of hydrogen. ‘This would mean that the second spectrum should show a Doppler effect—a result which is opposed to all evidence so far as yet obtained. In spite of a great many searches for such an effect by Stark +, Wilsar =, Rau, and the author under very varied conditions, it }as not been found. Under these cireum- stances it seems that the indirect evidence given by the experiments of Mr. Thomson is by no means conclusive, and it is my intention to direct the attention to certain facts which may give an interpretation of Thomson’s ex- periments without assuming that the positive rays themselves emit the second hydrogen spectrum. In a series of papers published in Ann. der Physik trom the year 1912 to 1917§ I have given results of investi- gations with regard to the laws governing the light-emission of positive rays, yand I shall briefly mention some results that have a bearing on the present question :— (1) The second spectrum of hydrogen is part of the “unmoved” spectrum of the positive rays, and the intensity of the lines of the second spectrum increases in the same rate as the unmoved spectrum of the series lines. (2) The ratio between the intensity of the moved and unmoved spectrum was found to vary very greatly with the velocity of the rays and with the pressure in the observation chamber. Thus when the potential varied from 8500 to 27,000 volts, the ratio between the moved and unmoved intensity of H¢ * Communicated by the Author. + J. Stark, Ann. d. Phys. vol. xxi. p. 425 (1905). t H. Wilsar, Ann. d. Phys. vol. xxxix. p. 1251 (1912). § L. Vegard, Ann. d. Phys. vol. xxxix. p. 111] (1912); vol. xl. p. 711 (1918); vol. xl. p. 625 (1918); vol. lii. p. 72 (1917). The Spectrum of Hydrogen Positive Rays. 509 decreased from 7°18 to 1:64, corresponding to a pressure in the observation chamber of 0°035 mm. Hg. For a given potential— 20,000 volts, say,—an increase of pressure from 0-035 to 0:10 mm. Hg produced a diminution of the ratio between moved and unmoved intensity from 3°22 to 1°24. Mr. Thomson has not separated moved and unmoved intensity, but only drawn his conclusions from the fact that under varied conditions he gets a variation of the ratio between the intensity of the line-spectrum and the second (band-) spectrum of hydrogen. But it follows, from what is already said, that this ratio can be greatly varied as an effect of variations of pressure and velocity, and with a hiyh pressure in the observation chamber and a large velocity of the rays the second spectrum appears on the plate very prominent relative to the line-spectrum, and this effect was found under conditions where no Déppler effect of the second hydrogen spectrum was to be observed. Now the hydrogen spectrum given by Thomson cor- responds to discharges in tubes of different form with a different length of the dark space, and probably also differences of pressure in the observation chamber ; and there is then every reason to believe that the changes he observes with regard to the ratio between tlie intensity of the series lines and that of the many-lined spectrum may be regarded only as an effect of velocity and pressure variations. “Mr. Thomson’s interpretation of his results could only have any weight when it was proved that the effective potential of the tube and the pressure in the observation ehamber were the same in both cases. In connexion with the question regarding the origin of the moved intensity, discussed by Mr. Thomson, I should like to point out that the same problem has been treated by the author in the papers referred to. From certain observations* I was able to draw the conclusion that the unmoved intensity was produced from the direct bombardment of gas molecules by the positive rays. Further, I made experiments that showed t tl rat at any rate a considerable part of the moved intensity is produced by the neutral part of the positive ray bundle, and that a complete ionization is not necessary to bring a neutral atom to emit light. I[ shall here briefly de sscribe. my experl- mental arran wement, and give results of calculations which I made some time ago, but which have not previously been published. * L. Vegard, Ann. d. Phys. vol. xli. p. 638 (1913). . + L. Vegard, zb¢d. vol. xxxix. p. 162 (1912), and vol. In. p. 86 (1917). 560 Prof. L. Vegard on the Spectrum I used « cylindrical tube of the form shown in the figure and of the very same type as those used in the experiments of Wilsar on the Déppler effect of hydrogen lines. The rays pass through a narrow boring in the cylindrical cathode to the observation chamber, which ends in a glass tube about 3 em. long and 0°75 em. diameter. This narrow glass tube was placed between the poles of a strong electromagnet, in the way shown in the figure, and the rays could be made to pass a strong magnetic field of 11,500 gauss for a distance of 4°2 cm. The luminosity was analysed and measured at the end of the glass tube by means of a spectrograph of high light-power and a suitable dispersion for ohacryations of ite Doppler effect. The direction of the collimator axis formed an angle of about 35° with the direction of the rays, and it was thus obtained that only light from the very end of the glass tube passed into the spectrog raph. During discharge a constant current of hydrogen was maintained by continual pumping. By a simple arrange- ment, the direction of the current of the gas through the boring in the cathode could be reversed ; and in this way the potential of the discharge-tube could be changed inde- pendently of the pressure in the observation chamber. ‘The * discharge-tube was surrounded by a thick cylinder of soft iron, to prevent the magnetism from exercising an influence on the discharge. The spectrograms were taken under the very same discharge conditions, with and without a magnetic field, and for some different pressures. When the magnetic field was put on, the positively-charged rays were driven into the wall as soon as they were formed, and the luminosity at the end was mainly produced by uncharged particles, and most of the moved inte ‘nsity was emitted from rays in the neutral state. By means of the idea of the mean free path of the poses rays introduced by W. Wien *, we can calculate the dimin- ution of light- -intensity which would be preduced by the magnetic field provided that the light was entirely emitted from the neutral rays. Let us consider a cross-section (A—B) of the bundle just before the rays enter into the magnetic field ; and let us suppose that the bundle at this place has reached statistical equilibrium, so that the ratio between the number of charged and uncharged particles would remain constant provided there was no magnetic field. * W. Wien, Ann. d. Phys. vol. xxxix. p. 519 (1912). of Hydrogen Positive Rays. d61 Let the number of the positive and neutral carriers that pass this cross-section in unit time be n; and ng, respectively. Now, if the “ moved intensity ” is due to the neutral rays vod! 1f9 Toé he Purp Sy N > GS i % S Q only, the moved intensity I observed without a field should be proportional to ne", where w is the coefficient of absorption and / the distance traversed from the cross-section 562 Prof. L. Vegard on the Spectrum to the end of the glass tube, where the intensity is measured, and the intensity I will be eS eee ite. When the field is put on, first of all the positive rays 7. present at (A—B) are deflected into the glass wall, and then the positive rays which on the way / are produced from the neutral part are also brought out of the field of view of the spectrograph. This gives for the observed moved intensity I, with a magnetic field : l ; ie = hyngeé Ly g— Ml Consequently : a i tates sis L, is the mean distance which a neutral ray moves before it takes up a positive charge, or “tle mean free path” of the neutral ray. If we know L,, we shall be able to calculate the ratio pal Now, according to Wien, “‘the mean free path ” changes comparatively little with the velocity of the rays, but very considerably with the pressure in the observation chamber. Wien’s measurements, however, are confined to fairly low pressures, ranging from about 5.1074 to 4.107? mm. Hg.,, while the pressure in my experiments varied between op 1077 and as mm_, le. Wien gives the following values :— For p = 0:0051 mm., the quantity Le — 5°25.10-5 em. 0 = 0039 EN rae b= 10.10-5 em. 0 Po is atmospheric pressure, and L is defined by the equation 99 en tae Dee Be The ratio L/L, is for the same pressures 6°1 and 2°6 respectively. i is “the mean free path” of the positively charged carriers. . “The mean free path” corresponding to the lowest pressures used in my experiments can be found fairly accurately from these values. In the case of the highest pressure 0-l mm., however, we have to extrapolate over a fairly wide range; so the value found for “the mean free path ” will be somewhat uncertain. of Hydrogen Positive Rays. 563 In the table are given “‘ the mean free paths?’ Ly; and L, as they are derived from Wien’s experiments :-— P ss (ot ressure, Wie L,. ae \ = 0-055 mm. 79 em. 2°78 em. 1:70 1:89 O:056>. oOe;, Reo. 2°31 2°33 ORO gs, oTlOn. UP Hes. 38] 2°91 The effective length of the magnetic field is put equal to that of the pole-pieces (4°2 cm.). Now we observe the light at a distance of about 0°5 cm. from the end of the narrow glass tube, and, further, some of the rays that acquire a positive charge during the passage of the last centimetre of the glass tube may come into the field of view. This will be equivalent to a diminution of the distance J. On the other hand, the spreading of the magnetic lines will produce some increase of the effective length J. To diminish these errors, the distance / ought to be somewhat longer than in my experiments. Comparing the calculated and observed values, we see that they show a fairly good agreement for the smailer pressures. In the case of the highest pressure the value taken for “the mean free path” is uncertain, and when L, becomes small errors in the distance / will have a comparatively great effect. At any rate, we can conclude from these numbers that the greater part of the moved intensity 1s emitted from the rays in the neutral state. Some moved intensity may also be emitted when the positive rays get neutralized; but this quantity should be small as compared with that produced through the bombard- ment of the neutral rays with the gas molecules. When we remember that the light-intensity measured by _ means of the spectrograph is proportional to the intensity per unit length of the ray bundle, the measured intensity should be given by an expression of the form : Ny No Ll=k(4,—- fever es Oradea X is the mean distance which a neutral ray has to pass between successive collisions which excite the ray to light emission. Ras : ae | a is then the number of such collisions which the neutral rays suffer in unit time per unit length. «. measures the probability that such a collision shall result in the emission of the particular spectral line considered. 564 Prof. L. Vegard on the Spectrum e . rt e In a similar way, — measures the number of times the Ly positively charged rays get neutralized per unit length of path in unit time; and if we assume that a neutralization . e e e e ny is always accompanied by light-emission, i also measures. 1 the number of times a positively charged ray is brought to emit light through neutralization per unit length of path and in unit time. x, measures the probability that a neutralization shall result in the emission of the particular spectral line considered. k is a constant, which depends on the apparatus and the units of the light-intensity. If we put the magnetic field on, the luminosity produced by neutralization becomes practically zero and we get: LO Maher and ae Ae LTE, i 1 i —_ = lis Susan é nis iis ee ee “ fea oN os d 1 My +r = le Sage kL fieeee Ko Ne L, / If the ray bundle at the section (A—B) is in a state of statistical equilibrium, we have according to Wien : 1, ee: Ue To Ls 1 l ana E WN aes = ‘al Te ) Ly IE Ko Lis As we saw, (I/I,)op;. came out nearly equal to e”%, and thus ae = should be a small quantity for H, and Hg. g Lip If the probability that a “light impact” of the. neutral ray shall result in the emission of Hg, say, is the same as the probability for a H, emission resulting from neutralization, then «,=K, and ir should be a fairly smal! quantity. Now if “the mean free path” 2 of successive “light collisions’’ of the neutral ray is of the same order of magnitude as the mean free path of the gas molecules in the observation chamber, we find, as a matter of fact, that tig is a small quantity. 9 of Hydrogen Positive Rays. 069 For the pressures used we find the following values :— Pressure. rN. i 0:085 ra, 0°39 cm. 0-050 0,056. 0°245 ,, 0:049 Olu: <; O35), 0:043 Probably the mean distance between successive light- emissions will be somewhat greater than “the mean free path” of the molecules in the observation chamber, and the true values of - somewhat greater than those here 2 given. The question with regard to the origin of the emission of light from a bundle of positive rays has been the subject of much discussion. In a number of papers@Stark has tried to prove that the moved intensity is produced by the positive ions, while Wien found that certain of his experiments were best explained by the assumption that the moved intensity came from the neutral part of the bundle. The results of my experiments, as I have stated in my previous papers, have so far contirmed Wien’s assumption, as, at any rate, the greater part of the moved intensity comes from the neutral ray. But, as we have already remarked, we ought to distinguish between the emission produced by the neutral ray on account of collisions with gas molecules and the emission due to neutralization of the positively charged carriers, and the last part will in so far be an effect of the positive rays as it will be removed when the positively charged rays are removed from the field of view. In this connexion I should like to mention another result of my experiments with the same arrangement, which is hard to understand without assuming that the positive earriers through the neutralization process are engaged in the emission of moved intensity. Jt was found that the ratio between moved and unmoved intensity was not changed by the effect of the magnetic field, or the changes were inside the limits of possible errors. As before, the moved intensity without a magnetic field is given by an expression of the form : Ty = (keyny + hans) e-™ ; and the unmoved intensity Jo: Ji (91% + gam) e~"” ; Phil. Mag. S. 6. Vol. 41. No. 244. April 1921. 2 P - 066 Mr. G. P. Thomson on the Spectrum and the ratio between the moved and unmoved intensity f te Ty one kin, + kon vO So Git + Gems When the magnetic field is put on, moved as well as unmoved intensity is produced by the neutral rays: 1 i = kenge eget el Vers — gonge Le" re lt ky Jn By Jin OF qo. Now the experiments give as the mean of four mea-- surements : Fel jm = 0:98, or approximately =1. This gives ky kynyt+keng _ fy G2 Gmtn. Now if we assume that the positive as well as the neutral rays, by their impact with the gas molecules in the obser- vation chamber, produce ‘‘ unmoved intensity,” then gq, is different from 0, and as ky and gy are different from 0 k, must be different from 0. As the positive hydrogen nucleus cannot emit the series lines, the positive hydrogen rays can only emit light at the very moment they pass into the neutral state, and 4,40 means that light is emitted as the result of the neutralization process. For other gases we have to reckon with the possibility that the ray may emit light also in the positively charged state us the result of such collisions with gas molecules, which do. not result in neutralization. Physical Institute, Christiania, October 28, 1920. XLIX. The Spectrum of Hydrogen Positive Rays. To the Editors of the Philosophical Magazine. GENTLEMEN ,— I WISH to thank you very much for allowing me to comment on Professor Vegard’s paper. He ascribes the differences which I found in the spectra of the hydrogen positive rays to chance variations in pressure and in the energy of Hydrogen Positive Rays. 567 of the rays. ‘This does not explain the marked correlation which I showed to exist between the ratio of the intensities of the spectra, and the proportion between the atoms and molecules of hydrogen in the rays as determined by direct analysis. In point of fact, the difference in pressure was in the reverse direction to that required for his explanation, the pressure being lower during the experiments when the second spectrum was obtained. While there were undoubtedly varia- tions in the potential difference used in the discharge, the resulting variation in energy of the rays was small compared with the range in energy at any instant, which, as determined from the electrostatic deflexion of the rays, was often 3 or 4 to l. The close connexion between the two resting spectra found by Prof. Vevard is to be expected on my theory as long as the positive rays are all of one kind—that is, as long as there is only one “ moved” spectrum. They presumably corre- spond, the series spectrum to dissociation with ionization (partial or complete) and the second spectrum to ionization only, of the gas in the observation chamber, with the sub- sequent return of the electron which is the actual source of the light emitted. As long as the particles forming the rays were of the same nature, the ratio of the numbers of collisions of these two types might be expected to be the same. As mentioned in my previous paper, the failure to find any “moved” second spectrum is a point which needs explanation ; but I still think that the use of cylindrical tubes of compara- tively small diameter is largely responsible, as in similar circumstances I too got a purely atomic beam of rays. Professor Vegard suggests no alternative theory as to the nature of the carriers of the second spectrum, and the mole- cular theory will seem much the most probable, especially in view of the recent work of Mr. Saba *, at least until it can be shown that rays which when analysed show the presence of appreciable numbers of molecules give no “moved” second spectrum. My experiments were not such as to throw light on the electric condition of the atoms which emit the series spectrum, and I fully accept Professor Vegard’s conclusion that light is emitted both when the positively charged atom is neutralized and when the neutral atom collides with the molecules of the gas. ae Yours faithfully, Corpus Christi College, G. P. THomson, M.A. Cambridge. * Phil. Mag. June 1920. ye oa a3" S| L. The Torsicn of Closed and Open Tubes. To the Editors of the Philosophical Magazine. GENTLEMEN,— YEXHE formule for closed and open tubes under torsion given by Dr. Prescott in the November number of the Philosophical Magazine have already been published by me in two papers: (A) “The Calculation of Torsion Stresses in Framed Structures and Thin-walied Prisms” (Brit. Assoc. Report, 1915, and ‘ Engineering,’ October 15th, 1915), and (B) “The Torsion of Solid and Hollow Prisms and Cylinders” (‘ Engineering,’ Nov. 24th and Dee. Ist, 1916). | : Formula (32) of Dr. Prescott’s paper, giving the stress in a thin tube, is stated in § 2 of paper B, and is a particular case of the theorem which forms the main subject of paper A, viz.:—If a hollow cylinder or prism, either continuous- walled or of framework, and having plane ends perpendicular to its length, be subjected to a twisting moment by couples in the planes of its ends, the total longitudinal shear is every- where constant and equal to the twisting moment multiplied by the length of the cylinder and divided by twice the area of one of its ends. This theorem was proved very simply from elementary considerations without using the equations of elasticity. As applied to frameworks, it was used in the calculation of the torsion stresses in the suspended span of the Quebec Bridge, and has also been applied to aeroplane fuselages. The formula (35) for the angle of twist of a tube given by Dr. Prescott is also stated in § 2 of paper B, and is deduced there from the work stored in the tube during torsion. | The formula (56) for the angle of twist of a thin strip (called an “unclosed tube” by Dr. Prescott) is slightly more general than my formula in paper B § 6 eq. 24, which is only true when the strip is of uniform width, but equa- tion (25) of paper B § 6 gives the extension of (24) to rolled sections. Dr. Prescott does not give explicitly a formula for the shear stress in a thin strip in terms of the torque, but, by combining his equations (53) and (56), it follows that which is the formula given by me in B §5 and extended to structural steel sections in $ 6. ” The Torsion of Closed and Open Tubes. 569 This result and formula (32) have been used in the design of the spars of the tail-planes of aeroplanes which are under torsion from the king-posts carrying the control wires. I also showed in paper B that formula (32) could be used to solve the general (St. Venant) problem of torsion by con- sidering a solid shaft as made up of tubes of shear. This method gives the same results as the St. Venant theory, without involving the use of the equations of elasticity or of conjugate functions. The approximate formule used in engineering practice for elliptical and rectangular tubes often give results which are greatly in error. This matter is considered in paper B § 7. Yours faithfully, Dept. of Civil Engineering ©. Bato. and Applied Mechanics, McGill University, Montreal, November 30th, 1920. To the Editors of the Philosophical Magazine. GENTLEMEN,— THE results in my paper on “The Torsion of Tubes ” were worked out about last March in connexion with a book I am writing. I had not then seen or heard of any of Professor Batho’s work on the subject, and I thought my results were quite new. Atter I had decided to publish my results in the form of a paper I saw in ‘ Engineering’ the paper (A) mentioned above, but this paper had so little in common with my own that I saw no reason for withholding publication. It was only after I had sent off the final proofs of my paper that Professor Batho’s paper (B) came to my notice. If I had seen this paper earlier I shouid probably not have published my results in a separate paper. How- ever, I now think that it was worth while to publish my results because our methods are so different, and especially as the correspondence in ‘ Engineering’ shows that there were people who did not believe in Professor Batho’s methods. Moreover, as he points out above, my formula for an open tube (or thin strip) applies to strips of variable thickness, whereas his applies to strips of constant thickness only. Professor Batho’s claim that he does not use the equations of elasticity cannot be upheld for his work on the thin strip, since, for this purpose, lie borrows St. Venant’s results for a prism of rectangular section. J. PRESCOTT Jane (th, LOZ LI. The Mass of the Long-range Particles from Thorium C. By Sir E. Ruruerrorp, F.R.S., Cavendish Professor of Physics, University of Cambridge™. T is well known that thorium C disintegrates in two ways with the emission of « particles of range 8°6 cm. and 50 em at 15°C. In 1914, Dr. A. B. Wood and myselk showed that a small number of particles—about 1 in 10,000 of the total—were expelled with the long range of 11:3 em. The range and number of these particles were determined by the scintillation method, and from the brightness of the scintillations 1t was supposed that the expelled particles were atoms of helium. In a subsequent paper the writer showed that the passage of # particles from radium C through nitrogen and oxygen led to the production of a small number of swift particles, which had a range 1°29 times that of the impinging a particles. Since the ratio of the ranges of the thorium particles, viz. 11°3 em. and 8°6 em., is of about the same magnitude, viz. 1°32,it was suggested that possibly the long-range particles emitted by thorium C might arise from collision or the « particles of range 8°6 em. with the oxygen of the mica, which was used as an absorbing screen to cut off the particles of range 8'6em. This, however, seemed unlikely. as the number of the long-range particles from thorium C were about ten times greater than would have been expected on this hypothesis. In order to make certain of this point, Dr. A. B. Wood kindly undertook to repeat the experiments, using aluminium instead of mica as an absorbing screen. This experiment, an account of which is given in an accom- panying paper, showed conclusively that the long- “range particles could not be ascribed to the oxygen in the n1¢a ; and at the same time a more accurate estimate was made ot the relative number of long-range particles, which were found to be 1/10,000 of the total number of particles from thorium C In the course of recent experiments the writer obtained some evidence that the short-range pariicles of the a-ray type of the range 9 cm. appearing in oxygen and nitrogen were not atoms of oxygen or nitrogen, but atoms of mass about 3. It was a matter of great interest to examine whether such atoms were liberated in radioactive changes in addition to atoms of helium of mass 4. If this proved to be the case, it would afford a more direct method of deter- mining the mass of the new atoms with accuracy, since they * Communicated by the Author. Mass of Long-range Particles from Thorium C. 571 would be emitted in number from the radioactive source instead of from the volume of the gas bombarded by a rays. : In the original experiments of Dr. Wood and the writer, the active deposit of thorium was used asa source of radiation, and with the active material available, the y-ray activity of the source of radiation was about 2/100 of a milligram of radium in equilibrium. In order to obtain about 20 scin- tillations a minute, due to the long-range particles, it was necessary in these experiments to place the mica within a few millimetres of the zine-sulphide screen In_ order, however, to determine the mass of the particles by observing the sph of their deflexion in a magnetic field, it was imperative to work ata distance of at least 5 centimetres, and to employ a source at least 100 times stronger, By the generosity of Dr. Herbert McCoy of Chicago, well known for ahs contributions to our knowledge of dhe align activity of thorium, I was presented witha quantity of radio- thorium of y-ray Agen ity equal to 24 milligragns of radium. In order to obtain powerful sourves of thorium C, this material, -after suitable chemical treatment, was slbiniaied mm By Gl lve solution of small volume. [am indebted to Mr. Chadwick for his kindness in preparing this solution in a form to yield the maximum amount of thorium C. This was obtained by exposing one side of a nickel plate of area about one square centimetre in the hot solution for one hour. During the -exposure the plate was kept in slow rotation by a small motor. By this method it was possible to obtain an amount of thorium C on the surface of the nickel plate equal in y-ray activity in- itially to about 8 milligrams of radium. The activity of this source decayed with the time according to the period of thorium ©, viz. to half value in one hour. This source gave a sufficient number of long-range particles of range 11°3 cm. to determine their bending in a magnetic field by the scintillation method. Two differ ent arr angements were employed. The first was similar to that described in the Bakerian Lecture (Proc. Roy. Soc. A, 1912), and illus- trated in fig. 1 of that paper. The a rays from the nickel plate passed between two parallel plates, 4 em. long and 2 cm. apart, placed in an exhausted rectangular box between the poles of a large electromagnet. The plates were distant 1:2 cm. from an opening in the end of the box, which was eovered by an aluminium plate of stopping power fora par- ticles equal to 5:4 em. of air. To examine the deflexion of the long-range particles, additional absorbers were added to stop completely the « raysofrange 86cm. For this purpose ~ D72 Sir E. Rutherford on the Mass of the the total absorption in the path of the rays was adjusted to 9°4 cm. of air. The zinc-sulphide screen was placed close to the aluminium absorbers, and the microscope so adjusted that,. on exciting the magnetic field in one direction, the @ particles fell over the whole surface of the screen viewed by. the microscope, and by reversing the field the scintillations were confined to the lower half of the field of view. The ratio of the number of scintillations per minute, usually 4 or 5 to 1, was determined for the two fields, and this gave a measure of the amount of deflexion of the rays. ‘This ratio was com-— pared directly with that found for the thorium particles of range of 86 cm. For this purpose a much weaker source: of a rays was obtained by dipping a nickel plate for a few seconds in a more dilute solution of radiothorium. The: experimental arrangement was identical with that described above, except that the absorption in the path of the e rays was: reduced to 5'4 em. of air. This stopped the « particles of range: 5°0 em. from thorium C, while those of range 8°6 cm. gave- bright scintillatidns onthe screen. In all cases it was found that the long-range particles were less deflected than those of range 8°6 cm. By determining the value of the ratio for- different strengths of magnetic field, the relative magnetic ‘deflexion of the two types of rays could be directly compared.. The different determinations made in this way varied betweem 1°08 and 1:12, with an average value of 1°10. _ In order to confirm these results, the deflexions were com-- pared by a more direct method, identical in principle with that employed previously to measure the deflexion of the swift H atoms set in motion by impact with @ particles (Phil. Mag. xxxvii. p. 563 (1919)). Therays from the source placed behind a horizontal slit of width 1 mm. passed through another: slit of equal width placed midway between the first slit and the zinc-sulphide screen. The distance between the source and screen was 8°3. cm. With the magnetic fields employed, the band of scintillation observed on the screen, due to « particles of range 8°6 cm. from a strong source of thorium C, was displaced 5:7 mm. by reversal of the field. The amount of deflexion of the long-range particles by reversing a current through the electromagnet of 6 amps. was directly compared with the deflexion due to the a rays of range 8°6 cm. from thorium C under similar conditions. For a current of 5 amp., the deflexion of the pencil of rays, range 8°6 cm., was °965 of the pencil of long-range particles with a current of 6 amp., giving a field 1:12 thatfor 5amp. For equal fields, Long-range Particles from Thorium C. 573 the deflexion of particles of range 8°6 cm. was thus 1:08 times that for the particles of 11°3 cm. range. If the particles of range 11°3 cm. are ordinary « particles, the relative deflexion of particles of range 8°6 and 11°3 cm. Oe AER: Si ne eee to be expected is fae =1:10, since the velocity of the « particles varies as the cube root of tlre range. The observed values by the two methods are 1:10 and 1:08, or a mean of 1:09. By making a number of experiments, no doubt the relative bending could be determined with more precision; but this was not thought necessary, as the agreement is sufhi- ciently close to indicate that the long-range particles from thorium C are ordinary « particles of mass 4. Itis of interest to note that if these long-range particles were atoms of mass 3 carrying two charges such as are observed to be released as a consequence of the collision of & particles with N and © atoms, the deflexion of the a2 particles of 8°6 cm. range should be ‘90 of that of the particles of 11:3 em. range instead of the observed value of 1:09. The data on which this calculation is based have been given in the Bakerian Lecture (loc. cit.). ; ; The experiments recorded in this paper thus negative the idea that particles of mass 3 are ejected from thorium C. It should be pointed out that the agreement of the atomic weights of radium, uranium-lead, and thorium-lead with the values calculated from the emission of « particles show that no particles of mass 3 are expelled in the main series of radioactive changes of uranium and thorium. Similarly, no certain evidence has been obtained of the emission of H atoms. From numerous experiments I have made, I am inclined to believe that most of the H atoms observed from a source of radium C under normal experimental conditions must be ascribed to occluded hydrogen. It has been generally considered that the expulsion of two distinct sets of a rays from thorium C isa proof that this product suffers a dual transformation. From analogy with the dual disintegration of radium (, it is supposed that . 35 per cent. of the atoms of thorium © break up with the emission of « particles of range 5:0 cm., giving rise to thorium D, which breaks up with the expulsion ofa 8 particle. The reverse process is considered to take place in the other branch, 65 per cent. of the atoms of thorium C first emitting B particles and giving rise to thorium ©’, which is very rapidly transformed with the emission of the swift @ particles ot4 Mass of Long-range Particles from Thorium C. of range 86cm. While it is difficult to give a definite proof of this scheme of transformation, the general facts strongly support it. | The emission of a particles of range 11°3 em. shows that the modes of disintegration are even more complicated than the above. There is no information to guide us as to the origin of these very swift particles, except that they appear to arise from thorium © and decrease in number at the same characteristic rate. The system of transformation may be similar to one of those outlined, or thorium C’ may break up in two ways with the emission of rays of ranges 8°6 and 11°3 cm. Suppose, however, 1/10,000 of the atoms of thorium C break up directly with the emission of these very swift particles. The atomic number of the resulting product, viz. 81, is that of thallium, but the atomic weight is 208 instead of 206 observed for ordinary thallium. It is of interest to note that Merton (Proc. Roy. Soc. A, xcvi. p. 393 (1920)) found that thallium from pitchblende residues gave a longer wave-length than ordinary thallium, indicating the presence of an isotope of higher mass. Soddy* considered the question whether thallium could be an end product ot the two main branches of thorium C, but found that the amount of thallium in a particular thorium mineral, Ceylon thorite, was far too small to admit of sucha possibility. The amount of lead, mainly thorium lead, in this mineral was 0-4 per cent. . If thallium results as the end product of the new branch, the amount of thallium should be about ‘00004 per cent., or for each 100 grams of thorium-lead, 10 milligrams of the thallium isotope—supposed stable—should be obtainable. Soddy states that some thallium was found in the mineral, but in very small amount not determined, but certainly less than :005 per cent. It would be of interest to examine whether such a thallium isotope is present in thorium minerals in amount to be ex- pected on the above hypothesis. IT am much indebted to Dr. Ishida and Mr. Chadwick for their assistance in counting scintillations. Cavendish Laboratory, Dec. 1920. * Soddy, ‘Nature,’ cii. pp. 356, 444 (1919). Va oa ae LIL. Long-range Particles from Thorium Active Deposit. By A. B. Woop. D.Se.* $1. FN 1916 Sir Ernest Rutherford and the author pub- lished a papert describing experiments which gave evidence that thorium active deposit emits a small number of high-velocity particles, of range about 11°3 em., in addition to the main group of a-rays of ranges 5°0 om. and 86 cm. At the time the experiments were made there seemed no reason to suppose that these long-range particles were other than a-particles. More recently, however, Sir E. Rutherford has shown { that when a-particles are fired into a medium containing light atoms, some of these atoms are enor mously accelerated by close collision of the e-particles with their nuclei, and consequently attain very high velocities and correspondingly iong ranges. Such high-velocity particles behave in a manner similar to that of the a-particles them- selves in that they can produce intense ionization and scintillation effects. It was calculated that oxygen atoms, for example, could be accelerated by close collision with a-particles of 8°6 cm. range, so that they attained the high velocity corresponding to a range of 11:1 em. in air. In the investigation of the long-range particles from thorium § a mica screen was employed to absorb the ordi- nary a-particles of ranges 5:0 and 86 cm., the long-range particles pentrating the mica and striking a zinc-sulphide oO sereen. ‘The maximum range of such particles was found to () be 11:3 cm., a value differing only slightly from that deduced by Sir K. Rutherford for the range of oxygen atoms accelerated by 8-6 em. «particles. Since mica’ contains a considerable amount of oxygen, it seemed not improbable that the “long-range a-particles from thorium” might conceivably be oxygen apenas originating in the mica absorbing sereen. A Peron of the long- range particles must undoubtedly have originated in this manner, since Rutherford has shown this to be so when RaC a-particles are fired through mica screens. On the other hand, the number of long-range particles observed in the case of thorium, about 1 in 10,000 of the total number of «-particles, is about 10 times the number obtained in the RaC experiments, viz. 1 in 100,000. It was considered desirable, therefore, to re-determine the * Communicated by Professor Sir E. Rutherford, F.R.S + Rutherford & Wood, Phil. Mae, xxxi. April 1916, t Rutherford, Phil. Map. xxxvil. June 1919, § Rutherford and Wood, loc. ert. 576 Dr. A. B. Wood on Long-range number of long-range particles from thorium, and to differen- tiate, if possible, between long-range a-particles and high- velocity oxygen atoms. § 2. Hxperimental Arrangements. The experiments carried out in 1916 pave an approximate value of 1 in 10,000 for the number of long-range particles relative to the total number of «-particles emitted by the source of thorium active deposit. Unless a very intense source be employed, therefore, it is not possible to obtain a sufficient number of long-range particles for purposes of in- vestigation. ‘To obtain this intense source of active deposit, the fiat tip of a copper wire, 0°8 mm. diameter, was exposed as negative electrode to a strong source of radio-thorium *, until the active deposit collected was in equilibrium with the emanation. The electrodes in the exposure chamber were arranged so as to concentrate the field on the tip of the copper wire, thus ensuring a maximum amount of active deposit collected. When sufficiently active, the wire was removed and mounted, as shown in fig. 1, with the active surface 4°5 mm. distant from a zinc-sulphide screen. Usually the solid angle of the «-ray stream from the end of the wire was limited by a small hole, 1:1 mm. diameter, through a brass plate 1°8 mm. thick. The absorbing screens employed to stop the 8-6 cm. a-particles were placed over this hole, and could easily be removed or interchanged without dis- turbing the relative positions of the active wire and zinc- sulphide screen. The latter was carried on an extension tube, fitted to the microscope so that it always remained in focus. This zinc-sulphide screen, prepared by Mr. F. H. Glew, was of specially fine grain, the area under observation by the microscope being completely covered with small erystals. The microscope, of magnifying power 50, covered a field 2mm. diameter. Under these conditions the maximum number of long-range particles observed was about 10 per minute. In cases where such a small number of scintillations is to be counted it is, of course, necessary to take all possible precautions against slight contamination of screens, etc., by radioactive matter. Consequently, it is essential to make “dummy ” experiments, in which the active source is re- moved, as a matter of routine. To ensure that the relative positions of the microscope, ZnS screen, absorbing screen, and active source always remained the same through an experiment, they were all * Very kindly lent for these experiments by Prof. L. R. Wilberforce, of Liverpool University. Particles from Thorium Active Deposit. D717 mounted on a common supporting base in such a way that the axis of the microscope passed through the centre of the active surface and the hole in the brass plate (see fig. 1). Jeneg, Ne Z7S. SCREEN. \*71CA OR Atuminiug SCREEN. Y PRE WR Ze ACTIVE WIRE. “WMONBEQ3 450.7. ABSORBING SCREEN. eee BRASS PLATE. MOLE [4 m7.77. DIAM. ACTIVE SURFACE. : O-S m7. OLA. (QaAwn TO ScALe.) The air-equivalents of the absorbing screens used in the investigation were determined first of all by weighing measured areas and deducing the air-values from data D78 Dr. A. B. Wood on Long-range supplied in a paper by Marsden and Richardson*. The values thus deduced were afterwards carefully checked by direct observation. When mica was used as absorber of the 86 cm. eparticles, two thin layers, each of air- -equivalent 4:3 cm., were superposed, the total “ effective’’ air-equi- valent from active source to ZnS screen in that case being 8°6 cm.+°45 em. or 9°05 cm. Thus any particle originating in the active material and striking the zinc-sulphide screen must have a range greater than 9 cm. In certain experi- ments aluminium screens were used to stop the 86 cm. a-particles. For this purpcese four layers, each of 2:0 em. air-equivalent, and one layer of 0°5 cm. air-equivalent, were ~superposed, the total effective air-value from active source to ZnS screen being in this case 8°95 cm. The methods of counting scintillations and experimental precautions necessary have frequently been described else- where, and need no further mention here. § 3. Relative number of Long-range a-particles and Oxygen atoms. In order to discriminate between the scintillations produced by a-particles and oxygen atoms, two methods have been employed :— (a) By varying the position of a mica absorbing screen between the active source and the zinc-sulphide screen. (b) By comparing effects with mica and aluminium absorbing screens. (a) In this method the distance between the active source and the zinc-sulphide screen is kept constant, whilst the mica absorbing screen (of air-equivalent 8°6 cm.) is placed (1) near to the active source, (2) near to the zine-sulphide screen, the number of “Jong- -range”’ scintillations being carefully determined for each position of the mica (see fig. 2). In this comparison the brass aperture (fig. 1) restricting the divergence of the X-ray beam incident on the mica is removed. It will readily be seen that if all the long-range particles come directly trom the active deposit, no appre- ciable difference in the number of scintillations counted should be observed as the mica is moved from position (1) to (2). If, however, the long-range particles all originate in the mica, it can be shown that the number of scintilla- tions observed in case (1) when the mica is close to the * Marsden & Richardson, Phil. Mag. Jan. 1913. Fi Particles from Thorium Active Deposit. ine active source, should’ be about 1°5 times the number observed in position (2) when the mica is placed near the zinc-sulphide screen and further away from the active source. Fig. 2. Zine SUANICGE SCREEN 4&-2am.m. . Mi CA SCREEN. ae AOSIFION. G) Oe ACFIVE SURFACE, (Qaawn TO Scace. } In a particular experiment of this nature the total distance from the active deposit to the zinc-sulphide screen was 5 mm., the distance from the mica to the active source in (1) being 1 mm., and in (2) 4:2 mm. Within the limits of possible error in experiment, the number of scintillations observed was the same in each case. Repeti- tions of this experiment always gave the same result. Whilst these observations suggest that the majority of the long-range particles come from the active source, they do not entirely preclude the possibility of a small fraction originating in the mica. Before discussing this result, therefore, it appears desirable to give the data obtained by the second method of long-range ‘particle analysis. (6) In the second method a direct comparison is made between the number of long-range scintillations observed when screens of (1) mica and (2) aluminium are used to stop the 8 6 cm. «-particles. A comparison here is made between the effects observed when the absorbing screen contains oxygen, as in mica, and when it is free from oxygen (except as a slight impurity perhaps), as in aluminium. If the long-range particles consist entirely of oxygen atoms from the mica, it is evident that the number of scintillations observed through mica will be far greater 580 Dr. A. B. Wood on Long-range than the number through aluminium. Alternatively, if all the long-range particles originate in the active source, the number of scintillations observed through aluminium will be exactly the same as the number through mica— always assuming, of course, that the aluminium and mica screens have the same air- -equivalents. If, however, both oxygen atoms and long-range «-particles are present, the result will be a compromise between these two extremes, i.e. the number through aluminium will be somewhat less than the number through mica. An experiment of this nature should therefore provide a crucial test of the origin of the long-range particles. It has already been mentioned that the mica screen consisted of two layers, each of air-equivalent 4°3 ecm.— total 8-6 cm., whilst the aluminium screen was made up of four layers of 20 cm. air-equivalent and one layer of 0°5 cm. air-equivalent—total 8-5 cm. It will then be seen that the ‘‘comparison”’ screens were, as nearly as practicable, of equal stopping power, the aluminium screen, if at all, having a slightly less air-equivalent than that of the mica. In carrying out an experiment, a careful comparison was made of the number of scintillations observed when the 8-6 cm. a-particles were absorbed in each of these screens, the relative positions of the active source and zinc-sulphide screen being fixed during an experiment whilst the absorb- ing screens were interchanged. As a result of such comparisons it was observed that the number of long-range scintillations was almost the same in the two cases. A slight excess, in favour of the mica, was however always obtained, this excess amounting only to a small fraction— less than 10 per cent. of the total. Thus it was found in one comparative test that the number of scintiilations observed through the mica screen averaged 10:0 per minute, whilst with the aluminium screen the average was 9-4 per minute. On another occasion the number through the mica screen averaged 6:0 per minute, whilst aluminium gave 5°) per minute. The ditferences are small, and fall very near to the limits of error of experiment. They are significant, however, in suggesting, what otherwise seems probable, that a aml fraction, 7 or 8 per cent., of the long-range particles comes from the mica, the ‘oreater proportion, over 90 per cent., coming directly from the active deposit. The two methods of analysis lead therefore to the same rresult, viz. that the greater proportion of the long-range Particles from Thorium Active Deposit. O81 particles originates in the active deposit. It seems probable, on general grounds, that these particles are «-rays, but this point can only be settled conclusively by some form of magnetic deflexion experiment, in which the value of e/m, and consequently the mass of the particles, is determined *. § 4. Number of Long-range Particles. In 1916 a determination was made, by two different methods, of the ratio of the number of long-range «-particles relative to the number of ordinary «-particles. The value then obtained, 1 in 10,000, was based on the results of a few observations only, and may consequently be not entirely reliable. Since the publication of Sir Ernest Rutherford’s paper on ‘‘ Collision of a-particles with Light Atoms’’t, it became of considerable importance to re-determine this ratio more carefully. The first method employed in the earlier determination of the ratio was briefly as follows. The number of long-range particles per minute was measured with the source fixed ata known distance from the zine-sulphide screen. After the active deposit had decayed in situ for 32 hours, the mica absorbing screen was removed and the number of z-particles from thorium C measured at distances from 6 to 7 cm. so as to include only the 8°6 em. «-particles from thorium C. Reducing both sets of observations to the same distance from the active source and correcting for the decay in 32 hours, a value of the ratio could at once be deduced. In the second method the source and zine-sulphide screen remained fixed at the same distance apart (about 5 mm.) throughout the measurements, a comparison being made between the number of long-range particles penetrating the mica sereen initially and the total number of «-particles (5°0 and 86 cm.) from thorium C six days afterwards (the mica screen being removed). Assuming the decay period of thorium active deposit to be 10°6 hours, the number of ordinary «particles at the commencement of the ex- periment could easily be obtained and the required ratio evaluated. Recent experiments have proved the latter to be the more reliable method, the resulis now given being based entirely * The magnetic deflexion of the long-range particles has since been accomplished by Sir Ernest Rutherford, who has proved conclusively that the long-range particles are a-rays. iP dediwl Mag. June 1919. Phil. Mag. 8. 6. Vol. 41. No. 244. April 1921. 2Q 582 Dr. A. B. Wood on Long-range on such observations. The following example serves to illustrate the method of determining the ratio :— (a) Long-range particles (through mica screen of air-equivalent 8°6 cm.). 2 Senne f Number of seintillations on {°° Rs tBMES (ne ob NAe DAM OTONnT Sp .1 servations each of 1 minute alae 4 oe duration ). (b) Ordinary a-particles (mica screen removed, source and zinc-sulphide screen undisturbed). Neowin e919. at. 8, Pe servations each of 1 minute Number of scintillations on (10:2 per minute (mean of 15 ob- (144 hrs. after (a)). duration). Let N be the total number of ordinary a-particles emitted per minute from the source to the zinc-sulphide screen on Nov. 24 at 8 PM., and assuming the decay constant » for thorium active deposit to be 0-064 hr.—1, we have 102m N whence N=102,000 per minute at 8 p.m., Nov. 24, 1919. (c) Ratio. Number of long-range particles 838 — 1 Number of ordinary a-particles 102,000 12,300 — 064 M4, The most probable value of the ratio, obtained as a result of all deter- minations similar to this. is 1 in 11,000, a value only slightly lower than that given im 1916, viz. 1 in 10,000. The ratio 1 in 11,000 was obtained with a mica absorbing screen. It has been mentioned, however, in paragraph § 3, that the value observed when an aluiini in screen was used was about 7 or 8 per cent. lower than this, viz. 1 in 12,000. Tf the difference between the two values is real. the result implies that the proportion of “oxygen atoms” from the mica to the total number of ordinary e-particles is in the ratio of 1 to 100,000 approx., a value of the same order as that obtained by Sir Ernest Mntlentee? when RaC a-particles were fired into the mica. Further experiments should be made, however, with a more active source before this point can be definitely established. § 5. Particles of Range greater than 11°3 cm. Hydrogen atoms. When RaC «-particles (7-0 em. range) were absorbed in mica, Sir EK. Rutherford showed that, in addition to the oxygen atoms of range 9 cm., a number of hydrogen atoms Particles from Thorium Active Deposit. 083: of range 28 cm. were always observed*. It might be inferred, therefore, that hydrogen particles would be produced in a similar manner by thorium C «-particles. A careful examination has consequently been made to detect particles of ranges greater than 11°3cm. For this purpose the source of thorium active deposit was covered with the 8-6 cm. mica screen as before, and with an addi- tional 3-6 cm. mica layer which, with 0°5 cm. of air, makes a total of 12:7 cm. as the air-eguivalent between active source and zine-sulphide screen. In one series of observa- tions only 2 scintillations were observed in 20 “ counts,” each of 1 minute duration—~. e., the average number was 0:1 per minute, whereas the number of particles whose range exceeded 9 em. was 8 per minute in the same experiment. On this basis, it appears that the number of particles of range greater than 12°7 cm. probably does not exceed 1 per cent. of the number of range greater than 9 em., 72. e. not more than 1 in 10° of the total number of «-particles emitted from thorium C. It is hardly necessary to explain that this value is only very approximate, depending, as it does, on such slender evidence. § 6. Summary of Results. The results obtained in 1916 relating to the long-range particles from thorium active deposit have been confirmed. A re-determination of the total number of long-range particles (ranges exceeding 8°6 cm.) to the total number of ordinary a-particies (ranges 5:0 and 8°6 em.) gives a value 1 in 11,000 as compared with 1 in 10,000 obtained in 1916. It has been shown that at least 90 per cent. of these particles originate in the active deposit, whilst the remainder are probably produced by intimate collision of «-particles with the oxygen atoms contained in the mica absorbing screen which was employed to stop the «-particles of 8°6 cm. range. Slight evidence has been obtained of the existence of high velocity particles, probably hydrogen atoms, of ranges exceeding 11:3 cm. The proportion of these relative to the total number of long-range particles is probably not greater than 1 in 100. Thus we have for thorium aetive deposit :— (1) Number of long-range particles (probably a-rays) relative to the 1 in 10* approx. total number of «-particles. * Rutherford, Phil. Mag. June 1919. 2Q2 584 Messrs. Cowley and Levy: Method of Analysis suitable (2) Relative number of high-velocity oxygen atoms produced by cision 1 in 10° approx. of a-particles with oxygen in mica. (3) Relative number of particles of range exceeding 11°3cm. (tay 1 in 10° approxs hydrogen atoms). Ttems (2) and (3) are only rough estimates based on rather slender evidence. In conclusion, I should like to express my warmest thanks to Sir Ernest Rutherford for many helpful suggestions during the progress of this research. Also to Prof. L. R. Wilberforce for his kindness in lending me the source of radio-thorium and the microscope used in the experiments. LUI. On a Method of Analysis suitable for the Differential Equations of Mathematical Physics. By W. L. Cow ey, ASC 1S86.. DALIC...and Tale Tonya. Moe oe Soe Part I. TTEMPTS to obtain solutions of the differential we equations of mathematical physics appheable to problems of practical importance are usually handicapped by the fact that the differential equations arising cannot be solved in terms of the simpler functions without the introduction of restrictions tending to invalidate or limit seriously the application. To the practical calculator, however, the derivation of a complete solution expressible in this form is of merely secondary importance, and does not in fact frequently weigh with him at all. Since the instruments at his disposal can at best only attain a certain degree of accuracy, an equally approximate solution, other- wise unrestricted, of the equations is sufficient. But when the investigation does not merely deal with the evaluation of some quantity for a definite problem, but with the selection of a particular member of a class satistying certain requirements, particular forms of solution, especially particular forms of expansion, are most convenient for the analysis. On general considerations the practical requirements that must be satisfied by such solutions are easily outlined. The eyaluation of the separate terms in the solution should be suitable for application in a drawing office ; that is to say, they should depend purely on graphical or arithmetical * Communicated by the Authors. for Differential Equations of Mathematical Physics. 585 methods demanding the use of a slide rule, a multiplying machine, or planimeter. Graphical processes depending on differentiation ought to be avoided, as experience shows that without a considerable degree of labour inaccuracy is unavoidable. In the second ‘place, if the expansion is in series the convergence should be as rapid as_ possible and the rate of convergence should be more or less evident at each stage. Of the common methods of expansion, the Fourier’s series type is perhaps the most suitable so far as these considerations are concerned. An ordinary power series, on the other hand, may or may not be rapidly convergent over the whole range according to the nature of the problem considered. The crux of the matter is reached from a third consideration, which requires that the form of expansion should be sufficiently expressive to allow of further analysis of the solution to determine the properties of, say, the class of problem under consideration and the particular member desired. Consider, for example, the problem where a series of struts whose law of cross- section is given for the whole class, say [=I,R, where I and Jy are the moments of inertia at any section # and at a standard section respectively, and R is a non-dimensional function of x specifying the law of variation in I along the length. Let F be an end-thrust acting longitudinally along the strut and / the length of the strut. Jet the problem be, to determine the particular member (that is to say, the value of /) of this class which when under given eccen- tricity will give a deflexion a at the middle, the strut being simply supported at the ends. The differential equation for the flexure of the strut is BI2Y dx? when the origin is taken at one end. This of course holds for all values of EI, F, and J, and therefore represents apparently at first sight a four-fold infinity of problems; but it is easily shown that this 1s not the case, but rather that the passage from one member of the class to another is brought about by the variation in value of one expression involving all these quantities, so that in reality there will exist me rely a one-fold infinity of members in the class. Writing v= la’ ,y=ly and inserting this in the differential cas oa we find, dropping dashes, + Ky = 0 d? y Geen (: Ie dx” at R where C= —— and R is a function of &. HI, ; 586 Messrs. Cowley and Levy: Method of Analysis sutable For a given form of R and definite end conditions the nature of the solution of this equation depends only upon the value adopted for C, and ail the members of the class are included among the one-fold values of this class variable. In fact, if the differential equation representing any physical problem is thrown into the non-dimensional form, one or more class variables, themselves non-dimensional, will “be derived in the manner shown above. The differential equation of the steady two-dimensional flow of a viscous fluid, for example, when thrown into the non-dimensional form Pees TOs AOS VE = — Ea =| v (eee ele, where € represents the vorticity, V the speed of the moving body, / its length, and v the coefficient of viscosity, indi- cates that the properties of the motion will centre round a consideration of the modifications in the solution as the non-dimensional quantity VJ/v varies. The general solution of differential equations involving such a class variable satisfying certain boundary conditions which may themselves involve the class variable may be written in the form f@eyQ=0. . . The ordinary power series solution presents (1) in the form Ay + ax + by + agu® + boy? + cgay + etc. = OU, Mah 3) where of course the quantities a, , etc. are functions of the class variable C. In this form each of these coefficients may be more or less complicated expressions of C, and the whole expression is not in a form suitable for investigating the relative properties of the members of the class, aithough it may be convenient for other purposes. There is, however, another method of presentation of (1) more suitable from the present point of view, viz. : Xo + X,¢+ Xoc?4+ ete. = y, . . ee where X,, X,, etc. are now functions of «. The investi- gation of the properties of the class and selection of a particular member satisfying certain optimum conditions can then be easily derived from this expansion. Whether or not the solution can be thrown into this form in general is a matter which cannot be discussed here, but will clearly depend upon the form of the differential equation and the boundary conditions. Hach case will require to be treated on its merits. jor Digerential Equations of Mathematical Physics. 587 Since the above expression for y may be assumed to remain true for an infinite number of values of the class variable, the boundary relations for each of the functions Xo, X,, etc. can be expressed in a particularly simple form. Suppose, for example, the boundary condition be that at a certain point, say @=a, y = ay + aC +a,0?+ etc. 3 then the boundary conditions for the functions Xo etc. are, at i ol, x = Ads Xs = &1, ete. In the same way, if the differential equation be one involving more than one independent variable, a similar series of boundary conditions for Xo, X4, etc., now functions of more than one independent variable, are easily derived. In those cases where the equations are linear and involve only one independent variable there is another method of presentation which bears a close resemblance to that given above. Taking an equation of the fourth order for example, the general solution may be written in the form y = Afi(«)+ BA(#)+Cfa(e) + DACs), . . (4) where A, B, C, and D are four quantities to be determined by the boundary conditions. Instead of, as in the previous method, supposing y expanded in a power series of the class variable, we may imagine each of the functions /,, /), f3, fs 80 expanded. If these expansions can be determined, the boundary conditions when inserted will provide similar expansions for A, B, C,and D. For certain cases, parti- eularly with linear equations of this type, this latter form of presentation is frequently the simplest. The first step in the analysis is to determine the form of the coefficients in the expansion. Unless it can be proved that a con- vergent expansion of this type is always possible, it will be necessary to check the result & posteriori. For illustrating these points the classical engineering problem of deter- mining the whirling speed of a shaft whose cross-section varies along its length will be treated in detail. It will be seen that all the conditions of rapidity of convergency and ease of calculation are most satisfactorily fulfilled. Generally the method that will be adopted will be to assume an expansion for the dependent variable as a power series in the class variable, to insert this in the differential equation, and equate the coefficients of the 588 Messrs. Cowley and Levy: Method of Analysis suitable various powers of the class variable to zero. A series of simpler differential equations will then be obtained for each of the coefficients, with simplifed boundary conditions, Consider the following example. The equation whose solution is required at whirling is, expressed in non- dimensional form, d? d*2 alt= hy = 0, es where R=I/I,)=the ratio of the moments of inertia at any point « to that at any standard section ; y =deflexion of the shaft axis at any point w. This quantity is assumed_so small that it is legitimate to $2 write oe for the curvature where / is the length of. the aa shaft. OS sel Ae ee rine p= —, | ose where E=elastic constant, —=mass per unit length, and 27/q =period of rotation of shaft. The boundary con- ditions that we may suppose imposed on the shaft are given as deflexions and bending moments at the extremities of the shaft. For the sake of definiteness, let these conditions be z=0 y=0, Bending moment = MEI), | 7) vi 1 ¥ as ) 9 at 0. J E ; ( The class variable in this case is clearly 0°, while R and mw are given non-dimensional functions of w. I[t will for the moment be assumed that y can be expanded in ascending powers of @* in the form Y = Uf +410 + y6 + 930". C ° ° ° (8) Inserting this in equation (5) and e uating coefficients of q powers of 6 to ZeLrO, & lL Nete (Riga) | Pan dx? for Differential Equations of Mathematical Physics. D89 Hence ey AUN os : al af i) mt (90) dl Gaue sal Zo) Gee 7 From (99) it follows that vd x 2 Tle aNea Yo =A “del tb def pt Ce+D 5 - 5 (10) therefore ,® é te "@ da Die w Va de =\ d| dar pur PAU we 0 RJ, «0 ae wv “a ot ‘a “e Z M d a A{ de = ie div pw a it v0 0 v0 0 ae Oa e dy ” *2 x +B dw\ = da| ude dx i 0 a 0 aw e 0 v x @ day "2 x a rae (2 aa(6 ra R def dana | ae) R peda e 0 0 0 0 v0 0 ECE es ee Wh UGH EMSs Gigee, les diag iy Qelead 7) pets this expression for yo y,... in (8), y takes the orm y = Ahlen) + Blo) + Of) + Dios + (12) where ade de a) = a — B40 | ade’ a) ar" pdr le if > eee. (S) 7) = f da We | de ("4 (ae | nae! a : aia (14) e/ 0 70 20 R iaic) = 06" “ae {* “Ale aed. LS Se i 98 HS ee ERE fia) = 1408) dn | ale ‘ie GE LN oct ed) UR em 0 0 CUA) The forms which these integrals adopt, although apparently involved, are in reality very simple. It should be remarked 590 Messrs. Cowley and Levy: Method of Analysis suitable in the first place that the law of formation is clearly bound up with the relation * a 2 dy ( #® ~ x Yn — { dx i dx AL LYn—1 0 a 0 v0 e/ 0 The four arbitrary constants A, B, C, D are as yet un- determined, but will shortly be found from the boundary conditions. Inserting the boundary conditions (7) in (12), the four following equations for A, B, C, D are easily derived : 0 = Af,(0) + BA(0)+ CK) +DA0) 0= ALG) +B,(1)+CA+DAG) t ae Mo= Aj! (0) + Bf"(0) +070) + D/O) | 0 = Af") + BA") +CA"0) + PA") J Referring now to the expressions for these functions, it is to be noted that | AiO) = fo(0) = 73.0) = 9, fa(0) = 1, AiO) = 0, f2"(O) = 1/Ro, 73") = 0, f’(0) = 0. Hence it follows that B=M,)Ry and D=0, where A and C are likewise easily determined. The interest of this problem does not, however, lie, from the engineer’s point of view, in determining the deflected position of the shaft under the applied bending moment HI)M) at the end, for any given period of rotation 27/q, but rather in finding those particular values of g for which the shaft whirls. Since the analysis is based on the assumption that the deflexions are small, whirling would involve a violation of the basic assumptions, and the deflexion would appear from the equations to be infinite. The condition that this should be so, and therefore that whirling should occur, is easily derived from (17), viz. OO). 00) Oa Flee) FC), | Rely Ome Of (0), (20) fr Oma ain E01), fed) which, when the particular values for the functions are inserted, becomes Ais =f OA), .°. 7. eee = 0, (18) vite aa iz de ‘ ‘ Gd © wd i= l. (" = +0'f a — ; da ( da da ( ” a R 0 90 Ro a Jo ig Us BA for Differential Equations of Mathematical Physics. 591 where a= 1464 (ar ( ude f ae \, a x wv ts a vde( a) dx mrdot.. 0 Ryf'(1) = a de {" padat... 0 For purposes of evaluation of the coefficients of the powers of @* in each of the above, itis to be noted that the first term on the right of (20) will. be automatically. evaluated in deter- mining the second term on the right of (21), which again will be evaluated in calculating the ‘second term on the ‘right of (20), and soon. The labour, therefore, in determining the coefficients in the above four series is really only half what it would appear to be at first sight. The geometrical outline of the shaft, and the loading, are of course introduced through the functions R and yp, and since all the integrals can be rapidly evaluated by means of a planimeter, or other wise, as will shortly be seen, the most general case will be almost as simple as any other by this method. As an illustration, however, an example will be taken which is capable of solution by another method in order that a comparison may be made. Huample.—Consider a homogeneous circular shaft simply supported at the ends and built of three portions, the middle portion being of diameter d and length //2, and the two ends adjoining the supports of diameter dj and length 1/4. Itis required to find the first whirling speed of this “shaft. The mid-section or the section of reference has a diameter d, and w, and Ip can easily be evaluated from this dimension. p will be 1/16 over the two end portions and unity over the middle ye while R will be 1/256 and unity respectively. The series f/;(@) and f,(2) can be worked out simultaneously as well as the two series f(x) and f(x), as can be seen from the expansions. ‘The steps in the calculation are evident. For example, in evaluating f(x) and f,//(w), «/R is first obt ined for various values of 2. Each figure is then divided by the maximum occurring in the column for convenience in plotting, a method that may be used throughout whenever a 992 Messrs. Cowley and Levy: Method of Analysis suitable curve is to be integrated. From the graph the integral curve, \ a can easily be obtained, the mean-ordinate 0 method being most convenient. Thus at 0: 05 the integral will be 0-05 times the average height of the curve between 0 and 0:05, at 0-1it will be the area up to 0°05 plus 0:05 times the average ordinate between 0:05 and 0-1, and so on. These integrations can rapidly be performed, and if it is necessary for the sake of accuracy to integrate thoroughly a portion of the curve where the ordinates are small, this portion can be plotted to a larger scale and allowance made accordingly. The actual details in the calculation will be omitted, but experience shows that the various steps in the process can easily be carried through accurately by anyone not even conversant with the calculus. We then find 0 = Al) fs 1) — AG) A") = $1(13:28 + 2°84 + 0°00127 6°)(0°1174 4 2:4 x 10°444 +5°54 x 107868) —(14+ 09585 4 + 1°033 x 107368 + 2°656 x 1077 x 10”) x (1+ 0°601 6* + 2°36 x 107468 +5°1+ 10780?”), te.» b— 0375.02 + 0-015 63 —0:00007202 2. Se This gives wygrl* gil, — = 3, from which gq is at once derived, and it will be seen that the term in @”? may be neglected, the series being so rapidly convergent. As a check, the whirling speed in the present instance can be found by an alternative method applicable when the shaft is composed only of portions of uniform section. The method consists in supposing the shaft simply supported at the positions where the section suddenly changes, as well as at the ends, and then utilizing the conditions that the reactions at these intermediate supports are zero. The portions between two neighbouring supports are uniform, and therefore the functions /;, fe, etc. are directly integrable algebraically*. This method gives C2 = 2) as against §* = 3. by the previous method. * Advisory Committee for Aeronautics Report, Q & M. 690. jor Differential Equations of Mathematical Physics. 593 It has become appar ent that for all practical purposes the series used in the foregoing analysis are rapidly convergent. In point of fact the rapidity of convergence can easily be investigated, for each one of the functions Alan. fale) is absolutely convergent. Let y, and R, be the greatest and least values of w and R over the range of w sorandion ed ; then, for example, Q we © ae x p | fa (2) (x) | without loss of generality, so that the fundamental equations may be re-written : eee Oy (s*) Ov 0 ory oe ey pda ot\dy/ dy d@Loy) de OY : BES Ly V7NG) Se et) orgs oe 3 ev Oy Ow Ox ve Ox — >) 2. Ware ° (5) * Phil. Trans. A, vol. 214. ORG 2 Tp Ou Ou. \Ow jor Differential Equations of Mathematical Physics. 595 The usual process is to eliminate the pressure terms from these equations, giving an equation in ee alone in the form pe ox! a ot 2 ie Se oe = EV ES Vt SS TT. (8 This 8 derived a differentiating (4) and (5), is one order higher than each of the original, and therefore an equivalent system to (4) and (5) must associate with (6), an equation of one order lower, otherwise extraneous problems may be introduced not contempiated i in the discussion. These extraneous cases may be excluded by the following process. Let A and B be two points situated in the Hae ‘Teast emities of any given curve ; then from equations (4) and (5), aA 2 li 1 vA \ a. (P,—Pp) = i © (ude+ ody) B ov a ( ou Ov A - c (Ned Ve Ody New wn ee le A(T) Hquations (6) and (7) are the equivalent of (4) and (5). Equation (7) is of course an expression giving the pressure difference between two points Aand B, and as such need not be considered in relation to (6) except in so far as it must be used for the interpretation of any externally applied con- ditions. It will then furnish information with which the solution of (6) must be consistent. For example, if points A and B are maintained through some external agency at a constant difference in pressure P, then equation (7) will furnish a condition which must be “satistied by the solution of (6) along every curve, lying wholly in the fluid, that can be drawn between A and B. The necessary and sufficient condition that the external forces can only be of a conservative A nature is that | (Xdw+Ydy) should be independent of the B path. It follows in that case that the path from A to B may be arbitrarily chosen, and one may be selected in any convenient manner, the integration along all other paths reconcilable to iis being che same. i general it will be found most Caavoniene to select as a portion of the path the contour of the boundary along which the velocities are specified. Consider a contour completely enclosing one of the rn 596 Messrs. Cowley and Levy: Method of Analysis switable boundaries ; then the terms on the left-hand side vanish, and the equation (7) becomes i (p+ 5 dy) = yp t (Viude+ V7vdy) Dé D = av | Seas 4)" en Any such contour is reconcilable with a contour round the actual outline of the boundary. Equation (7) has three particular forms, when no external forces exist, of interest to the present problem. Round a body at rest, where wee e Lt, e etc. are zero, it takes the form t e) aL ( 2 oy 5 as = (0.0 2.3. v body "On Round a body moving with constant velocity U in the direction «, | HOS typi ek (Ue 2 )as, ee e/ pede On Ne eaOr on Along a boundary at rest, such as a wall of a channel where the fluid is under a given pressure-head P between the ends maintaining the flow, it becomes t 20 § Cael. On p “ boundary Before a solution of equation (6) can be accepted, therefore, where the expression for w satisfies the appropriate boundary condition of the problem, it is necessary to satisfy the various forms of (7) where the integral is taken round each boundary of the fluid. The boundary conditions that are to be inserted are of course those that involve the statement that there is no slip. § 3. Steady motion in two dimensions.—Consider the case where the boundaries of the fluid are in steady motion, and where in addition it is assumed that the fluid everywhere moves steadily. As far as experimental evidence shows, these two assumptions would appear to be entirely independent of each other; for in all known eases above a certain value of vl/va steadily moving body appears to give rise to a periodic eddying in its wake. Below this critical value, however, these two assumptions appear to be perfectly consistent. If steady motion of the fluid is not possible above this critical value, for Differential Hyuations of Mathematical Physics. 597 an indication of this would be expected to be given from the form of the solution in the steady case. It will be convenient to throw the equations of motion into the non-dimensional form by the following substitutions :— =the’, oo where L is the length of any particular part of the moving body, and 2’ and y' are now merely variable numbers ; “w= W's | (es MUI where U is the steady velocity of one of the boundaries, and uw and v’ also variable numbers. Under these circumstances vp = UL’. Inserting these in the equations of motion and in the boundary conditions, we find, omitting dashes, p+ Of 0% 9 cy, 9% 9 op te at vp eof Se 2 yy SE oy bio, . . (8) where C= UL/». In the case where the body is in motion with uniform velocity in the direction of x there are in addition ee | round the moving body; . (10) OU i nnd when the boundary is at rest SOUMCNbOCyeatenostw ys.) (LL) The integral condition in these two cases becomes 2 08 ds = C ( 8 as round the moving body, (12) On n body © body lh ae ds = 0 round body atrest. . . . . s (13) n ; body It is in this non-dimensional form that the equations may Plil. Mag. 8S. 6. Vol. 41. No. 244. April 1921. 2K 598 Messrs. Cowley and Levy: Method of Analysis suitable be more conveniently analysed. Reverting to the experi- mentally known facts that the value of UL/» determines for any particular problem the nature of the flow, we see that this finds its counterpart in the presence of that quantity as a variable parameter C in the differential equation. It is clear, in fact, that mathematically as well as physically the whole problem may be made to centre round this parameter ; and since we are more directly concerned in the modification in the state of the flow over the whole field as it were as C varies, rather than with a comparison with the state of flow at one point in the 2, y plane with that of another, the most natural form of solution that is suggested would be a solution asa Seriesin powers of C. Whether or not such an expansion is possible as a convergent series is a matter which will be entered into shortly, but for the moment it is proposed to assume that the expression for wW the stream-function may be written in the ferm y= wotwCtwt i... 2. HA) [It may here be remarked that if the problem of the motion of a body had been solved for a given value of C, the solution for the same body moving backwards may be derived by writing —C for © in this expression.| Inserting this in equations (8), (10), 412), etc., and equating the coefficients of each power of C to zero, since the equations hold for all values of that variable, the following system is obtained : V ivbo == 0, 7 . PMO. AC!) (8a) ovo — 0, and oe = 1 - (10a) | Coefficient of co, round the moving body | fae CW es O- - (12a) eon _ 0% 0 Ov, O sn Viab = a ae Ao OE ae to, | | (30) ey, re, | aay Ser ae | . : (106 0 ie oy r Coefficient at c, ) round the boundaries 0 0 Ov» | 2)», (a on hha {as On On’ JO tS ae a ete., etc. In the above the expanded expression for yr has been inserted not merely in the differential equation, but.also in the integral and boundary conditions, and in all cases the coefficients of jor Differential Equations of Mathematical Physies. 599 powers of C equated to zero. It will be seen that there are thus sufficient equations and boundary conditions for each of the functions Wo, Wu, ete. to determine them uniquely. It is not proposed here to prove the uniqueness of these expressions, but rather to point to an analogous series of physical problems with which each of these equations correspond. § 4. Parallelism in the theory of elastic plates. Hlexure of a flat plate --lf a flat plate of flexural rigidity EI per unit width, Poisson’s ratio o be loaded laterally with an intensity Z per unit area, the plate being supported in any given manner along the edges, then if yw be the deflexion at any point measured relative to a given horizontal surlace, the equation determining vf is Darran aires erat ia ene (LD) where D=EI/(1 —o”) and [=2/3h3, h=thickness of plate, while the shearing force N per unit length of any curve drawn upon the plate is in the direction perpendicular to the plate ; ¥ fo) 9 ee Wy, e ° ° ° (16) when n represents the normal to the curve. A complete parallelism may now be established between the problem of the flexure of a flat plate under lateral loading und each of the problems involved in the determination of the functions Wo, Wj,..., the boundaries of the flat plate being the same as those of the fluid. For the evaluation of the function W in that case the following equations and boundary conditions were required to be satisfied :— Vien = Pho Wi eee Wn) 3 on = 0, Bes oe Q along every boundary. ote as 1 (2 VEN US a ) body body For the evaluation of Wo, Vivo aS ovo iL ovo = () alone the moving boundary ae Ow 2R 2 G00 Messrs. Cowley and Levy: Method of Analysis suitable If now the value of y for the deflexion of the flat plate be identified with the stream-function component bo, 1b 95a simple matter to interpret the boundary conditions in terms of those for the plate. Consider first the determination of ro. The fundamental equation Vivo = 0 indicates that it may be taken to represent the deflexion of an unloaded flat plate. Along the curve representing the moving boundary, how- ever, the two component slopes are given—viz., oY = 0, oa =—1; and this curve must therefore be Pe to te these slopes at each point. Although this determines the relative elevation of each point of *this boundary, it does not fix the elevation of the whole curve with reference to, say, that of the other boundary. ‘This is immediately derived from the interpretation of the remaining condition, oe: fs ‘| any ro) ds = 0. From equation (16) this implies that ( Tg == 02 ea & that is to say, the total shearing force round the curve corre- sponding to the fixed boundary must be zero. Consequently the absolute elevation of the curve corresponding to the moving boundary must be so adjusted that no resultant shearing force is brought to bear on the curve corresponding to the fixed boundary. The conditions can be realized with comparative simplicity in practice ; but for the present it is sufficient to note that under the circumstances described a measurement of the elevations of each point on the plate will determine the function Wo directly. By inserting this value for yo in the expression on the right-hand side of the equation (8 0) an expression is derived which, when interpreted in the light of the parallelism explained above, determines the lateral loading which must be imposed on the same plate as before. ‘The boundary conditions can be interpreted with equal ease and simplicity, and by a direct measurement of the deflexions yy is at once derived. A repetition of this process leads successively to the evaluation of the functions Wy, Ws, ete. for Differential Equations of Mathematical Physics. 601 § 5. The parallelism with the flat plate demands that if the solution is to be obtained by an experiment of this nature, the lateral loading will vary in magnitude and in sign at different parts ‘of. the plate, a condition diffieult to realize in practice. By a procexs of graphical integration, however, the problem may be reduced to that of an unloaded plate with modified boundary conditions. Consider, for example, the equation ViZ=f@,y)3 then a particular integral is Jy = JS derdys[log re {flog ms founder dy} This equation may be evaluated graphically, or, if convenient. analytically, and by the or dinary | transfor annert the equations whose ev len 3 is required can be reduced to V*Z = 0. This reduces the problem once more to an unloaded plate, the method of fixing at the boundaries being determined quite simply from the original conditions and the derivatives of Z,. § 6. Without attempting to enterinto too great detail or to evaluate the results in a given case, the nats of the disposition of the stream- eae can be seen on general grounds in any particular problem. ‘Take, as illustration, the problem of the motion of a cylinder about its diester of motion moving at uniform speed down the centre line of a channel. To determine the first term in the expansion outlined above, we must consider an unloaded flat plate bounded externally by the parallel walls of the channel and internally by the section of the cylinder. Along the walls the velocity of the fluid is zero, and consequently the plate must be clamped horizontally along these boundaries. Round the inner boundary the slope at each point with respect to a must be zero, where « is along the channel, while with respect to x it isunity. At the same time there must be no total reaction on each boundary in consequence of the integral condition. This implies that the ean must be.placed under the influence of a pure couple about the ae of w cf such magnitude as to produce unit slope. Any slope, provided it is small, may be taken as standard. This does not imply that the velocity of the body is small, but rather that the analogy of the flat plate will only be \ sae when the deflexions of the latter are small. The stream-lines corresponding with the first term in the expansion can now be found by plotting 602 Messrs. Cowley and Levy: Method of Analysis suitable ‘the contours of the plate. On general grounds it is clear that these will be of the nature shown in fig. 1, where points A and B, the positions of maximum and minimum deflexion, are stationary points at the instant relative to the walls of the channel. If the body be considered at rest and the planes be supposed in motion past it, the problem can be re producec by tilting the whole system in the previous case until the section uf the cylinder is once more in a horizontal plane. If the cylinder is symmetrical about a plane perpendicular to the axis of the channel, for example, a circular or elliptical cylinder, the system of stream-lines obtained above would be symmetrical about the lenothwise and crosswise plane. A consideration of the next term in the expansion, however, indicates how asymmetry about the crosswise plane arises This is due to the fact that the expression on the right-hand side of (8D) is of the same sign in opposite quadrants. The result of this is, that whereas the loading still maintains symmetry about the w axis, asymmetry occurs about the y axis. The stationary points A and B are thus, as can be seen by a simple inspection, similar to that aire ady indi- cated, moved further back towards the rear of the body, the extent depending upon the value of VL/» with which yy, is associated. It is evident that the exact expressions for the various functions could be obtained by experiment, and until that is carried through, no general conclusions can be drawn, but the general nature of “the stream-lines is clearly Sav henced in the foregoing discussion. It is hoped in a future paper to indicate how the system of equations (80), (8c), ete. can be solved graphically. $7. Simple cases for verification As a check on the analysis the case of a cylinder rotating in a viscous fluid inside a concentric cylinder is here worked out. It should be noticed that in this case the body is not moving forward with constant speed but rotating with constant angular g velocity; but the necessary modifications and boundary for Differential Equations of Mathematical Physics. 603 conditions can easily be obtained.” The result can then be compared with that obtained by the ordinary methods. Consider the case of a circular cylinder of radius a spinning with uniform angular velocity @ , and coaxial with another cylinder of aide b, the region eww een being filled with a viscous fluid. ‘ The boundary conditions are then, writing C= a and using non-dimensional quantities, Eons 7 — 1 ae OMe at Pe ea Ges fon) Q7 ; ( OVo hve 0 b Omen.) Paty alone: 7 = — + The equations for y, etc. are V oro = 0 plea aa er \—% | ay and the equations for yy ete. all reduce to the same, the right side vanishing in virtue of the fact that CHG eee oY oO v7? . ae ar v) Hence the expansion in C consists of the terms in fv only. The solution of (17) is tro = Ar logr+Br?+Clogr+D, . . (8) where the four constants are to be derived from Wo=0, Ov) _ and 2 Shee | oe ==); {ave (\77ahro) = 0 at # =. O/ as 604 Messrs. Cowley and Levy: Method of Analysis suitable Now, round r=6b/a "20 Iar b dO (a2 v9 a { a E (VW) | ae. [2 ww] =o. r=hja Inserting these four conditions in the expression (18), for Wry we derive finally att 1 (iit b BIeD | to = 3a ae mane ds Hence the angular velocity at any point is 5 0 aon Ee r Or a? — 1)? ie where R is the actual distance from the centre to the point (cf. Lamb’s ‘ Hydrodynamics,’ § 333). Justification of expanding in powers of C near C=0. The foregoing analysis is based fundamentally on the assumption that a convergent expansion for wv in ascending integral powers of C valid over at least a finite range from C=0 is possible. It has been customary in dealing with problems of fluid motion to impose tentatively on the equations the restriction that the inertia terms are negligible for slow motions. In effect this is equivalent to a neglect of the second group of terms in (&), the assumption being that if a solution of Vie =0 can be found, the inertia terms oo ete. will at most be of the same order as V4; and since C is. small in comparison with unity, that group of terms may be neglected. That this is justifiable a@ posterior’ is clear when we remember that, on the analogy of the flat plate, the equation 7 Vito = 9 will represent the deflexion of an unloaded flat plate where the boundaries are compelled to satisfy certain conditions as regard slope ete. In order that the analogy may be valid, these slopes, although reckoned as of the order of standard slopes, are small. It follows that all the derivatives of ro which occur in the equation are approximately of the same order, and therefore those involving the factor ( in the equation may be neglected in comparison with Vi. On for Differential Equations of Mathematical Physics. 605 this physical argument it:seems justifiable to assume that there is a solution for the differential equation in the neigh- bourhood of C=0, approximately given by v = Wo where 0 V bo = Let the solution therefore be written v= votx for C small, and let us assume that y is also small. Inser ting this in the differential equation and boundary conditions, we find | ey = Oo 0 a Go ov 0 vy? vo} Ou O Oy 0. av [ Ovo > ) { Ox ) 2 \ 6 se oy a x ete. 20 ae ou x etc. [- (20) Assuming x and its derivatives are small when C is small, an assumption that will be checked a posterior, we must therefore write to a first approximation Vis ae iJ Ovo 0 Vv? Wo ovo 0 Vivo}. Ou OY Oy du where y must satisfy the boundary condition OK ON. arm oy round all the boundary, and the correspending integral conditions. The quantity y therefore represents once more the deflexion of a flat plate loaded with a definite finite distribution, but with density proportional to C. It therefore follows that we must write y=Cw,, where y is finite and independent of C and wy, satisfies the differential equations already used in determining y, in the Leen assumed expansion. By referring back to equation (20) it is now evident that we were justified in neglecting the remaining terms on the right-hand side, for all the derivatives of from the flat-plate analogy are of the order (. This step-by- step process may be continued along the same lines where we find that on seeking to determine the finite function, which isa solution of the differential equation for small values of C, the power series previously assumed is derived, Wo, Wi, Wo, ete. being determined from the same system of equations as before. In point of fact, the expression for x so found is a-‘Taylor’s series regarding C as the variable. 606 Differential Equations of Mathematical Physics. ~ Theorem. There is no expression, when Ul/v is small, for the stream function of the viscous steady-motion equations in two dimensions, expressible in a finite number of terms in powers of Ul/y other than those explicitly independent of Ul/v. For if so, let bp = lene and insert this as before in the differential equation. Equating the coefficient of Ct in the equation to zero, we find for Yn Orn O ; TUR = "30 Uy Oy oe with the conditions that On and = and {a 2 (*) y b fol ody OF NOL are zero round the boundaries. Hquation (21) is clearly the limiting case of our funda- mental equation (8) when v tends to zero. There is, however, this difference from the common conception of the perfect fluid—that here the no-slip condition is maintained by the presence of some distribution of vorticity. In tact, (21) may be written dv» [Oy _ 9 ws ee Oy Ow aa on, Wn So Wns which says that the lines of constant values of V7, are coincident with those of constant Yr, 3 2.é., V7? Wn ar I (na), where f is a function to be determined from the boundary conditions. We may regard equation (21) as the limiting case of the flow of a fluid of exceedingly small viscosity ; but since there are no externally applied forces or pressures causing motion of the fluid. and no motion of any boundaries, there cannot possibly be flow of any nature. Under these circumstances yy, will be constant, and may accordingly be ignored. ‘This clearly applies to all values of x down to n=1 when the boundary conditions are now no longer zero. It may ve remarked in passing that the solutions of problems in steady motion independent of Ul/v must Ware Propagation over Parallel Wires. 607 therefore be such as to satisfy the two equations : Vine = 0 and We le) where 7 is arbitrary, along with the appropriate boundary conditions. Such problems, of course, give a system of stream-lines in virtue of V7*w, which are not changed when the direction of motion of the body is reversed. The only known solutions of this type so far obtained are those of the steady motion of a fluid between the walls of a channel and the steady relative rotation of two concentric cylinders. It may be remarked, in conclusion, that certain classes of problem, although violating the ordinary condition of OY A steadiness, viz. ry, and oy) =() everywhere, can still be reduced to that of a steady-motion case by the super- position of a uniform linear velocity or angular velocity upon the axes of reference. [For example, if every point of a cylinder of any given shape describe a circular path inside and concentric with a given circular cylinder, the motion may be reduced to that of a steady case. LIV. Wave Propagation over Parallel Wires: The Proximity Hifject. By Joun R, Carson, Department of Development and Research, American Telephone and Telegraph Company®. Tl. [ntroduction. HE importance of the problem dealt with in the present paper—wave propagation along a conducting system composed of Um similar and equal parallel wires— has been emphasized by modern developments in telephonic transmission such as the carrier wave system of the American Telephone and Telegraph Company, and the utilization of loaded cable circuits in which the wires are in very close juxtaposition. For such systems, where the frequencies employed are relatively high and the wires very close together, considerable theoretical work has been found necessary to reduce the solution to a form available for immediate engineering use, in spite of the previous * Communicated by the Author. 608 Mr. J. R. Carson on valuable researches of such mathematicians as Mie* and Nicholson f. In the present paper the analysis of the problem starts with Maxwell’s equations, but one simplifying assumption is introduced ab initto—namely, that the exponential proj agation factor is a smal] quantity. The approximations involved in this assumption are fully justified in all physicai systems which could ‘actually be employed for the trans- mis-ion of electrical energy; so that from a practical standpoint the restriction thus imposed on the generality of tle solution is purely formal. By aid of this simplifying assumption the determination of the current distribution in the wires is essentially reduced to a two-dimensional problem, which is solvable from the boundary conditions satisfied by the tangential magnetic force and the normal magnetic induction at the surfaces of the wires. With the current distribution in the wires thus determined, the expo- nential propagation factor y is solvable by applying the law curl H=—yd/dt H to an appropriate surface bounded by a contour which includes line elements in the surfaces ot the wires. By this means it is shown that the propagation factor satisfies an equation of the form lipK=2Z + ipL, where K is a electrostatic capacity between wires, Z the “impedance” of the wire ver unit length, and L the in- ductance aren toe to the magnetic flux between the wires. This equation is of exactly the same form as that derivable from the telegraph equation, but differs therefrom in that Z and [are both functions of the frequency p/27 and the parameter f& (ratio radius of wire to interaxial separation between wires). As formulated in the present paper, the actual calculation of Z and L involves only the computation of Bessel functions. The inethod of solution sketched above and worked out * G. Mie, Annalen der Physik, vol. 11. pp. 201-249 (1900). In this paper the problem is attacked in a fundamental manner. The results. arrived at are, however, limited to a restricted range of frequencies and the parameter k (ratio radius of wire to interaxial separation ). Furthermore, Mie’s method of attack does not admit of extension to other types of transmission systems in which the surfaces of the conductors are generated by lines parallel to the axis of transmission. ud. ¥V' Nicholson, Phil. Mag. vol. xvii. p. 255 (1909), and vol. xviii. p. 417 (1909). In these papers “formulas are derived for the resistance and reactance of parallel wires which are valid for a very wide range of frequencies, but which are applicable only wken the ratio of the radius of the wire to the interaxial separation between wires is a relatively small quantity. Wave Propagation over Purallel Wires. 609 in the following section of this paper has one substantial advantage which gives it an interest extending beyond the specific problem : it is quite generally applicable to problems in wave propagation along conducting systems in which the surfaces of the conductors are generated by lines parallel te the axis of propagation. For example, it has been successfullv applied by the writer to the problem of wave propagation along a wire parallel to the plane surface of a semi-infinite solid of finite conductivity ; the corresponding practical problem is, of course, transmission over a ground return circuit. Again, it The been applied to quantitatively investigate the effect of a concentric ring of iron armour wires on submarine cable transmission. From an engineering standpoint the most important effect in parallel wire transmission is the dissipation of energy in non-magnetic wires. Consequently formulas for the alternating current resistance of the wire have | been worked out in detail, and the functions involved have been computed and graphed, the data being collected in section IIL of this paper. The a.c. resistance of the wire is expressed in the form eR, where Ry is the a.c. resistance of the wire when the return conductor ts concentric (and is therefore calculable from well-known formule and tables), and C is a correction factor which formulates the modifying effect of the current in the adjacent wire. This is termed the prowim’ty effect correction factor, tollowing a usage suggested by Kennelly *. The correction factor C approaches an upper limit C,,, chi h is a function of the parameter é¢ only (ratio of radius to inter- axial separation between wires), which it appreaches in accordance with an asymptotic formula derived from the asymptotic expansion of the Bessel functions involved. By aid of the data of section III the calculation of C is reduced to a very simple matter. II. Mathematical Analysis and Derivation of Formule. The conducting system under consideration, as stated, consists of two long similar and equa! parallel wires of circular cross-section, in which equal and opposite currents are Howing. ‘The radius of the wire is denoted by a, its conductivity and permeability by \ and mw respectively, and ua Kennelly, Laws, and Pierce, Proceedings A.I.H.E. 1915, pp. 1749- 1813. 610 Mr. J. R. Carson on the interaxial separation between wires by c. The co- ordinates of any point in the system with respect to the axis of one wire are denoted by 7, 01, and the co-ordinates of the same point with respect to the axis of the second or return wire by 72, 02, as shown in the sketch herewith. / Zz 4 7 / ea a / / oe / = UZ \e x ase , Ly a 4 me BS 2 / S a / 8 Za Ze \82 \ \ / Before proceeding with the analysis of the specific problem, a very brief discussion of the fundamental field equations - will be given, in order to indicate the significance of certain important simplifying assumptions employed in the sub- sequent analysis and the restrictions thus imposed on the generality of the solution. It may be remarked that these simplifying assumptions are quite generally applicable to problems in wave propagation where the surfaces of the conductors are generated by lines parallel to the axis of propagation. The discussion starts with Maxwell’s equations in a continuous medium: eurl EK = —pwip H, a curl H = (40+ Kip) E, | chive =_40; ( rion lee hiveniay -=—=s()) ) In these equations E and H denote the electric and magnetic forces, while A, w, and K are the conductivity, permeability, and specific inductive capacity of the medium. Itis assumed throughout the following that elm. c.g.s. units are employed. The axis of propagation will be taken as the axis of Z, and it will be assumed that the electric and magnetic forces vary as exp (ipt—Y2) 3 consequently the frequency is p/27, y is the propagation factor, and the operators d/dt and 9/0z are replaceable by ip and —y respectively. All six vector com- ponents (Ezyz, H2yz) satisfy the wave equation (B92 +97/dy7) = —(m?+y, . . (IL) where ne = (47Apip— (n|v)?) and v= 1/\/ Ku. Wave Propagation over Parallel Wires. 611 It will be found oprsanc ane to write the field equations 1m the form: 2a? pp i, oy ° 1) (m?—y")H, = oh oe Be Ve 2° per le) ghee a m* 10) Hy o 7) (m?—y?) H, = hp Sa ay ees are ( 2) (m?—9) EK, = ne EH, + pip os lal (3) Oz Op fe 1 O) (m?—y’) HE, = Ty H+ pip a: Ee (4), From equations (3) and (4), ape 0 0 : coed jae —— al iD eae eae ° e ° ° ° eo Ff Pip D2 y Oy Zz (9) We now introduce the assumption, essential to the sub-. sequent analysis, that y and p/v are very small quantities of comparable orders of magnitude. That is to say, they are very small compared with unity and also compared with the value of m in the conduetors. The justification for these. assumptions and their immediate corollaries, introduced ab initio, resides in the fact that the solution obtained by their aid actually satisfies the necessary conditions in trans-. mission systems of ordinary dimensions, even if the frequency exceeds a million cycles per second. From equations (3), (4), and (5) it follows that the electric: force in the plane normal to the axis of propagation is of the order of magnitude of y/(m?—4”) compared with the axial component H,. In the conductors this is a very smal! quantity of the order of magnitude of y/47Aup, while in the dielectric it 1s a large quantity of the order of magnitude of 1/y. Consequently in the conductors the electric force in the plane normal to the axis of propagation will be ignored in comparison with H,; in the dielectric, however, the former is large compared with the latter. By corresponding considerations the axial magnetic force H, is very small compared with the magnetic force in the plane AY, ‘both in the conductors and in the dielectric. As a consequence of the foregoing, the magnetic force w- the conductors is derivable from wip H. =—2 i, oC) ai ag ieca girs elt te) eRe = 5, oe Se ee (7) which replace (1) and (2). 612 Mr. J. R. Carson on We are now prepared to take up the analysis of the problem of wave propagation along parallel wires; in the course of this analysis the significance and utility of the simplifying assumptions will become more apparent. Irom the general solution of the wave equation in olar co-ordinates and the special conditions of symmetry which obtain, the axial electric force in wire $1 is given by the Fourier-Bessel expansion = SA, Jno) bos 16), 4) oe 0 cand in wire 22 by Kk, = aS (—1)"AnJ n (pe) COS 700, Sh ee Ava (9) j=) where pi =r, V AmApip, Po = iV 4ardpap. In these equations J,(p) is the Bessel function of order n and argument p, and the coefficients Ay...A, are to be determined from the boundary conditions at the surfaces of the wires. In either wire the magnetic force is then -derivable from a 0) Pere © oa where 7, 0 denote either 7), 0; or re, 6, according as wire #1 -or Wire #2 is under con-ideration. From the symmetry of the system, however, the satisfaction of the boundary con- ditions imposed at the surface of one wire insures their ‘satisfaction at the surface of the other. In the dielectric the electric and magnetic forces -are expressible as Fourier-Bessel expansions, the Bessel functions, however, being of the ‘ external” or second kind. In accordance with the assumption, however, that is a small quantity of the order of magnitude of p/o, it follows that so long as pe/v (where ¢ is the separation between wires) is a small quantity compared with unity, the Bessel functions in the neighbourhood of the wires may be replaced by the limiting forms which they assume for vanishingly small arguments. In particular, the mag- snetic forces H, and H, in the neighbourhood of the wires Wave Propagation over Parallel Wires. 613 are expressible as es we a sin nes Gla ee DS Be I) i = =B, oi "9 Seve a sh eC cs nde H, tr x6, ry ry” ap | \ Le ie The magnetic force in the dielectric is thus expressed in terms of two symmetrical waves centred on the axes of the two wires respectively. From the equation div. H=0 it follows that 0 TT 0 Stay e differs from zero only by yH., which is a very small quantity since both y and H, are small. uy ith very smail error we may therefore write “0 fo) Sal sy, = 0), which determines the relation between the B and C co- efficients of (11) and gives = SB, { sin BO (—1)" sin ie 1 a) n=l Dee Lo, f | ry B ae a An Ge): n=! . 2 In the dielectric the electric forces satisfy the wave equation II, and are therefore expressible as two Fourier- Bessel expansions oriented on the axes of the two wires. In accordance with our assumption, however, that y and p/v are very small quantities, the Bessel functions are replaceable in the neighbourhood of the wires by the limiting forms which they assume for vanishingly small arguments. The same result is arrived at if we take EH, and H, as satisfying the equations (07/02°+07/dy") H, = 0, (37/82? + 92/04) By = 0. Furthermore, from the relative magnitudes of EH, and the electric force in the plane XY in the dielectric, the equation div. #=0 may with very slight error be replaced by fo) 0 K+ Ow v o Plul. Mag. 8. 6. Vol. 41. No. April 1921. 28 ‘i = 0), 614 Mr. J. R. Carson on These equations are satisfied if we introduce a function V which satisfies the equation (07/00 +0%/3y*)V = 0 and then derive E, and EH, from V in accordance with Were EB, = —3.V, adapts y= —sV Now, at the surfaces of the wires the tangential electric force in the plane XY, which is continuous, is very small compared with the normal component. Consequently very small error is introduced if in determining V it is taken as constant over the circumferences of the wires in the plane XY. It follows at once that V = Voce“), 5. 4 aes where Vo ts the electrostatic potential and the surfaces of the wires are equipotential surfaces. The determination of E, and H, in the dielectric is therefore reduced to a two- dimensional electrostatic problem, in which the surfaces of the two wires are equipotential surfaces. The solution of our problem—namely, the determination of y and the coefficients Ay... A, and B,...B, of equations (8) and (12)—is obtained by formulating and satisfying the boundary conditions which ovtain at the surfaces of the wires. These are that the tangential electric and magnetic forces are continuous. ‘The current distribution in the wire, which carries with it its alternating current resistance, is, however, determinable by a less general statement of the boundary conditions ; namely, that the tangential magnetic forces and the normal magnetic induction are continuous. With the current distribution in the wire determined, the propagation factor y is determined without difficulty, as is shown subsequently. Before proceeding with the determination of the co- efficients Ay...A, of equation (8), the alternating current resistance of the wire will be formulated. let the value of p, at the surface of wire 21 be denoted by & = biVi = iaV Arduip, ale mane ROR Gael) 203 ee Wave Propagation over Parallel Wires. 615 ‘Then, omitting the subscript in 6, the axial electric force at the surface of wire #1 is AN Ia( Gl a(©) cos Go) (6) cos 26-4...) (15) and the value of the tangential magnetic force Hy at the surface of the wire is by (10): ACG +hJ,/(E) cos @ +hoJo'(E) cos 204+... ). (16) ‘Since 47 times the total current I flowing in the wire is equal to the line integral of the magnetic force H, around the circumference of the wire, it follows at once that f SUG ee ek mc, (li) Hip which determines the fundamental coefficient Ay in terms of the current in the wire. The resistance R of the wire per unit length is con- veniently defined as the mean dissipation per unit length, divided by the mean square current. The dissipation W in the wire is very conveniently and simply formulated by Poynting’s theory of the energy-flow in the electromagnetic field, which, applied to the present problem, gives a 20 = 4, |, Beds, Pane ese Gls) where E, and H, are the values at the surface of the wire, as given by (15) and (16). If these series are substituted for E,and Hg in (18) and the value of Ag is taken from (17), and if the resulting expressions are realized, it follows without difficulty that RR {141/23 |hen |? 2 ! ! UnUVn Un Vp V 2119) ! e Ug Vo — Uy Vo J’ where Witt = lin (G) == Jnr(bivi), d wigs uy +v,/ = ap bb Vi ); Ry denotes the a.c. resistance of the wire when the co- efficients h,...h, are all zero; that is, Ro is the resistance of the wire where the return wire is concentric, which is ealculable from well-known formule and tables. The functions u, and v,, it will be observed, correspond precisely with the well-known ber and bei functions, which are 258 2 616 Mr. J. R. Carson on similarly derived from the Bessel function of zero order and complex argument (i¥i. From (19) the prosimity effect correction factor C is given by 5 ! Lee tupUn —Un Un Cr 1/2 >) | dale (20) nm By aid of formulee (19) and (20) the a.c. resistance of the wire is calculable, once the coefficients 4;...h, or A,...An are determined; to this determination we now proceed. As stated and discussed above, the harmonie coefficients are determined by the continuity of the tangential magnetic force and the normal magnetic induction at the surfaces of the wires; that is, by the continuity of He and nH, at 7,=a,and of He and wH,». at r=a. From considerations. of symmetry, however, these boundary conditions need be formulated at the surface of one wire only, and their satis- faction at the surface of either wire insures their satisfaction at the surface of the other. To formulate these conditions. at the surface of wire $1, we require that the tangential and normal components of the magnetic force at the surface of this wire be expressed in terms of 7, and @, only, whereas. H, and H, of formule (12) and (13) are expressed in terms of both 71, 6; and 72, @. As a preliminary, we therefore require the expansion of H, and H,, as given by equations (12) and (13) in terms of 7, and 6, alone. This is effected by the following transformations :— Uplo — Uo Uo - a1 (7/¢) cos peleee (17,/c)? cos 20; cos sO, __ ay a: s\(s+1)(s+2 ; | ae (s) ( 3 : ) (77,/¢)? COS BiG) oo ond (21): s GG) — (a/c. sin 0y— ~—,—’ (14/c)” sin 20 sin S0y il ( ile le ; Dit (m/e) i SPE ey oes. IG ap : 3 : ( He) : tee Guess sin 30, 2... 5 azdiy (It may be remarked in passing that these transformations. may be very advantageously employed in calculating the ' capacity coefhcients of a system of parallel cylinders.) If these transformations are substituted in (12) and (13), H, and H, in the dielectric are expressed entirely in terms of », and 0, or, omitting subscripts, in terms of 7 and 6. If we now employ the relations H, = H.cos 0+ H,sin 0, H, = H, cosO—H, sin 8, Wave Propagation over Parallel Wires. 617 we get, after rearrangement and simplification, the following infinite series for the tangential and normal components of the magnetic force in the dielectric at the surface of wire #1: Ho = — B,/a—cos 6 (B,/a?— Xo) ge oi 1L (COS 20 a (a/«) -) =) — cos 30( By/a! oN aio. Sy. ) eR Ra irene (3D) Be sin OC Boat Se) : +sin 20 ( B/a® —= (afe\ 3s) if : +sin 30 ( B,/a* 194 (a/c) >) hic (20) In these expressions the >’s denote the following infinite series :— Xo = Bri Gem lanl C71 a) C1 a1, 21 = By/e—2B,/c? + 8B;/c? —4B,]c* — Xo me, 1.2.B,j/co—2.3.B,/?+3.4.B/e— oe eg 1) + 2) Zn =n! Bye— pen Lio ee 240 ok - B/e— . = a ee aS No as —_ From (10), (15), and (17) the tangential and normal components of the magnetic force at the surface of the wire are, in terms of the interval solution and the current I in the wire: = (21/a)(145! = a COS a+" fel = (21/a)(1/EJo') (J hy sin 6+ All Oe sin ONG aN aie ae (26) fie Cos 20 cay (25) where the argument of the Bessel functions Jo...d, and ae. dn 1S A esis The boundary condition of the continuity of H» at the 618 Mr. J. R. Carson on surface of the wire gives by direct equation of corresponding terms of (221 and (25) : Qiila, = —— by aE and di, 1 n Mla) he = — Bosal Gy (16 Ean 28) n = 1, 273 eee Similarly, the boundary condition of the continuity of the normal magnetic induction applied to (23) and (26) gives : , dn ya N/a) Bier he = Bras Gay (Sam (28) nm Sal 22h From (28) and (29): Day dN (30) EJ and Edn! + npedn — n-1 (I/a) EJ 0 i = @=aly ! (ale) Yn—l: . (31) It is now convenient to ae the following notation = a, = (Ebn'—npidin)/ESo , > —— J,/— Jn ( ie Jn | Gn i ght, | GHG == Ih J In terms of this notation it follows from (30) that B= —e@"¢k Lee If the B coefficients in the > functions as defined by (24) are replaced by their values as given by (27) and (33), it is easy to show that equations (31) may be written as § an 1 eee B70 ie ! Oe (=) Zp,k" — 2 =) 1PM fe kqy— ( ae kdo+ ne which may conveniently be written as 1 n Cam ( a ep ae = 4 j Pr” >, (9), 74 ¥ (34) Wave Propagation over Parallel Wires. 619. Equations (34) constitute an infinite system of equations in the infinitely many variables q,... 9, and on their solution depends the determination of the harmonic co- efficients hy... An. The solution of (34) is to be obtained by some process of successive approximation. For example, a formal solution is gotten by taking g,...gn as the limit of the sequences : 0 1 4 3 TOE GRO HOS GO OO Oe @ e e ® se ® @ 8 (0) (1) (2) (3) (s) Vn I Gn qT, ? Tn nb 3 where the successive terms of the sequences are defined by the relations : OY aa (—1)"2p,k", i I, 2, Ouicis and : Lt 1)" i 7 ee an (—1) 2p,,k an (n— : ! Pak 3, (q®). The method of solution results in a convergent sequence provided the parameter & is less than its limiting value 1/2, and for values of & likely to be encountered in practice a very rapidly convergent sequence. Another method of successive approximations which may often be advantageously employed may be termed the method of successive ignorations. ‘This consists in first ignoring all the variables except g, and determining its first approximate value —2p,k from the first equation of the system. A second and higher approximation is then gotten by retaining qg, and gq» and evaluating them from the first two equations. A third and still higher approximation results from retaining 4g), g2, and gs and solving for them from the first three equations. This process is to be continued until the convergence of the sequence is evident. This latter method of solution likewise results in a convergent sequence, and works very weil in practice unless the parameter is too close to its limiting value. While some such process of approximation is to be employed in the general case, and indeed has been success- fully applied by the writer to several similar problems, a simpler method of solution fortunately suggests itself in the special case of greatest practical importance—namely, when the wires are composed of non-magnetic metals and, in consequence, the permeability mw is equal to unity. The resulting formule have the added advantage of being 620 Mr. J. R. Carson on asymptotic in character, and consequently give the values of g,..-.g, With increasing precision in the practically important range of values. It should be remarked that the formule now to be derived constitute asymptotic solutions also when mw is greater than unity; they cannot, however, be safely applied when the permeability is large unless the frequency is very high. Restricting attention, therefore, to the case where w=1, we observe that the functions p, and a, of (32) may be written as ) : Tr, aas Jailda Pr oF eae pri aca These identities follow from the definitions of equations (32) and well-known properties of Bessel functions. We know also that when the argument & is large compared with the order n, the function p, becomes closely equal to its limiting value unity. We are therefore led to consider the auxiliary system of equations in the auxiliary variables Pi-+-P,, Which is obtained from (34) by replacing the functions p,...p, therein by their common limit unity. That is, we define the auxiliary variables p,...p, by the following system of equations :— p= (Dee esp), 2 (i =="), 2, oS Now, since the functions p,...p, approach the common limit unity as the arguinent & approaches infinity, it is evident that p,...p, are simply the limiting values assumed by the variables qg,...q, when the wire is of infinite con- ductivity. From the known surface distribution of magnetic force in this case the following solution of equations (35) at once suggests itself, and may be readily verified; the variables p,...p, are simply the Fourier coefficients of the expansion 1 1+ 2k cos @ 7 = K(1+p; cos0+p.cos20+...). (36) From (36) it is easy to show that P1 = —2ks, Py = (1) 2kre", | . Com Wave Propagation over Parallel Wires. 621 where s is the series ratio Se Oo Ok Ole Irae + ae 2 | a ; ie he Ord has Den CoN ee aes 1-1 -(2k) (2h)? ek) Having thus solved equations (35) for the variables Pi---+P,» it is easy to show from (34) and (35) that if we write 3 qn ie Pr a d,s the variables d,...d, satisfy the system of equations: ee d,, a \Wa= Oe 1 Pr y > (d). : i (39) (eas } The system of equations (39) in d,...d, admit of solution by successive approximations, as discussed in connexion with the solution of the corresponding system of equa- tions (34) in g,...¢,. For the important case of non- magnetic conductors, however, a very close approximation to the exact solution is obtained by replacing (39) by the approximations : d,, = (p,—L)p,+(—L)"np, kot). Pare (40) This gives to the same order of approximation ES aaa Oar WO) corey Meron ieee he ov eed® (41) ‘ ae i oka peas Since by (82) h,=q,/c,, this gives for non-magnetic conductors ; Ji, ie ai ( he? J : = fl ¢ u u 9) —— ) Sees 1 — 2 — ) ° ° e e 42) te a fee gv—l E 0: ( We are now in a position to formulate the proximity effect correction factor C of equation (20), which involves the harmonic coefficient h,...h,. From (42) to the same order of approximation as (40) 2 2 Uy" +v ; Sy ae — (1+ 2ngh?/s"—*), reall a ean where g denotes the function oc iis —w) , b Uo? + Vo? mith | he. i2 =p,” 622 Mr. J. R. Carson on It will be remembered, of course, that b = av 4rrpp and J,(iW ib) = Un tin. If the foregoing is substituted in (20), some easy simplifications give C= eee (Si—2gk"8.), . - . (44) where 5, = S De Es . oh ea. n=1 Sy S nae, f2rs2ns" +1, Pe A 1 LD Ugo — Up V fv, — Las \ 2 2 Y e e ° e e (47). TT Uy =P Oy = resistance of wire with concentric return, and w. is the auxiliary function I iy / Ub) tH 'y nn He Ry en eens ae tty. (48) n—1 n—l Hquation (44) is the formula for the correction factor C for non-magnetic conductors, the evaluation of which is. discussed in section III. For the purposes of numerical calculation, the following asymptotic expressions are useful. For values of the argument 6b equal to or greater than 9, le ro = aR Va. . |. ee Consequently for b=5, C2 142(¥2—1/0)(8, eae Ss). . . (50) From the asymptotic expansions of J, it is easy to show that for b=n? Wy 2 fA 2—(2n—1)/20. . . . 92 Ge Wave Propagation over Parallel Wires. 623: Tf this is substituted for w, in 8S, and 8, of equation (44), some easy simplifications give Cr Ch AO ye aneatie (52): where 1+ k’s? i Ce == 7 ee ° (53) and eas f:2 5? ‘ 1 — ks? 2 : A= 2V2-— | 1+24 a.) |: 6S) The limiting value of the correction factor is therefore C,,, and this is a function only of the parameter k=a/c. The asymptotic formula (54) can be used under the following conditions and in accordance with the following rule:—lf the series = h?"s?” converges to a required order of approxi- mation in a finite number of terms n, then the correction factor C may be calculated from (53) provided the argu- ment 6 is such that Se Ds The correction-factor formule need not be further con- sidered here, as they are fully discussed in section III. We shall therefore now proceed to complete the solution of the problem by formulating the propagation factor y. In this discussion it will be assumed that h,...h, and g,...qn have been evaluated in accordance with the methods fully dis- cussed above. For non-magnetic wires they may be calcu- lated from (41) and (42), while in other cases any of the methods of successive approximations discussed above may be applied to equations (34). The propagation factor y is determined by applying the law curl E=—ypop H to any appropriate surface, the contour of which includes a line segment dz in the surface of each wire and two lines in the dielectric joining their corre- sponding ends. The most convenient surface to take is a plane surface in the plane of the axes bounded by the bs Wire # 2 Wire #4 — eee” - OO elements dz in the inner or adjacent surfaces of the wires and the straight lines in the dielectric joining the corre- sponding ends of the elements dz, as shown in the cross- section sketch herewith. 624 Mr. J. R. Carson on To apply the law curl H=—ypipH to this surface, it is only necessary to calculate the magnetic flux through the surface and the line integral of the electric force around the contour. The contribution to the line integral from the elements dz is, bv (15) and (17), _ 2eips oy } a dz an Che hy inlay Bee e' | which may be written as Di Lda — 2ZyLda( 1M, + eae ae -), -) ast hp Jo Jo where Zy = 2uipdo/EJo’ and the argument of the Bessel functions is =iaV4rApip. Z, is the “internal” or “* self-impedance” of the wire when the return wire is either concentric or at such a distance as to make the proximity effect negligible. To calculate the contribution to the contour integral of the lines in the dielectric joining the corresponding ends of the segments dz, it will be recalled that the electric force in the dielectric in the plane XY is derivable as the gradient of a scalar or electrostatic potential, as given in equation (14). Consequently the contributions of these lines are simply (v CO, —Vo) dz = —yVodz Oe 0 0 Yv 0G-, where Vo is the electrostatic potential between the two wires. If Ke denote the electrostatic capacity between the twe wires, then AEN Ny: a ) = ip if . ~ . 5 ° (56) and a yy? , —vy\ oz = Skee and the total line integral of electric force around the contour is oN ys ( 20 Jide. ol ee The calculation of the electrostatic capacity K involves merely the solution of the two-dimensional potential problem in which the surfaces of the wires are equipotential. Wave Propagation over Parallel Wires. 625 We have now to calculate the magnetic flux through the surface ; it is © = a, "Hdr, which by reference to equation (13) becomes —2de | Pee Be Bie: pom -B,(+- =) Ne Clana: a Cc—& From (27) and (33) this reduces to Do = ade J 2 log (=) — Fag Ene | Ne OF ey! is e—a J +4m[1-(22) |+--.) - Ds eye The law curl H= — pipH now gives at once Vel ogi | i ke SRE LN UG sal} 4a) where Z = Lo (L—hyJq/Jo + hod o/Jo—. ° sg) ° ° ° ° ° (60) ke Bi NZ ate, its l—k 1—k iG = Lo( 1—2¢; aay Ee Le ; + ee ), ° and Ly pte ey ay ie if. (62) Te tlog(*F"), Lc See) Z may therefore be regarded as the impedance of the wire, and L the inductance corresponding to the magnetic flux between the wires; Z and Ly are their limiting values when the parameter & is vanishingly small—that is, when the proximity effect is negligibly small. While it is convenient from this standpoint to regard L as the inductance per unit length of the circuit, it must be carefully borne in mind that both Z and L are complex. Consequently the- 626 Mr. J. R. Carson on true resistance R* and reactance X of the circuit are defined by the relation . R+iX =2Z4+iph, ... 2 eee Having calculated g,...g, and h,...h, in accordance with meods discussed shows: it is a straightforward process to calculate Z, L, R, and X ‘from (60),... (64), the only operations involved being the evaluation of the Bessel functions appearing in the formulas. For very low fre- quencies, Z and L approach the limit Zp and Ly respectively. On the other hand, when the frequency is sufficiently high they approach upper limits corresponding to qn a Pn rat (— Dye 2k" 5" InJn/Jg ~ 2k"s” when n is even. ‘Consequently, 1+ hs? (14 heJa/Jy+IudufJo-+--.)~ pgs = Jms ‘where C,, is the upper limit of the correction factor C. -It may also be shown that 2Z + ipL~2C mZ,y + 4tp log (;. ), ‘The limiting values of Z and L correspond to the surface -distribution of currents which would exist if the wires were of infinite conductivity. The calculation of Zand L from the foregoing Me ios -and tables of Bessel functions is not a difficult matter. The writer, however, hopes when time permits to prepare nume- rical tables and the theoretical data for the computation -of Z and L, similar to those given in section III for the -eorrection factor C. The latter function is, however, of smuch the greater engineering importance. Til. FormtLa ror CorRection Factor C ror Non-MaGnetic Conpuctors. Last of Symbols. = radius of wire in cm. = interaxial separation between wires in em. = ratio a/e. = conductivity of wire in elm. c.4.s. units. ih permeability a o> 2a times frequency in eycles per second. 5= v—l. b = a WV Amdpp. -* The circuit resistance is, of course, twice that of the wire. ene Wave Propagation over Parallel Wires. 627 Jn (62 V2) = Un + Une = = Bessel function of order n and argument bivi. R = resistance of wire per unit length. ip ms es nm 5, with concentric return. C = Proximity Effect Correction Factor ECR en eo CT) The auxiliary functions involved are : 1 “D Ud.! — Ud! v Resse eee petals ORLOV ee SEAN a manic Tru we+v,? ”’ (1) 7 ae Olt) ae) Ug? + Vo" ek Ss The formula for the correction factor C is then aa =144 nV 2 Si—2gh5,), =. (IL) where i > TE oe BNE pole Nba hay at CO.) Se = $ nw,,k?s?*}, ENE Ti a 2, DN at Co Ni For values of the argument b=5, -1/6)(8; Ee MS), oe LE) For larger values of the argument 0b, the correction factor C approaches an upper limit okie hs? Tw 1 k?s? in accordance with the asymptotic formula COMMON. oe ak 1—k’s nays Eh fee (EN) Cr (1V.) where Praextmum vetve of Conrection LOGaay soon Om aT ROG mOS 1.0 Valves of 2k = 2a Fig. 2. _ eyste ee 1.20} 04 0.6 Yarues of 2k Wave Propagation over Parallel Wires. 629: The correction factor © may be calculated from the asymptotic formula V instead of II or III under the fol- lowing condition and in accordance with the following rule:—If the series 1+7s?+h's*+... converges to a required order of approximation in a finite number of terms n, then formula V may be employed, provided that been =). The auxiliary functions involved in the foregoing formule have been computed and are plotted in the accompanying curves, the accuracy of which is believed to be sufficient for Fic. 3. —- rues Els pos Ve/ues of b= av4naAup all engineering purposes. An example of their use will now be given in calculating the correction factor C for the following representative case: b=5 and 2k=2a/e=0°75. From the curves of fig. 3, On es) (OGLE es 0-410 i = 05239 ws = Oar. Phil. Mag. S. 6. Vol. 41. No. 244. April 1921. PFA 630 Mr. J. R. Carson on Fig. 4. sare) ! 2 3 4 5 6 Fie. 5 >: Values of 5=avAnAyp Wave Propagation over Parallel Wires. 631 From the curves * of fig. 4, ] Dp FF Sh PDN ian aC ene we = 1°225. Oe ee ane alo TH i From the curve of fig. 2, Se= IL), Consequently, Wiese = 1229 VO eee) as LAA) wo Ss) OLOSI Zk sa =) O2S0k Hees == - OOLIA2 3w3k®st = 004065 ee OUO2Kem Aw, k's? = -0009058 Sy) eS) SEED So oo SL Va/ues of Fk cient for practical purposes Higher orders can be calculated from the Bessel function recurrence formule, yd Wie. 632 Wave Propagation over Parallel Wires. Substitution in formula IT gives CG = 142:45 (1415 —-01223) == Ikcally From the curve of fig. 1 the maximum value C,, of the correction factor for this case is 1°51, and from the curve of fig. 6 the value of the auxiliary function A is 0:76. The asymptotic formula V therefore gives : C ~ 151 (1—0°76/5) = 1-28. As would be expected from the magnitudes involved, the asyinptotic formula therefore gives a result which is ia error by a small amount. : 110 1.04 3 i. 3 6 q 8 To give an idea of the variation of the correction factor C with frequency, it has been computed for £=0°25 and plotted in the curve of fig. 7 as a function of the argument b. This case is of practical interest, since the corresponding conductor spacing is that of telephone cable circuits. For this ratio of a/c the value of ©,,=1-155 ; this limiting value Separation of Miscible Liquids by Distillation. 633 is however approached quite slowly, as is evident from an inspection of fig. 7 To facilitate ine calculation of Ry, the ratio Ro/Ra. has been plotted in fig. 5. Rg denotes, of course, the d.c. resistance of the wire. For values of the argument b=>5, Ro/Rae b/(2,/2—2)b)> which makes the calculation very simple. April 17th, 1920. LV. The Separation of Miscible bias by Distillation. By A. F. Durrox, B.A., Frecheville Research Fellow, Royal School of Mines * 1. PN the search for a perfect apparatus for the separation of mixtures by distillation the greatest advances have been made in industrial practice. M. Sorel, one of the leading French authorities upon the distillation and the reeti- fication of alcohol, in reviewing the principles underlying the construction of stills, writes :— “La plus grande partie des données dont nouns avons beso peuvent ¢tre determinées dans la laboratoire du physicien. Malheureusement bien peu de savants s’en sont occupes, soit que le sujet leur partt peu important, s soit, que Vimpossibilité jusqu’ici reconnue d’arriver a des lois mathé- matiques les ait rebutés. I] faut done que le constructeur se transforme en expérimentateur. . .” “C’est ’aveu france et net,” to quote M. Chenard +, “ d’un empirisme certain.” The extent to which laboratory practice has been out- stripped may be seen by coupons the still designed by Coffey ft, of Dublin, in 1832, or Derosne’s still, came give in continuous distillation on the large scale the strongest spirit whieh can be obtained, with the various laboratory still-heads examined by Dr. “Young § in 1899. The only continuous laboratory still appears to be an experimental one devised by Lord Rayleigh ||, which consisted of a long mee (12 metres) of copper tubing 15 mm. in diameter, and similar one described by Carv eth J. * Communicated by Sir E. Rutherford, F.R.S. t+ Chenard, Bulletin de Association des Chimistes de Sucrerie et de Distillerie de France, 1915. { ‘Chemistry as applied to the Arts and Manufactures.’ Vol. I. Alcohol, § Young, Journ. Chem. Soc. 1892, p. 679. i| Rayleigh, Phil. Mag. (4) 1902, p. 536. “| Carveth, J. Phys. Chem. vi. p. 253 (1902). 634 Mr. A. F. Dufton on the Separation The invention in 1918 by Dr. 8S. F. Dufton * of a labora- tory still-head of small working volume and not inconvenient length, giving perfect separation of simple binary mixtures, e.y. benzene and toluene, renders possible the examination in the laboratory of the rationale of separation by distillation. This examination is the object of the present research. The problems presented by close isomers and by the recently discovered isotopes render desirable every possible refinement in methods of sepa ration, anda knowledge of the physical processess taking place in a_ still-head should render possible both the application of distillation to more difficult separations and the elimination of the empirical element in design. To minimise the empirical element in the design of still- heads by determining more clearly the phy sical processes involved, calculation thas been made in the following pages of the requisite flow of liquid and of vapour at each point in a theoretically perfeet column, and indication has been given of the corresponding supply oer withdrawal of heat involved. A quantity, termed the TsHERMaL Erricrency, has been defined to afford a comparison of the performance of columns yielding a pure distillate and a measure of the approach towards theoretical perfection. 2. F. D. Brown f has pointed out that when a mixture of two liquids which mingle in all proportions is heated to ebullition, it evolves a mixed vapour containing a certain proportion of each of the two substances, the liquid mixture and the gaseous mixture being mutually related and existing together in a state of equilibrium. It is upon this equilibrium between liquid mixture and gaseous mixture of different composition that separation by distillation depends. Ideal discontinuous distillation consists in the evaporation of liquid, in the removal of the vapour produced and in the return to the still, in liquid of the same composition as that in the still, of the whole of the less rolatile constituent. The evaporation of 2°32 grams from a large mass of a mixture of equal masses of benzene and toluene, for example, yields as vapour 0°66 gram of toluene and 1°66 gram of benzene, and the return of the 0°66 gram of toluene with 6°66 gram of benzene as liquid involves the separation of 1°00 gram of benzene. The amount of evaporation necessary and the corresponding quantity of heat required may be calculated mathematically. If in a system of two substances A and B, which mingle in * S$. F. Dufton, J. Soc. Chem. Industry, 1919, p. 45. + Brown, Journ. Chem. Soc. 188], p. 531. of Miscible Liquids by Distillation. 635 all proportions, V, and V, be the masses in the vapour at any moment and L, and L, the masses in the liquid in contact with and in equilibrium with the vapour, and if Ef be a mass evaporated and C a mass condensed, the criteria for equilibrium are OL, 3B _Y, Oly ms V;, ron) 0 Lia (OC au uP and Ale mare a0 If the mass L, be conserved, as is the case in discon- tinuous distillation, ee ap et ae SC a0. =i()) These three equations show that | 7) O Lia = on pat a+ Sede} =-{ te oe ab+ Oe. ac Pee eM ACS 2 lie where Me ke one the proportion of liquid which is A, Ne ¢ L, and where — Ta. Equation (1) signifies that the separation of unit mass of A if involves the evaporation of a mass 4 1+ acs of the liquid. If Q be its latent heat of evaporation, the quantity of heat required is i a Q4 1+ aa} . ° . . . (2) 636 Mr. A. F. Dufton on the Separation The THERMAL EFFICIENCY of a still-head for such a binary mixture may be defined as the ratio of the mass actually separated by this amount of heat to unit mass, or, equiva- lently, as the ratio of the theoretical quantity of heat required for a separation to the amount actually expended. It is expressed conveniently as a percentage. The value of :—that is, the relative composition of vapour and liquid phases in equilibrium-—must be determined experi- mentally. Attempts have been made to express & in terms of the physical constants of the components, and some success has been attained ; thus, for alcohol and water, a mixture of considerable industrial interest, Duclaux* has established the relation A a He gae where p is a constant, and where a represents alcohol per cent. by vol. in the liquid, é 99 water 99 99 99 A er alcohol ‘ e condensed vapour, E a water 8 a condensed vapour. For substances which are chemically closely related to each other a simpler relation exists, viz. : k= hs where P, and P, are the vapour-pressures of the pure com- ponents at the temperature considered +. Fig. 1 shows for mixtures of benzene and toluene the composition of liquid { and vapour phases in equilibrium at atmospheric pressure, the composition of the vapour being calculated from that of the liquid by means of this relation. As in the design of an efficient column a knowledge of the flow of vapour and of liquid is desirable, the minimum flow in an ideal column mustbe calculated. Ifin the column * Duclaux, Ann. de Phys. et Chim. 1878. t Brown, Journ. Chem. Soc. 1881, p. 304. Zawidski, Zeit. f. Phys. Chem. xxv. p. 129 (1900); ef. also Young, ‘Stoichiometry,’ p. 256 (1918). Ostwald, Lehrbuch d. Allgem. Chem, (2 Aufi.), iii. p. 618. {Spielman & Wheeler, Tables of Chem. and Phys. Constants, H.M.S.O. 1919. of Miscible Liquids by Distillation. 637 the net flow be oe of substance A, the net flow of substance B being zero, the flow past any point must satisfy the equations dVatdba=dA and dV,+dlL,=dB=0 ; Biewk Mixture of Benzene and Toluene. The Composition of Phases in Equilibrium at Atmospheric Pressure for different Temperatures. Ce Lake) 100 \ VAPOUR 90 LIQUID we stl PS RY Ay ue WIRY fe 80 50 100 BENZENE 100 50 r@) TOLUENE "65 PER CENT moreover, since the flow of each component in the liquid or vapour 1s proportional to the amount present in that phase, lias 8 Tee dl; ma La : AV sf Va and dV» a V, 638 Mr. A. F. Dufton on the Separation From these four equations it follows that dA dV pane —]| a Ripa allan i j Va ly dV b rer V, oh eis ad Lu, ee i, of : a V>. by =1—-A, if there be equilibrium between liquid and vapour: di liaiie 1 Sra eG a(k —1) 3 . ° ° e > . (3) dl . where —— ae is the total flow of liquid at the point dV 1 and ak = 1+ Zeal 5 - . ° : 3 (4) Ty, / where - is the total flow of vapour. The upper portion of Table I. shows the necessary flow for the separation of one gram of benzene from a mass of a 50-per-cent. mixture of benzene and toluene. Auresina IE CA Grams Vapour Grams Liquid alaaip. per cent, passing up flowing down 2 inliqe C,H,+C,H,=total. C.H,+C,H,=total. 802 © 100 1-64+0:00=1-64 0:64-+0:00=0-64 81:8 90 1:65+.0-67=1:72 0-65 -++0:07 =0°72 83°5 80 1°65+0:16=1°81 0°65+0°16=0°81 85-4 70 1:66 +0:28=1-94 0°66 +0:28—0'94 87-6 60 1°66 +0°44=2°10 0-66 -+0:44—1-10 90-0 50 1-66 +0°66=2°32 0 66+0-66=1°32 930 40 1:-67+1:01=2°68 0°67 +1:01=1°68 96-4 30 1°68 +-1:57=3-25 0°684 1:57 =2°25 100°4 20 1:70+2:80=4°50 0°70 +2'80=3:50 105-0 10 1:72 +6-43= 0:72+643=7:15 110°6 0 175+ © = © O75+ o0 = w of Miscrble Liquids by Distillation. 639 It will be seen that condensation of toluene takes place throughout the length of the column. The lower portion of the table shows that an increased amount of evaporation and condensation must take place as the mixture becomes poorer in benzene. It is of interest to observe that in the removal of a trace of toluene from “pure”? benzene by distillation 1°64 gram must be evaporated for each gram - collected. The heat which must be lost in the column during the distillation of one gram of benzene from a mixture may be calculated from Table I. At the top of the column 0°64 gram of benzene must be condensed at its boiling-point. This in- volves a loss of 61 calories. It is convenient to divide the remainder of the column into sections, each corresponding to a temperature difference of two degrees, and to evaluate the loss in each. for these sections the loss of heat is shown in Table IT. DPasiE Lf, Sean Grams of Vapour Calories lost an condensed. (approx.). SOSEs2 0:10 9 82— 384 0-12 10 S+t— 86 O14 12 "86-88 0:16 14 88-— 90 0-19 wi 90- 92 0:23 20 92— 94 0:29 25 94-- 96 0:37 32 96- 98 O51 44 98-100 O78 86 100-102 1-18 101 102-104 1:60 156 108-1106 oo we It is important to observe that for perfect thermal efficiency the liquid and vapour need not be in equilibrium at any point other than the top and bottom of the column. The particular case in which the whole of the necessary loss of heat is effected at the top of the column is of interest. It gives the maximum flow for perfect efficiency. This flow for the distillation of one gram of benzene from a 50-per- cent. mixture of benzene and toluene is shown in Table III. The loss of heat is 117°5 calories, 7. e. the 208 supplied for 640 Mr. A. F. Dufton on the Separation necessary evaporation less 93°5, the heat of evaporation at 90° C. of the benzene separated, plus 3 calories, the heat given up by this benzene in cooling to 80°2 C. The loss of this 117°5 calories means the return down the column as liquid of 1:24 gram of benzene. VA Bice ale (Crile Grams Vapour Temp. Temp. Grams Liquid per cent. passing up of of flowing down in liq. C,H,+C,H,=total. Vapour. Liquid. CO,H,+C,H,=total. ° ° 100 2°24+-0:00= 2°24 80:2 80°2 1-24+0:00= 1-24 74:3 1:95+0°33=2:28 86°4 84:5 0°95 +.0°33= 1°28 50 1:66+0°66=2°32 90°0 90:0 0:66 +0°66=1°32 It will be seen that in the middle of the column the vapour is considerably hotter than the liquid. With benzene and toluene, owing to the approximate equality of the latent heats, this method of working gives a practically uniform How throughout the column. In the determination of the thermal efficiency of a column used to separate benzene from a mixture of benzene and toluene, calculation must be made of the quantity of heat which would be required if a perfect column were employed. The quantities necessary for different mixtures are given In Table IV. and plotted in fig. 2. TaB.e LV. Liquid Benzene in evaporated : Latent heat Calories required mixture grams calculated from to separate 100 a pex cent. j ] } components. 1 gm. of benzene. Walt ieee Wh U az -—-\) 0 00 84:0 oC 5 15°6 84:5 1320 10 8:15 8a°1 698 20 4°5 86°3 388 30 3°29 874 288 40 2°68 88°6 237 50 2°32 89:7 208 60 2°10 90°9 191 70 1:94 92°0 (179 80 181 93°2 169 90 1-72 94:3 162 100 1-64 93°5 156 of Miscible Liquids by Lrstillation. 64] 3. In the experimental investigation of the thermal efti- ciency which can be obtained in the laboratory, a mixture of 40 c.c. of “crystallized” benzene with 10 c.c. of reputed pure toluene was boiled in a 100-c.c. vacuum flask as still. For iors: The Heat required to distil Benzene from mixtures of Benzene and Toluene. CALORIES 1500 BENZENE SEPARATED HEAT PER GRAM OF | Gem ene SOME is VENA SH $00 ad BENZENE 1090 50 0 TOLUENE MASS PER CENT the first experiments a Dufton column* was constructed in a glass tube, 105 cin. in length and 0:46-cm. bore. Heat was supplied electrically by means of a platinoid spiral immersed in the liquid and the rate of heating was determined by measure- ment of the current and the potential difference, calibration being effected by heating a known mass of water in the flask. The benzene condensed was collected in a burette and observations of its volume were made at five-minute intervals. A thermometer placed in the top of the column showed the temperature of the vapour and indicated the * S. F. Dufton, Joe. cit. Wn 80-9 642 Mr. A. F. Dufton on the Separation purity of the distillate, a rise of one degree above the boiling- point of benzene corr responding to 2°78 per cent. of toluene, The thermometer was calibrated in position in the apparatus by the distillation of pure benzene, the boiling-point, cor- rected for atmospheric-pressure variation, being taken as 805-210: After some preliminary experiments, a series of four was made in which the only variation was in the rate of heating. In the first, with a heat supply of 170 calories per minute, pure benzene was separated, and when 39:5 e.c. out of the 40:0 ¢.e. of benzene had been collected, the thermometer fell and distillation ceased. At 178 calories per minute, in the next experiment, a trace (0°5 per cent.) of toluene passed over during part of the run. Distillation ceased when 39°3 cc. had been collected. With a heat supply of 193 calories per minute there was a little more toluene in the distillate, and at 206 calories per minute still more. Fig. 3. Effect of Excessive Heat Supply on Purity of Distillate. ZO6 carories /min 193 CALORIES / MIN '70 CALORIES / MIN 5O i1ad0 In these two experiments, after 40 c.c. had been distilled the temperature rose suddenly and toluene began to pass over, ‘The temperature-time curves for the four experi- ments are plotted in fig. 3. The experiments show that for a given column there is an upper limit to the rate of neating, above which the distillate is uot pure. 178 cavories / MIN MINUTES of Miscible Liquids by Distillation. 643 For the first of the experiments just described Table V-. shows the volume of benzene collected, the percentage of benzene in the mixture remaining in the apparatus, the volume of distillate collected per ininute and the thermal efficiency. The slow rise of temperature and the gradual increase in the rate of distillation and in the thermal efficiency during the first 50 minutes are attributed to the expenditure of heat in warming the column and the thermo- meter. When a steady state was attained, the thermal efficiency was sensibly constant at 44 per cent., falling slightly as the mixture became poor in benzene. TABLE V. ae .c. Mixture be Thermal minute, lected. — in, Mhengene, = °C. pet cont 0 0-0 aS &0 68°8 ee 10 16 O:23m 79 796 25 =) 52 0°39 78 79:9 34 30 OES) 0-45 76 801 38 40 13°8 0°46 72 80°2 42 50 185 0°45 68 80°2 42 60 23-0 0-45 G3 yw) he 80:2 44 70 20-3 0°43 56 80°2 44 80 31°4 0°35 46 80:2 43 90 34:8 0:29 34 80:2 39 100 37°3 0°195 2) 80:2 38 Ai) 38°6 0-085 nae 80°1 120 39°1 0-035 ua 80°0 130 39°35 0-02 Sie 19:2 140 39°5 0-01 aes 17:3 The composition of the mixture given in column 4 is obtained from the original composition by deducting the amount of benzene collected. When the composition falls to 20 per cent. of benzene only 2°5 c.c. of benzene remain, and the calculated percentage ceases to be trustworthy. Experiments made with slower rates of heating showed that the greatest thermal efficiency was obtained w hen distil- lation was at the maximum rate to yield pure benzene. At 170 calories per minute the thermal efficiency was 44 per eent., with a heat supply of 146 calories per minute it was 30 per cent., and at 128 calories per minute it was only 5) per cent. 644 Mr. A. F. Dufton on the Separation In the experiments described above, the column was lagged with cotton-wool to reduce the loss of heat, so that the liquid condensed was not sufficient to ‘ flood”? the column and to obstruct the upward flow of vapour. Hxperiments were made to determine the effect of further reduction in the amount of heat lost. The amount of lagging was increased until the maximum rate of heating to yield pure benzene was 151 calories per minute. In an experiment with a heat supply of L58 calories per minute the distillate contained a trace of toluene. For 151 calories per minute the thermal efficiency was 38 per cent. With a heat supply of 143 calories per minute it was 25 per cent. These experiments show that the thermal efficiency is reduced if loss of heat in the column be unduly prevented. A shorter column, 25 em. in length, was constructed with a bore of 0°3 em. at the top and 0:7 cm. at the bottom, the wire for the spiral being the same throughout. With this column it was necessary to reduce the rate of heating as the proportion of benzene in the mixture diminished. Table VI. shows the results obtained in an experiment with this column. The thermal efficiency for a mixture containing 75 per cent. of benzene was over 70 per cent., and for mixtures con- taining more than 50 per cent. of benzene the column was more efficient than the 105 cm. column. (ean Welle Nees Ge Mixture aa Calories Thermal miinates, Colcol Oe Oa 0 0:0 a 80 a 105°5 10 46 0:54 78 ASE 105°5 78 20 a9 (44 75 80°2 93°5 72 30 137 O°375 (2 80:2 88°5 66 40 17°28 0°335 70 801 84°5 62 50 20°5 0°32 66 80°2 84°5 61 60 23°6 9°29 62 80°25 84°5 dT 70 26°25 0-235 58 80°2 775 52 80 28-4 0-195 D4 80-2 75 47 90 30°2 0:185 50 80°2 75 46 100 318 0°145 45 80°2 69 4] 110 33°05 0:108 4] 80°2 67°5 32 120 039 0:06 38 80°3 63 20 of Mrscible Liquids by Distillation. 645 For mixtures containing less than 40 per cent. of benzene this column did not yield a pure distillate. For such mix- tures a longer column must be employed. 4, The exposition which has been given of the primerales underlying the art of discontinuous distillation and the con- firmatory experiments which have been described are an essential preliminary to the investigation of continuous distillation. In the continuous distillation of a binary mixture both components are constantly removed and the liquid does not become increasingly difficult of separation by reason of the accumulation of one component. ‘Table I. shows the mini- mum flow of vapour and of liquid during ideal discontinuous distillation. Table VII. shows in similar manner the mini- mum flow in an ideal column during the continuous distilla- tion of a mixture of equal masses of benzene and toluene. The upper portion of the column behaves as in discontinuous distillation, condensation taking place throughout its length, hut in the lower portion evaporation of benzene occurs and the requisite quantity of heat to produce this evaporation must be supplied. This heat may be supplied throughout the length of the lower portion, as in the still devised by Lord Rayleigh, or, 1f minimum flow be not essential, by sup- plying a quantity of vapour of the less volatile constituent at the bottom of the column, as in the Coffey still. TasxLe VII. C,H, Grams Vapour Grams Liquid Temp. per cent. passing up flowing down 26; in liq. C,H,+C,H,=total. C,H,+C,H,=total. Top of column. 80°2 103 1:64+0:00=1°64 0'64+0 00=0°64 81:8 90 1:654+0:07=1°72 © 0:65 +0:07 =0°72 83°5 80 1-65+0'16=1°81 0:65+0:16=0'81 SEE UNO 1-66 +0-28=1-94 0°66-+0-28=0-94 87°6 60 1:66+0:44=2°10 0°66+044=1:10 90:0 50 1:66+0°66=2'32 0:°66+0°66=1°32 2 grams liquid supplied. 166+ 1:66=3°32 93°0 40 1:12+0°68=1-80 1:12+1:68=2°80 96°4 30 0:73+0°70=1:48 O-738+1:70=2-43 100°4 20 0:-43-+0-°71=1-14 0-438+1-71=2:14 105:0 10 0:19+0°72=0°91 0-19+1-72=1°91 110-6 0 0:00+0 75=0°75 0:00+ 1-75=1-75 Phil. Mag. 8. 6. Vol. 41. No. 244. April 1921. 2U 646 - (apie pal in which the first member represents the scalar product; according to Grassman’s notation, the equation n GH) == 0 | expresses that the complex quantities a, b are perpendicular to one another. Let us consider a sequence, a, 6, ¢,d...n of n complex quantities, of m' coordinates satisfying the following equations : vig==(), hi a—(). Gli) | Gi.) oan Ve a brea Qe diag—O, di 0s) diGa=s "| ‘from which we obtain by derivation ia Dian | Jla=~—c/a’, c/b=—e/l’, t Gi) d'/a=--d/a’, d'/b=—d/v', d'fc=—d/c’, | / i e e e e e e . . Catena} tthe derivation being with respect to a variable ¢, of which the said quantities are assumed to be functions. The geometrical meaning of the equations (1.) and (ii.) is an extension of that which they obviously have when the * Communicated by the Author, 2U 2 648 Mr. F. Tavani on complex quantities are complex of second order or vectors— that is to say, the equations of group (i.) say that each complex is perpendicular to all those which precede it in the given sequence; while the first equation of group (il.) expresses the projection of 6’ upon 4a, the second equations. express the projections of ¢’ upon a and b, the third equations give the projections of d’ upon a, 6, ¢, and so on. The quantities a, b, c, d... may be assumed further to verify the Cannons: ae B= ee ee and a : C sie) mod.a'? mod. ¢”? | a! me ae | 7 moda’? “" mod.b'? mod.’ po) ab a b' -¢! | ~ mod.a’? ~~ mod.b”? ° mod.e”’ d= aan e! ; which are obviously equivalent to those of the ee group : G—imodeaen ac —emadaee a@—emod.a. bi=\cmod.b/. dmemoi.d,t- (111.3) Let P be a point moving in a space of n dimensions as a function of a variable ¢ (time), so that mod. P’=v (speed of P) and ? a= —, an equation analogous to those of the group (iii.). VU Let also m7, 7)73...7, indicate points, such that the complex (7,—0) has the direction of P’ and passes through the fixed point o. In the case of complex of second order, and P moving in ordinary space, 7, describes, while Pp. moves, what is called the spherical indicatrix of the tangents to the path of P. We assume the point o to be such that 7,=o-+a, and in a similar way T2—-O-+0, %3—O0+0, %,—O0-+-d.) so pee In the case of complex of second order, vectors, and P’ * The meaning of these equations is obvious: they express that the. complexes are of modulus =1, and each perpendicular to its derivate. Motion and EHyperdimensions. 649 moving in ordinary space, 73 describes, while P moves, the second spherical indicatrix. From (iv.) we obtain by derivation mod.7,'=mod.a’, mod.7./=mod.b', mod.73/=mod.c’... (iv.;) Let us now introduce the notion of curvature p; through the relation ds ads, ; Pure in which ds is the portion of the path described by P, and do, the portion of the path described by 2,, while P describes ds. Then we have 7 as oe a) v ea) Peace = ~ mod. mod.a’’ a) Therefore mod.a'= —, and with a similar reasoning we obtain Pi mod.b'=", mod.c!= ey mod.d'=—..., (v.) P2 P3 P4 where - py, Ps, Pq-- have a meaning similar to that of p, established in a similar way by extending the same reasoning to Oi Wee EE It is easy now to find the expressions of the projections Bae eed upon the axis) a: ¢, ab) a, a, 0, c, €, taken as systems of reference ; by replacing 1D the second members of the equations (ii.), the values of a’, 0’, ¢', d’.. hs given by (iii.,), and making use of the equations (9 (v.), we obtain : Ce eye ind yal | (1)4 Pi |b '[e= — be’ = — }/b mod. ¢ = a L We ui perk Weis v | c'/a=—e/a' = —c/e mod. a’ = — 7 (vi) '/b= — ¢/b! = —e/e mod. b' = — — (2) c/ c/b c/¢ mo z e'|d= —c/d' = —c/e mod. d'= Las i) 650 Mr. F. Tavani on Therefore the expression of 0)’ through a,c, taken as a system of reference, is ; ye ge ei - P1 P3 | ; d vil.) co eagle : Pi See J and in general m=p+l u a coo ¢ ee > G pace v,,) aed) Vi 5. Gye) P m=1 Pin Po valid from p=2 upwards, where V denotes the quantity of the general type defined HOIP UP OO Cheese Let us make an important application of the relation (viil.). I From «a= = we obtain P’=va, then by derivating P'=va twice with respect to ¢t, and replacing in the second member to a’ its value, v a'=bmod.a'=—), P1 and to b' its value given by vii. (1), we arrive at 3 Oy aaretl 2 3 p= (var C2 — epee «Gy 1 1 it 1p3 We can repeat the derivation of (ix.), and replace to a’,b',c' their values given by (viii.), so that in the second member appear only terms consisting of the quantities a, b,c, d, with toeflicients which are elementary functions of v and the curvatures 1 p2p3,.... If we indicate by $(c. p) such coefficients, we can write in general OL ale ze fe —- = > b,(0,p)Na . 2 en (dt)” y= where: Vio, Ne—bVe—c...!. - * This equation is given by Prof. Peano in his ‘Analest uyfinitesimale, vol. ii. § 325, by a method which in this paper I have followed and generalized in order to obtain the relation (viii.). T This relation is also given by Prof. Peano, loc. cit. Motion and Hyperdimensions. 651 Thus we have established the relations (viil.) and (x.) which are the generalized forms of vii. (1) and (ix.), the latter being of particular use for the case of three dimensions, viz. of the physical space. We are going to make an appli- cation of it tor the study of the motion of a system of reference in a space of three dimensions, a similar study being susceptible of extension to spaces of any dimension. Let us consider the equation d?P ee DCO nor v? a=” alee PTT Te Jo- at p ye. Pips If we assume a,b,c to be three vectors normal to one another so as to forma system of reference with the origin P, then the analytical expression of P as function of time, (¢), represents the law of motion of the system, and ¢’(t) and @'/(t) the velocity and the acceleration of the motion. Extending the meaning of ¢', @', we may consider ¢$'”, g'''’..... as the hyperaccelerations of the system of order three, four ..., the reality of these quantities depending on the law of the motion itself. We can now express the meaning of the above equation through the following proposition :— “If a system of three vectors normal to one another, of origin P, is in motion with a given law P=¢(t), so that the axis of X coincides with the tangent to the path of P, the hyperaccelerations of the system represented by to a system of axes perpendicular to one another, provided that the dimensions of the orthogonal system of reference are taken in number equal to the number of the order of the acceleration.” The meaning of this proposition can also be expressed by saying that “the virtual displacements due to the hyperacceleration of an orthogonal system, moving with its axis of X in the direction of the tangent to the path described by the origin, are analytically expressed through a system of reference, of a number of dimensions equal to the order of hyperacceleration.” [ 652 ] LVI. Force- Transformation, Proper Time, and Fresnel’s Coefficient. By Prof. FREDERICK SLATE *. OR electronic conditions, Newtonian dynamics and relativity based on a Lorentz transformation are reducible to parallelism as mathematical schemes. The former introduces variable effective inertia where the latter treats inertia essentially as constant t. Moreover, relativity’s method here can be ‘assigned to widely inclusive grounds. Any attempt at a detailed physics by resolving further the data of energetics must countenance some flexible factoring of energy and of energy-flux in its tentative dynamics. Lagrange’s equations are known to admit such alternatives ; and no cogent reason exists for bounding the range of that proper freedom by their algebraic type t. As an implication of the present analysis, we achieve a broader outlook, the gain of whose perspective is worth seeking. | ~ Let an energy-transfer (W) of calculable amount be associated with a working speed (v) at the close of an interval (0, 7), the frame being one among a “legitimate group.” An interval like (wv, v) is covered by a difference. Then it is mathematically permissible to express (W) variously as a doubled kinetic energy (2H), and to prepare thus for corresponding mechanical analogues. Accordingly, write a series of equivalents; not exhaustive, but meant to exemplify useful forms : W =2E=(m,)(2v)?=(m2) (202) = (m2?) (v) =W(O=(#S)(@.- 2 C The first parentheses in each factoring separate an assumed inertia-coeflicient from a squared velocity ; auxiliary velo- cities (v\/z, vz, c, etc.) are then one inherent feature of any such series. The last two wacwiher employ a reduction to standard (or terminal} velocity §; (m,) is a constant; * Communicated by the Author. T Slate, Phil. Mag. vol. xxxix. p. 433; vol. xl. p. 31; vol. xli. p. 96. These papers are cited as (I.), (II.), (III... t Generelized velocity and Bee eat es are defined, and have been used practically, to realize this possibility. Incidentally, Abraham’s early success in extending Lagrange’s equations to the electron may find partial explanation here (Theorie der Elektrizitét, vel. ii. p. 177 {1908)}. § Introduced at eq. (9) of (II].). Notice also eq. (10, 11, 12) and the application in eq. (20, 21). otherwise; its partner being Force- Transformation and Proper Time. 653 {<) must be in general a variable ratio. . Whereas no parti- cular factoring can modify the total energy-flux (dW /dt), certain coordinating conventions will reserve some latitude about details under each adoption of factors. A momentum (Q) involves inertia (m), velocity (v’), and tangential fores (T), while (E) is invariant* : 2H =Qv; Q=m'; T = (aeye veh (2) lt Regarding (E) as a function of (m, v'); and defining i —— U5 Q, == Us OF = pie = BY 3 8 ‘f (3) the indispensable connexions among the group derived from equation (1) can be symbolized by dW aly yi OK dm oe ee IN, oh ics as dt S05, To ab 2 t om : elie OL dv ‘ = Hora Te ‘ (4) The definite (constant) inertia (m,) is uniquely linked with (T,), for which alone the partial (QE/Om) vanishes, and the principle of ws viva remains valid in the sense belonging to rigid dynamics. ‘The force (T,) is unique (c), the complementary partial (OH/dv') is suppressed. There is a second unpartitioned absorption of energy, also into kinetic form, but with an accompanying variable inertia (w'). This may be viewed as another sense of the vis viva principle. The recurrence of (c7du'/dt), with differing plausible values for (w’), has made itself noticeable throuchout previous developments regarding electronic energy f. The two abbreviated forms in equation (4), together with the two general members, yield the following relations among others : | ! Ty! ey, dv dm 1 AM av Wet (ms ap a)= Gee ooh, mu, ——.) (0) dt dt Chae le A a n 1 : Q e,e ) This amounts to establishing a transition between two activities (energy-fluxes), derived in turn from a variable inertia and from one that is constant. Hither value of the * Cf. (IL1.), p. 102 ; and eq: (12, 18). + See (1IL.) passim ; it is plainly one goal of relativity’s combinations. An important effort to construct a physical meaning for this expression is added by Sir J. J. Thomson, Phil. Mag. June 1920, p. 679. 654 Prof. F. Slate on Force- Transformation, activity may be favoured by physical evidence ; but on whichever one the preference thus falls, equation (5) or some simple equivalent shows how to calculate it in terms of the other, with due aid from a correcting partial derivative. Presented in forms like 0 = mvP—p'c? = (mr, 4 ic) (v1 +20) | = (mv t+ iv) (v, +720) [Real terms], (6) equation (1) sets in relief, first the idea of conservation (equal gain and loss ata transfer), and secondly the needful pairing of each alternative momentum with its own velocity- factor, when the complex product is expanded. All these aspects of the above more comprehensive situation embrace essentials of that correlation between Newtonian and “ non- Newtonian” dynamics upon which this discussion turns. Beside the frequent appearance of (y', w) just referred to, keeping the last members of equations (1, 5) somewhat at the front, mathematical prominence is assured to them through the Lagrange function and its derivative*. But this must not exclude the third member of equation (5). Not cnly was its type put forward earlier as a central necessity of general statement +, but it happens to offer also for the present phase rather direct contact with the systematic use of “proper time” and of “local time” }eculiar to relativity. This distinction is largely superfluous. for our Newtonian plan, since without according it a place, the main dynamical relations resting upon it have been reproduced. That composite scheme of time-variables must be truly secondary, if it be indeed carried into the funda- mental equations through a constancy of inertia made primary. Yet the newer doctrine takes so seriously what centres upon an entire parity of time and coordinate, that. more adequate review of these points is in place, for which the activity (v,T,) opens the way. A step or two in broader terms can be added, before limiting ourselves by the Lorentz. electron’s assumptions. Define now : ty | m=mz=m, v =w/2; then Q' = me Vz = m3”. * The conception of kinetic potential has this consequence. Some hie coincidences have shown themselves already: (LII.), p. 104; (1T.), eq. (47). ie n (I.), eq: (10); (I1.), eq. (3) Proper Time, and Fresnel’s Coefficient. bod fe ) With these values, equation (5) gives ee He Oe TE From two factorings of (Q,), an important duplicate expression of T, follows : \ — vl du dn ji a Midz (2) (7) Rar, ee ay doy any i h=h (742) — ole = nr dt Re an ° 0 (8) Since an activity of importance for the electron grows out of (vT,), which equation (7) connects with an energy- transfer (W/z), the latter quantity will presently cJaim attention. Next define m=mz2=m"; with v'=v, and Q"=m''v. These lead at once to r dv ae 1 nan ev 2 T, — oT" —mye?e — (=e [ 2evTy— 90” —ams0 Fe, (9) the factor (z,) being arbitrary. Finally it is evident that ( dm! ai dine 22 | vl, —c? — | =c? | —— SO esa nh (aL [oy ' a : ea dt | o The supposition is continued, that c.a.s. units of length and time (fluxion time) fix values for all observed velo- cities ; either (v,) in the standard frame (F), or (v,') in any trame (U). ‘Therefore (dt=dé,) in the defining ratios aX, bs ALN 2M aah Oy Saar 8 dt, dt, the accents add only a helpful indication of the ‘‘ observing- frame.” This will not bar auxiliary time-differentials, tor convenient expression ot auxiliary velocities, originally determined by ¢.a.s. units. Thus, if in relation to (1, U) (a Co dt —s dt ° Ak, 14) d Ax, — di — au e Mee a? as ip) de = ad. (12) dtr’ = dt! 0 dato. =_ o!y Uses Vv ! d io _— dey’ —a dey | ; Mee a rin. Wei inh Maa fin dai) dts Consequently, multiples of previous forces and activities may be drawn upon for the algebra, from parallel series like Eh, a2 ii dit’g (m ds te)? | ad W 2 dm! ate A eG) | “ifs cto leap) : dt, ad cH a ae Z Without elaborating every detail, this outline goes far enough to be convincing : it retraces essentially the “step up snd step down” with the factor (y(v)) which reaches Newtonian activity in relativity’s procedure—attainable brevity or directness is not for the moment an issue. The repetition with quantities belonging to a frame (U) is so nearly routine that it is omitted. On any line of analysis, the decision lies in the resistance problem, whether at transfer a reduction factor shall be applied to values first written for (F) or for (U). The turning-point is located in (25) 2 * Moreover, this “ invariant function ’ is plainly an offshoot from the fundamentals of the resistance problem. Proper Time, and Fresnel’s Coefficient. 659 the answer to the question: Which relative speed (v) is phvsically responsible for the resistance (m,kv?) ? The manifold bearings of the force (T,) justify adding a word about its impulse; and transitions between frames (F, U) where it eccurs. As in equation (23), take the interval (u,v) in (F), important in corresponding to the interval (0, y,—w) or (0, v-’) in (U). Adjust a velocity (vw) in (U) to meet the condition making momentum-changes permanently equal; as (v,/) affected an energy-change : MVn = M(Y(Co)Yo—y(u)e). . - . » (26) Then the equal a entail diry lvy Se - (102) = oa see tee ca Cite) the tangential accelerations and their forces thus measured in (I, U) are equal ; and the activities as well, Cea (Us toy ee a 2) 28) Consequently, Tat) = Tye = y@)Ti[Leu—(—u) | ; ‘| (29 Dieu Tiy(u)(te—u) = Liro(y(vo) —y() ). eer ) Understanding that (mp) in (T,') is replaced by (m,) for this oceasion, the work- Meeeion appears * : : a Ud Us Tieu dt,—y(w) A he Ve dty =i Tivol y (vo) Say (w)) dt, e/( e 0 U oF? dp! "0 dm! Fi - i, —ey(u)| 7 dt,. (30) This result does something to enlarge command of inter- dependence between (I, U). But its better service, perhaps, is to enforce again two dynamical ideas that pervade flere investigations : “Binet that the attacks through energy and through momentum, though reconcilable, are not entirely congruent ; and secondly, that the sneantnen of variable inertia breaks away from what suffices for constant inertia. Hach term in the first member builds upon its own equal acceleration (dv,/dt,, dvfdt.) in (F, U). This is not on the surface true of the second member, since (T)v,) belongs to the third member of equation (8), and (Tv) to its fourth member f. The above expansion repeats for an obseryation- * Relying on (I.), pp. 486, 488 ; or on (II.), p. 44. t+ Cf. the earlier comment ; (IL.), pp. 38, 39. 660 Prof. F. Slate on Force- Transformation, frame (U); with its distinctive notation, and introducing for (F) a companion to (v,) similar to Gi iniGUy: Some mention has been made already of a “ Fresnel coefficient ” («), and of a meaning for it as “‘ inertia-drag,”’ when associated with the frame (I). There proves to be, however, a pair of such coefficients («, «’), symmetrically related to (I, Uj, like (dt, dt’) of equations (12). Some- thing remains to say about this pay connected with our WMetodien forees (I, T., ‘I.’, Ta}, including ow under those symbols values for either the more general (mj) or the more particular (mp ), as the context may indicate. We can quote for observation-frame (I) *: | a ee KL a = myo) ne KT, = myy(w)y(ve) de ’ Basing frankly on symmetry foe the defining ratio (but suc eceeding members are demonstrabie), write then for an observation-frame (U), noting (u'=—u): ! Vo. a Up Weim Vo! ee Tee ce Ver Vo Ve (32) n= a = — = = be : u! u C+ UUs. 6? = WU, In the special view of relativity, («, x’) become equal. Or here visibly through the coincidences that are mentioned below equations (16). But our Newtonian scheme, in the several instances enumerated, gives enlarged reciprocity to frames (I, U) ; endependent phenomena originating (we may say) in either are convertible into terms of the other. The contrast with treating the same phenomena indifferently in all frames of the group is certainly not to Newton’s disadvantage. Under the definition of («'), the proofs are direct that — Ales Bo. “( ‘a (oe ca) = my(Vo ae oa) = eee a Be (,! - ut) = (miy(e6')) E These round out the symmetry because, allowing always * From (IL), pp. 38, 43. Eq. (5, 18, 24, 26) there define (T,, T,, Va) questions, when the energy is assi Proper Time, and Fresnel’s Coefficient. 661 for the weighting factor (y(u)), and remembering (dt,=dt.'), the last equality “stands. in exact parallel with Av, d | : Spek jee == 73) > iv 6) dé, + (Y— 1) dt, (myy(vo)) ee C { 4) Next use («, «’) with those multiples (Kp, Kp’, K.’, 1.) of (LER eae eo a), which have been subjected to a “ Minkowski transformation ”; they are allof the same type as (To) itself f. The PB imations disclose another phase of symmetry : aa dv! } «Ke = yiu)n uy(ve ) oe ae an, = y(w)mnyy(v ) ae 13 iste | Ive : ; Rinne ge Kyl = y(winy(ve) a = 3 KK, = ry (w)inyy/( 0) 7 | Alo a y(u)=y(v’), the common factor ye) sionalizes again the fully reciprocal relations of (I, U) to transfer back and. forth. The effect otherwise of («, «') is to cancel from each force’s companion the term depending on variable mertia f., Put together equations (17, 33, 34), bringing out once-more the vital divergence from relativity through employing a time-variable uniformly in all frames. Whatever the influence in other directions of such comparisons, they will not prejudice judgment to undervalue the mental vision which detected this fitness of local (or distorted) time to restore an invariance in magnitude, as well as preserve a mathematical type. By drawing upon the sequence of equations (17), the original equation (9) can be employed to recalculate for transfers among frames (U). That repetition passed over as nearly self-evident, new conclusions remain to extract from the comment attached to equation (5), that codrdinates one activity (v, T,) with an adjustable set of activities (v'T). To grasp its full scope, and to weigh habitually all its alternatives, may tend to clear perplexing or obscure igned in advance, to which that equation shall be accommodated. Exhaustive inquiry * Laid down by way of fundamental premise in (II.), eq. (12). t See (II.), p. 48. They will not be confused with “ Minkowski forces,” like (2T,, 2'T,') in eq. (18) above. { Returning closely to the assumption of (IL.), p. 37. This point gains in importance, when the intimate bearing of Iresnel’s coefficient upon the physical action is conceded. Phil. Mag. S. 6. Vol. 41. No. 244. April 1921. 2X 662 Prof. F. Slate on Force- Transformation, into the combination jam ot = (2v)T, +40? =. ae di? should be generally fruitful, under suggestion from the successful analysis for Abraham’s elepea aie Bringing in electr omagnetic activity (dM/dt,) at the ie which mainly controls the reasoning here, aM i T m' dv, q mn ie Tore abe Pech | (36 a) its divergence from (v,1,) is not such a discrepancy logically, as leaning on (v%,T,) exclusively would imply. Nor is its removal at once compulsory, by inventing some release of the electron’s internal energy, for example. But concession must be made to two possibilities: first that (z) is not exactly equal to (y(v.)) : ae secondly, that the total energy-flux is greater than (v,1',), on account of imperfect conversion into electromagnetic ae We can illustrate both points by returning to equation (9), put into these terms: TY lv dm’ SeeAG AS aR pee a a) 4 fi ~2 1 = 2y, 1,1, — = 0, oe ee ne des Pa oat one If (m,) and the last parenthesis be associated, the larger energy-fiux denoted by the first member would <« appear as due toa graded increase of effective inertia. The idea read into Fresnel’s eoofaciant is revived. Proceed next to revised magnitudes of (y’, w), determined by an imperfect conversion that leaves (7 %.7/2) of (literal) kinetic seis : Hse, MU egos |e, m=; m'=5(14%5) Hence c dv é Vv," ds Qy(2 aa | —~j)=—:. . Meee = tT 3) ,)? iE ete and the set (I, #1’, #1) grow out of (Ty, #', #) by intro- ducing the same essential factor, which equations (38) make * In (ILL.), eq. (19) to (24). We are continuing the substitution of, (2,) for (mo). Proper Time, and Fresnel’s Coefficient. 663: characteristic of that imperfect conversion. This becomes significant for comparison with previous results*. Con- tinuing along the same line, we find for the assumed time-rate of electromagnetic energy, i CHE eae aa Mi d0s (T+ Te) = (a7 + im!) Ge Avo an Rigo V2 +4n' | Vo aie ° (40) where the brackets again set off an effective inertia. The facts can be summarized into saying that a composite operator (scale-factor) must be applied; each element in it to the proper quota of (m,). Equation (9) fixes, for the general ratio (z) of equation (1), a mid-way point between the activities (energy-fluxes) denoted by (2,1, T'’v,/y(v.)). But specializing (z) into (y(v)) throws the assigned electro- magnetic activity unsymmetrically into that interval. This outcome conforms reasonably with building on the basis afforded by the last member of equation (7). Provided that ne __nB d rwye nk) _ 2d _2Bde ET bee ZENG OR a Bei zeua@arew venta | = SU see = vl, +m Vv on a dt? | : | (41) d ees faa 1 av | dt |< hye aie | Hota) The preceding treatment of electromagnetic activity is in the first instance empirical, let it be granted. Jor one thing, it adheres to an activity-value whose exact validity is perhaps not yet beyond question. Supposing, however, this datum to remain unshaken under renewed critical exam- ination, the foregoing “ cut-and-try ” result can be ration- alized, at least partially, by comparing it with the routine in terms of (uw, u) and of (w,', 4). Let us lay out the main steps of that analysis, by way of conclusion. Consider first the activities for the speed (v,), based on equations (1, 38), d mL dv a ay. dv v1 7, (cm) =a oe So are VY dt (Ch) = 0 a — Chay = (42) * Particularly (III.), eq. 29; and the routine of eq. (11, 12) there. 2X 2 664 Force-Transformation and Fresnel’s Coefficient. For present conditions, on adding in the second case the _ ‘unconverted activity,” (2v,T,) appears repartitioned thus : A= oS (cp) = 20, 1,—o,11; : ] | . (43) 1 dv Ay mi (Cm) tre gy = nh pe" | lt Hence 2 A—A, m' ae 1 Sie Me ae | en 7 es eB , av (44) reat a aaarey a” or g tam iva. : This agrees with one particular (empirical) rearrangement of equation (9), after reducing the working-speed to (v); exploiting the flexibility of a zero-difference : (b) [’ Ki m' dv U a dv | yd cee \- pies A SS a oe 5) 201, Dail ) Or E (0) ap tne | i gee And the magnitude of the last item coincides with the time-rate of the corresponding electromagnetic potential. The quoted value of electromagnetic energy-flux rejects, we might say, both extreme suppositions, of (A, Aj); it is determined symmetrically between them. How far does this support the assertion of mass in the Lorentz electron ? Proper candour can admit this whole system of equations to be finally inconclusive, and yet hold to their present usefulness. So long as aggregates only are accessible, the search for their physical constituents will grope more or less blindly. The close of that period will be hastened by first enlarging the list of possibilities, and at last weighing them impartially. The simple thoughts of this paper do no more than exemplify a method, it is true, without exhibiting it in formal terms cr delimiting it. But it can scarcely be doubtful that such a widening of Lagrange’s plan is of good promise for the further discussion of energy-fiuxes in terms of mechanics. University of California. 665, 7] LVI. Some Problems relating to Rotating Fluid in the Atmosphere. By GEORGE GREEN, D.Sc., Lecturer in Natural Philosophy in the University of Glasgow *. CCORDING to the modern view regarding the con- stitution of cyclones and anticyclones the characteristic movements of the air are in planes parallel to the earth’s suriace. Upward currents may occur in certain parts of the system, but they do not form the essential feature of the motion. This view may, or may not, prove to be correct. It is therefore a problem of some interest to determine under what conditions horizontal motions of the type generally associated with the normal cyclone and anticyclone are possible in the atmosphere. Certain aspects of the problem of the travelling cyclone have already been dealt with by Dr. Jeffreys t, and by the late Lord Rayleigh {, and by Sir Napier Shaw §. The present paper deals with the same problem in a different way. Its purpose is to show that certain motions of the type generally associated with the normal cyclone are consistent with the hydrodynamical equations of motion and with the condition of continuity of the fluid, and are therefore possible motions of the atmosphere. ‘The difficulty pointed out by the late Lord Rayleigh regarding the boundary within which motion takes place, is not touched upon in this paper. et us assume that the air near any point O at the earth’s surface is in uniform rotation relative to the earth about a vertical axis OZ, drawn through O upwards. This point O may be at rest or in motion relative to the earth while the fluid layer in contact with the earth rotates uniformly about O as a centre of rotation. We can now refer the motion of each particle of fluid to three rectangular axes defined below drawn through the centre of the earth O’. O'X is drawn parallel to a horizontal line drawn due Hast from point O. O'Y is drawn parallel to a horizontal line drawn due North from poiat O. O'Z is drawn parallel to a vertical line drawn through point O. * Communicated by the Author. + Phil. Mag. January 1919, p. 1. t Phil. Mae. September 1919, p. 420. § Geophysical Memoirs of the Meteorological Office. 666 Dr. G. Green on some Problems The coordinates of any point referred to point O at each instant we shall denote by (a, y, z) and the coordinates of the same point referred to the three parallel axes through O’ are then (2, y,¢+R), where R denotes the radius of the earth approximately. For our present purpose we may treat the earth as a perfect sphere. The three axes which we have chosen through O' are at each instant rotating axes. We shall accordingly represent by @,:—the angular velocity of rotation of axes O'Y and O’Z about axis O'X, with w, and w, as corresponding quantities for the other axes respectively. The corresponding rates of change of these quantities are then denoted by @;, wy, w, respectively, and the components of the velocity of the fluid relative to the moving axes specified above, at each instant of time f, are denoted by u, v, w respectively. Any possible motion of the fluid must now satisfy the following system of equations :— (a) The equations of motion : du : : >} — —20,v+ 20,w—o.y+o,(2+ R) —(0,7+0,)e=— Lon dt p Ow : 20,w+2o0,u—@,(2+ R)+ @,2—(@/+o/ yo 22 \ (A) dw i 1 Op | —s Sas 2 vo — 2 2 — — — = | i yt + 20,0 — @yt + Wy — (@,? + w,?) = —9 mr) dio ols fo) fe) where. = a ai DASE ae 1 DSS (b) The equation of continuity of the fluid : us Ou 00 +82) = A le Wee ©) (c) The equation determining the physical nature of the fluid :— p=kp for an isothermal atmosphere, p=kp’ for an atmosphere in convective equilibrium, where vy denotes the ratio of the specific heat of air pressure CN to the specific heat volume constant. relating to Rotating Fluid i the Atmosphere. 667 (d) The boundary equations;—which we may take to be those represented by the conditions, wee “<0, Abe —Omame: at i, h being the upper limit of the troposphere which we assume to be a fluid of finite depth. For convenience we have taken p=kp throughout; the results obtained in the paper can, however, be easily modified to suit other physical conditions than that of an isothermal atmosphere. These equations are simplified to some extent by the assumption which we now make that the motion of the fluid in each layer parallel to the ground at point O is entirely . : dw i horizontal. This makes wand —— each zero. Moreover, in at the special cases of motion to be considered in the present paper the terms @, and , are also zero. The values to be assigned to the terms @,, w,, », depend in part on Q the rotational velocity of the earth about its axis, on ¢, the latitude of the point O, and on the motion of the point O relative to the earth. If we denote by (U, V) the com- ponents of the velocity of point O in the directions due Hast aud due North respectively, we have : o,=—o=— ve wy= 0 cos b+ Fy a @,=Q sin @. R’ oe o,=—QOsin¢d. b+ To obtain the values of (@,? + @,”)2, (@,? + o.’)a, (w,°+@,")y to be used in our sqmerilous we omit all terms containing (? which appear in »,? and w,”.. These terms are already allowed for in treating gravity as a force uniform in direction over the whole field around point O. That is, the terms referred to are compensated for in the variation in the direction of gravity around point O, and do not attect the motions now being considered. o2— 0) cosd.d. oO, = — Case I.—The first motion of the atmosphere to which we shali apply the above equations is that corresponding toa uniform rotation of the atmospheric layer in contact with the earth about a vertical axis through a point O which is at rest relative to the earth’s surface. ~ We assume that each particle of fluid in any of the upper layers describes a circle 668 Dr. G. Green on some Problems with uniform angular velocity @ about a centre in its own plane. This type of motion is represented by Uu=—a(y—Bz); v=o0L; ies @O;— 0); w,=Ocosd; o,=Osin d. The equations (A) and (B) of page 666 then take the form 5) — (@? + 200 sin 6) = 4 See | — (+20 sin 8 (y—B2) =k log p (1) +200 cos d(y—Bz) =H (Fue +L Vlog p=0. Were ty: (4), The integration of the first three equations gives k log p=4(@?+ 2@0 sin d){a?+ (y—Bz)?t—gz+C, (8) as the general equation determining the density and pressure at any point in the neighbourhood of O. The constant C represents the value of k log p at point O itself. In order that the above integral may satisfy the third equation of motion, @ must be chosen according to the equation ; 2Q cos & = o+20 sing” 1 eas The continuity einabion is then fulfilled also; and the equation (3) corresponds to conditions of pressure and of motion which are possible in the atmosphere. When o is taken positive in the same direction as the earth’s rotation, the atmospheric motion described above comes ponds with that obtaining in the stationary cyclone. When @ is taken negative the en corresponds with that obtaining in the stationary anticyclone. In each plane parallel to the surface of the earth fhe air is in motion about a definite centre determined by tle beight of the plane above the earth’s surface at O. Referred to the point O, the centre of isobars drawn on the earth’s surface, the line of centres of rotation lies in the meridian plane ZOY and is inclined. to the vertical line drawn through O towards the North in the Northern hemisphere at an angle 7 given by 20, cos 7 @+ 20)sin' Gi) 1) aa (5) iani=p= relating to Rotating Fluid in the Atmosphere. 669 We may regard the line of centres as an axis of the cyclone or “anticyclone. In a cyclone, with » small, the axis is very nearly parallel to the axis ue rotation of ihe earth. Corresponding to larger and Jarger values of @ positive, the axis tends more and more towards the vertical line drawn from O thé centre of isobars. In anticyclonic motion on the other hand, with small, the axis is very nearly parallel to the axis of rotation of ae earth ; and with increasing values of negative w the axis tends towards the horizontal line drawn due north from O the centre of isobars. The greatest admissible value of w negative, according to the Paone equation, is clearly given by Ope 20) Sin Dana varins eyewear 11% (6B) Thus for an anticyclone the maximum angular velocity at the equator is zero and at the pole it is 2Q. This result would imply that no anticyclonic motion could be observed at the equator; at least no motion of the type under con- sideration in which the motion of each particle is in a horizontal plane. ‘The absence of permanent or semi-per- manent centres of high pressure from the equatorial belt of the earth’s surface is in avreement with the theoretical result obtained above, and may be regarded as some confirmation of the idea that in the lareer cyclonic and anticyclonic movements horizontal motions predominate. The assumption that the motion is horizontal cannot of course be considered as one likely to be strictly fulfilled in the equatorial region. Lhe relation of the upper winds to ground winds a any point of the earth’s surface within the area covered by a cyclone or anticyclone can be readily determined for a stationary cy ‘clone or anticyclone from the equations already given. The inclination of the axis of rotation could also readily be determined from observations of the ground wind and upper winds at one or two points within the area. We may take for example two points of observation—A due north of O and B due south of O. At B the upper winds continuously increase in velocity with height above the surface. At A the velocity of wind honed ads with height until the axis is reached where the wind velocity is zero. Further increase in height is accompanied by increasing wind velocity in the opposite direction to that of the wind at the ground. If our points of observation do not lie due north or due south from O, upper wind velocities change in direction and amount with height j in a manner Fae on the position of the point of observation relative to O. The 670 Dr. G. Green on some Problems change of wind direction with height increases 1 some Gases to nearly 180 degrees. Case /J.—The next question which arises is this: What is the smallest modification of the conditions of motion described in Case I. which will correspond with a uniform motion of O, the centre of isobars at the earth’s surface, relatively to the earth, without alteration of the motion of the fluid relative to O? Let us suppose first that O has a steady velocity U in a due east direction, other conditions of motion being as in Case I. This type of motion is represented by —o(y— Bz); v=a2; w=! U : o,=0: =Qeospt ps w-=(sin ¢. Fmaa() 5 Oy, =0e @ 3. Since U is to be regarded as a small velocity, we may af | Uae Ul cies neglect terms of the order a or O “pon the equations of p. 666 which tien reduce to fe) } — 2 2 Oe as fe : (w* + 2o0Q sin $) a eS JOE | 5 | —(w? +- 200 Sin i) Oras els role seg > (7) | +20 Deo $+ 7) (y—B2)=—g— k 2 tog p | fe 0 fe ‘= ene os log p=0. . > 2°. aes The integration of these equations again leads to k log p=3(@? + 2@0 sin g){a? + (y—Bz)?}—ge+U. (9) as in Case I., but the constant 8 has now the value given by 2 8 cos é+ ie) — ee ahi ot Lee ae ) o @+2Q sin db oe It will be seen trom this relation that the essential difterence between this motion and that considered in Case I. relating to Rotating Fluid in the Atmosphere. 671 g RONG / is that the axis of the cyclone or anticyclone is now inclined to the vertical line through O at a greater angle than before, for the same value of (positive) in each case. The axis still remains in a meridian plane through O at each instant. When the inclination of the axis to the vertical line OZ exceeds the value given in equation (5), the centre of isobars moves towards the Hast ; when the inclination falls short of the value given in equation (5), the centre of isobars moves towards the West. A westward velocity equal to the velocity of a point on the earth’s surface brings the centre of isobars to rest in space, and the axis of rotation 1s then vertical. (See also Case IV.) Case III.— We proceed now to consider what modification of the conditions of motion described in Case I. would correspond with a motion of the centre of isobars towards the North as well as towards the Hast. It we take U, V as the velocity components of O east and north respectively, and if we assume that the motion of the fluid relative to the point O is of the same type as that of the previous cases of motion, we can represent the conditions of motion by u=—w(y—Bz) ; v=o(L—az) ; w=: V U : Oe cama gee @y= 0 cos p+ R? o,=( sin d. o,=0=—¢: @,=0; @,=Qcos db . d. Owing to the existence of the component of velocity V towards the north the angular velocity of the axes OX and OY about OZ is continually undergoing change. If we regard the total angular velocity of a particle of fluid about OZ as constant, while the relative angular velocity w changes, this gives the conditions QqOisinidi=— Constante Vy. i. i.) CLL) and o+Ocosd.fd=0. AU RR aN OR Sa rag (B28 As the immediate intention is to determine the effect of a small modification of the conditions of stationary cyclonic RO UE V motion, we may treat terms of the order, 0°, —,-, On, Ue ial ; : R?? pe? 2s negligible. To the order of approximation stated, 672 Dr. G. Green on some Problems the equations to be fulfilled then iake the form a \ — (w? + 200 sin ¢) (a Bo eae | — (o? + 200 sin o)(y—Bz)= -12 - log p . (13) i) 1,0 Ss 2 D5 ee Sy —/; ° oO 2(Qeos$ + 7 )o(y—Be)—206(w—22) 5. eee | ire) O SORe | 7 iS aay al ==?) 5) 08 —= () = . - . (14) An equation similar to 3 and 9 ean readily be obtained by integration, namely :— klegp=$(@? + 20Qsin gd) {(z—2a2)? + (y—Bz)?$} —gz + C, (15) where C is again the value of ‘log op at each instant at O, the centre of isobars at the surface of the earth. In order that the third equation (13) above may be fulfilled, « and B must be chosen so that 224 ee The values of « and 8 so chosen are not constants but functions of the time. Their variations with respect to time, however, are of the order of magnitude of the terms which we have agreed to neglect. The continuity equation is also in this case not exactly fulfilled, though its fulfilment is secured to the desired order of approximation. From the equations which we have obtained for this case it appears that the axis of rotation does not remain in the meridian Ene when the centre of isobars has a motion towards the North or South. The inclination of the axis to the meridian plane through O is towards the west side when the motion of O is towards the north, and towards the east side when the motion of O is towards the south, in the Northern hemisphere. According to equations (16) and (17) above, a knowledge of the angle of inclination of the axis of the rotating aoa is all that is necessary to enable us to determine the rate at which the system moves to fe North or Kast. relating to Rotating Fluid in the Atmosphere. 673 In view of the approximate nature of the solutions con- tained in Cases II. and III. it eannot be said that the motions discussed are possible steady motions in the atmosphere... If a motion of the type indicated were established it would be subject to gradual modification owing to the cumulative influence of the sinaller terms which have been omitted in the discussion given above. The chief interest to us lies in the fact that these motions retain all the characteristics of the motion of the air in the stationary cyclone or anticyclone, while indicating the nature of the slight disturbances w Mek would procuce ‘motion of the system as a whole. Case IV.—It is of interest to find also a steady motion which would correspond with the moving cyclone or anti- cyclone and which would be free from the limilations as to the smallness of U and V which have been imposed in Cases II. and III. This can be done in the case where the centre of isobars at the earth’s surface has a uniform motion in a direction due East or due West. Let us consider the conditions of motion represented by u=—ol(y—B-) ; V=02; (== U a)"; Sie COND + - R? @-= Osin d. o,=0; ani o.=0. These conditions of motion must satisfy the system of equations — (ow +: 2’ Osin g)r— (20 cos p + v=) | Moyen ae Ty oa oe —(@w’ +200 sin d)(y— Bz) = kL log Si CaS) 20 (Qeos¢ + BWA Bs) — p( 22c0sp-+ 9) )= hd =—g—k— loge ! fe) fo) fo) (sp +5, t?5, )loz e=0. Pvp G19) 674 Problems relating to Rotating Fluid in the Atmosphere. The equation derivable from these to represent the distri- bution of pressure or density throughout the fluid consistent with the above motion is fh F y2 Hog p=4( oo! + 20/0 sin abs OQ cosh+ z) a u +1400’ + 2@0 sin ) (y— Bz)? + Le (20 cos + el >2 92+ U5) eid le where C is the value of klogp at point O, the centre of isobars drawn at the earth’s surface. It is easy to show that the required conditions are fulfilled provided Df ae we ot =alo+ 7 (20 csb+ p), ba ines 2(2c039+ 3) @ ZO sith AE MN From these equations we learn that the relative path of each fluid particle about the centre of rotation in its own plane is not a circle but an ellipse whose axes are due east and due north respectively, these axes being proportional to 1/,/o' and 1/,/a@ respectively. We fall back again upon our former case of a circular path if Fr = — 2208 $3 es that is, if the centre of isobars moves relative to the earth uniformly west at a speed equal to twice the speed of a point on the earth’s surtace coinciding with it at any instant. It can readily be verified from equation (20) above that the isobars are ellipses concentric and coaxal with the ellipses representing the paths of the fluid. Hence isobars are also lines of flow in all the cases considered. If the motion of the air in cyclones and anticyclones is, as we have assumed, mainly horizontal, it may be possible to obtain from actual observations some confirmation of the conditions of motion and corresponding pressure distributions indicated by theory in the above cases. As we have already indicated for the case of a stationary cyclone or anticyclone, the inclination of the «axis, or line of centres of rotation in the upper layers of the atmosphere, could be determined by observations of the wind velocity at various heights above the ground taken at one or two points within the area and 6= Precision-Measurements in the X-Ray Spectra. B75 covered by the rotating fluid. The variation of pressure with height above the gr round indicated in the various cases may also be capable of observation, although this would no doubt prove a much more difficult matter. The solutions obtained above deal only with a very special atmospheric motion in which the angular velocity of the rotating fluid is uniform throughout the whole mass of fluid and horizontal temperature gradients are neglected. This may also prove to be not in accordance with actual observation. Other cases of interest would be those in which the angular velocity of rotation varies with height above the earth’s surface and with distance from the centre of rotation. Perhaps the most interesting point established in connexion with the cases considered above is the very great importance of the angular motion of the earth about its axis in relation to cyclone and anticyclone motion. In this connexion it would be of interest to determine the effect of small vertical movements of the air in tending to bring the axis of the cyclone more towards the vertical ; and this would probably lead to results more in neonedenves with actual observations. than those which we have obtained above. I hope, how- ever, to continue this investigation in a later paper. LIX. Precision - measurements in the X- Fay Spectra. Part [V.—K- Series, the Elements Cu—Na: By Puts HJALMAR * iG a former part of this paper + Prof. M. Siegbahn has given an account of a precision-measurement of the Kz cine in the region Cu— ‘Cl. I have continued these investigations (1°) on the K Q)-line for the same elements ; and (2°) on the K a,- and K £,-lines in the following domain, —wNa. The first of these examinations was made with ealcite as analysing crystal, the latter with gypsum. Tue apparatus used, the new vacuum spectrograph with a metallic X-ray tube, was described by Prof. Siegbahn + and specially by Dr. W.Stenstrém {. The slit between the tube and the spectrograph was for the measurements in the region * Communicated by the Author. t Phil. Mag. xxxvii. June 1919. { W. Stenstrom, “ Experimentelle Untersuchungen der Rintgen- spektra,” Diss., Lund, L919, 676 Mr. Elis Hjalmar on Cu— Cl (A<5°3 A.U.) covered with a thin foil of alumi- nium, thickness 7; in the following researches on S—Na (5°33:1A.U.) will give too small values of the wave- lengths, but as the deviations are very small (< 0-01 per cent. of X), no corrections may be made. — In the second and third orders the deviations are larger, exactly as it was established by Stenstrém. The results of the investigations in the domain S—Na are collected in Tables VIII.—XIII. Here I have discovered* TABLE VIII. TABLE IX. 8. | P. Line. cee = Line ae EXaU z= i heer 5360°66 169-99 CA Re ie 6141-71 14837 Ca Sunes visible, not measurable. | CF ae visible, not measurable. Wel ceests 532833 171-02 Ga Aree 6102719 149°33 Gir ae 5321-75 iets} Git cae 6095-00 149-51 (a5c.) 5277 172°68 B.S 282040 156°56 eis se 5047 180°55 a Was tiae See visible, not measurable. Crane 5019°18 181-55 rey CHa a wee 5785:13 15751 * In a photogram of Ca they could also be observed. Precision-Measurements in the X-Ray Spectra. 681 TABLE X. | Tasie XI. Si. | Al. Line. Ngee: = Ta, yp BE = Ee 7109°17 128-18 ee teas Sanaa 831940 109°53 Tpiltane Sogn 7083 128-65 Can Le 8285°60 109-98 Cis eee 7063°82 129-02 COR ne 826460 110:26 od es. 7053-72 129:18 Pipe eres se 8253:00 110-41 EE Das ok 7014 129:92 tat 8205'80 111-05 ees “1003 180-12 Gane ee 8189:20 111-27 (Sh ancl 6793 13415 Seen es 8025 113-55 |S’ Papal 6744-20 135°11 [Crm sdecee visible, not measurable. [Coenen 6739°33 135-21 ON Ye FMI ee 7940-50 114-76 TasLe XII. | TapLe XIII. Mg. | Na. Line. Ag X.U. = | Line. No xGUEY = CAI RAGES 986775 92°34 (ean rt 11883°6 76°68 eee 9826°(5) 92:73 Gs. 11835 76°99 (age 979940 92:99 Cpa Sie 11802°4 21 ee re 9786:20 93°11 CN ek 11781‘4 7734 Baer: 9730 93°65 CO a visible, not measurable. a, 9712 93°85 | Cy Layee Rs is a (Gene cen 9647 94°46 (See 11591 78°62 Be Me visible, not measurable. [Cay corte 953450 95°57 some new faint lines (a’, #5, x, 8’, and 83). The mutual position of all the lines can be seen from the figure. The lines a, and a appear here unseparated as one line, in the tables indicated by a. The line 83, however, is to be looked upon as the continuation of 9’ in the region Ni—Ti. Diagram of a K-Spectrum. ‘ ; ‘ Fa, eB B, a 0 a A ee ee _dudging from this investigation, a complete K-spectrum would have the appearance which is shown in the figure. It is, however, to be noticed that with no element do all the lines occur. Physical Laboratory, University of Lund, June 1920. LX. Notices respecting New Books. Ou en est la Météorologie? By ALPHONSE BERGEY. Svo. Pp. 303. Gauthier Villars. Price not indicated. —— work is one of a series intended to give general accounts of the present positions of various scientific subjects. Asa summary of our present knowledge of meteorology it is excellent, and often remarkably up to date, even including results published in the early part of 1920. The chapters on the constitution of the atmosphere and its optical and electrical properties are parti- cularly good. Full treatment of particular points cannot naturally be given in a small work covering so much ground as this, aud some important matters are omitted; but it isa notable achievement, in so diifuse a subject, to have made the striking omissions as few as they are. The inclusion of references to original papers and an index would have increased the value of the book considerably. Dr. Berget is distinctly hostile to Germany in general and German science in particular, and expresses his views somewhat freely, and often soundly. But one may disagree with his ob- jection on p. 155 to the complexity of Koppen’s 11-compartment classification of climates, when he himself requires 6 classes to deal with France alone. He also disapproves of millibars and absolute temperature. It may be noted that on p. 84, “‘ par centi- métre carré ” should be “‘ par métre carré” ; on p. 90 the pattern of wind-screen shown is not that used in Britain ; on p. 179 there is no indication of the unit of velocity ; and that on pp. 275-6 “‘weather ” is three times misprinted. The section on the temperature of the stratosphere makes no mentionof Gold or Humphreys; that on the lunar tide in the atmosphere, none of Chapman; that on the variations of wind, none of G. I. Taylor; those on the structure of cyclones and methods of forecasting, none of Bjerknes; and if there is any mention in the book of Hann, Shaw, or Dines, it is not easily found. The account of the general circulation is based on the conventional theory of the geography books, but bad as this theory is, it presumably had to be included in a work on the present position of meteorology, as itis still the only one available. The blame for this state of affairs rests on meteorologists in general, and not on Dr. Berget in particular. But in spite of these defects, most meteorologists will find in this book much that is new and interesting to them, and the definite mistakes are very few. Hod. a EB hh) LXI. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. {Continued from p. 160. | March 10th, 1920.—Mr. R. D. Oldham, F.R.S., President, in the Chair. T ‘HE following communication was read :— ‘The Lower Paleozoic Rocks of the Arthog-Dolgelley District (Merionethshire).’ By Prof. Arthur Hubert Cox, M.Se., Ph.D., F.G.S., and Alfred Kingsley Wells, B.Se., F.G.S. This paper gives an account of the ceology of the country between the Cader Idris range and the Mawddach BSE The stratigraphical succession is as follows :— ? LUANDEILO ...... Lower Basic Volcanic Series. ( Cefn-Hir Ashes. | Crogenen Slates. LOWER 4 Bryn Brith Beds. LLANVIRN ...... | Moelyn Slates. _ | ‘China-Stone’ Ashes, \ meee } Pont Kings Slates. | Lower Acid or Mynydd Cader | | ) Volcanic Series. Lower Ashes. ARENIG §...00...5.-. Beane { vi. Upper Pencil-Slates. PUnconformity. | v. Upper Diectyonema Band, J iv. Asaphellus Beds. ( Tremadoc Slates. |} iii. Lower Pencil-Slates. fatsne | Dolgelley Beds. ia soe ee Band. i. Nio eds. CAMBRIAN ...... 4 Ffestiniog Beds. Wea : | Maentwrog Beds. _Not seen south of the Mawddach Estuary. | The Upper Cambrian beds are similar, both lithologically and faunally, to the corresponding beds on the north-east and north- west of the Harlech Dome. Discontinuity between the Cambrian and Ordovician Systems is shown by the marked contrast in lithology between the uppermost beds of the Tremadoc Slates and the suc- ceeding arenaceous Basement Series of the Ordovician System. The Ordovician succession includes in its lower portion two distinct volcanic series; the lower one immediately succeeds the Basement Series, and consists mainly of rhyolitic ashes, with occasional rhyolite-flows which thin out westwards. Coincidently with the thinning of the volcanic rocks, slate-bands with an Arenig and Lower Llanvirn fauna appear interbedded with the rhyolitic rocks. The higher voleanic series 1s separated from the lower by the main mass of the Bifidus Beds, and includes a great thickness of ‘andesitic’ ashes and agglomerates, together with abundant spilite- flows often occurring as “pillow-lavas. Intrusive igneous “rocks in the form of transgressive sills and laccolitie bodies occur at all horizons in the succession. ‘They are of two main types: (i) diabases of normal Welsh types, and (ii) granophyres (eurite type). Basic intrusions are the more 684 Geological Society. numerous, but the acid rocks which build one large sill repeated by strike-faulting, have the greater bulk. Progressive deep-seated, differentiation of the eranophy ric magma has given rise to a series of small intrusions near the main sill. The rocks in these minor intrusions show all transitions from eurites, through markfieldites, to quartz-diabase, but all agree in showing granophyric structures, and they are usually arranged in an upward order of decreasing density. ‘The main sill is the youngest intrusion in the district, and is of uniform composition throughout, except that differentia- tion in place has resulted in the production of narrow basic selvages at both the upper and the lower margins. Petrographical and structural considerations unite in justifying the assumption of a pre-Upper Bala date for all the intrusions. Faulting and folding were already in progress before the in- trusion of the hypabyssal rocks. The faults and the igneous rocks, both bedded and intrusive, decrease in importance westwards coinci- dently with a change of strike. The early movement acted along the same lines as the later, more powerful ‘ Caledonian ’ movements. April 21st.— Mr. R. D. Oldham, F.R.S., President, in the Chair. The following communication was read :— ‘The Devonian of Ferques (Lower Boulonnais).’ By John William Dudley Robinson, B.Sc., F.G.S. : In the Lower Boulonnais, between Calais and Boulogne, lies a small tract of Devonian rocks. ‘They form a lnk between the Devonian beds in Belgium, France, and Germany, and those of England geographically, and also. geologically, since they appear to have been formed in a narrow strait which joined the open seas extending towards the Atlantic and over Germany and Russia. They have been regarded as undisturbed beds lyimg on a Silurian land-surface, covered confor mably by Carboniferous Limestone, but separated from it by a gap due to cessation of deposition. Detailed mapping shows that the Carboniferous and Devonian are separated by a thrust-fault and that the Devonian has been riven into blocks by dip-faults or ‘tears’ caused by the stresses set up by the thrust-faulting. These blocks are seen to have been pushed farther and farther to the north as one proceeds westwards. The failure to recognize this structure has caused beds to be inserted in the succession which are, in fact, merely parts of other bands. The paleontology, therefore, must be reworked for correct zoning. The structure thus shown indicates that the Devonian coast-line may well have taken a northward trend under the Straits of Dover and passed north of the Kent Coalfield and the Thames. The more complicated structure of the Devonian thus revealed indicates a corresponding increase of complexity in the Carboni- ferous strata, which may be of importance in unravelling the structure of the Kent and Boulonnais Coalfields. The Devonian beds of the area described are practically unaltered and fossiliferous in many parts. Some of the types of rock described have previously escaped notice. Sees J 6 TH i LONDON, KDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCLENCE. ea aa ey ff \ BRASS [SIXTH SERIES.) we tas ee) AY 73 ae : Cea d eRe Cals, SSI & 2) ey Por MOA Ye 1921 ENT ort LX. The Disappearance of Gas in the Electric Dis- charge—Il. By The Research Staff of the General Hlectric Company, London *. (Work conducted by N. R. CAMPBELL. ) OBSERVATIONS ON DIFFERENT GASES. N the first portion of this paper (Phil. Mag. xl. p. 585, Nov. 1920) there were discussed (1) the electrical con- ditions in which gases disappear from a discharge-tube with a hot cathode, (2) the chemical actions involved in the dis- charge, especially those connected with the disappearance of carbon monoxide. We shall now consider the disappearance of other gases and the destination of the gas that disappears. It has been found that some of the minor conclusions under (2) of the previous paper are erroneous; consequently, carbon monoxide will be included again among the gases to be discussed. 12. Hydrogen.—Langmuir, in a very complete investi- gation ft, has shown that hydrogen will disappear from a vessel containing a hot tungsten filament, even when no discharge passes through the gas. The disappearance takes place even when the temperature of the tungsten is as low as * Communicated by the Director. + I. Langmuir, Journ. Amer. Chem. Soe. xxxiy. p. 1310 (1912). Phil. Mag. 8. 6. Vol. 41. No. 245. May 1921. 2 Z, 686 Research Staff of the G. E.C., London, on the 1300° K. and the greatest potential difference in the vessel as small as 10 volts; in these circumstances there is no glow discharge. No other gas is known to disappear in the absence of the discharge, unless, like oxygen and chlorine, it can react chemically with the tungsten to form well- known compounds, or unless the temperature of the tungsten is so high that there is rapid volatilization of the metal. Consequently, the phenomenon differs trom all those which form the: main object of this investigation. But since hydrogen is one of the gases the disappearance of which is of most importance, and since the form of discharge dis- cussed here involves the presence of an incandescent cathode which would cause the gas to disappear even if there were no discharge, the matter requires attention. The experiments made on the disappearance of hydrogen in the absence of a glow discharge confirmed completely those of Langmuir. If the gas has access to glass cooled in liquid air, some of the gas is liberated again when the cooled glass is warmed to ro.m temperature (the filament being now cold), and is not re-absorbed when the glass is cooled again. In a few experiments all the gas that had dis- appeared was liberated on warming the cooled glass ; in others only a part of it reappeared. When only part reappeared, a further portion could be liberated by heating to about 600° K. that part of the glass which had been at room temperature during the disappearance. It was always impossible to recover all the hydrogen by this method ; possibly the residue was adhering to portions of the glass which, for mechanical reasons, could not be heated, such as the connecting-lubes and the taps or cut-offs wh ich were necessary to isolate the part of the apparatus in which the disappearance occurred; but the general results of the work here described indicated that this is not the only explanation. Increasing the temperature of the oven from €00° K. to 730° K. (when the glass began to soften) did not increase the quantity restored. By baking the vessel before warming up the cooled glass, then warming up the cooled glass, and, lastly, baking again, it could be shown that some of the gas liberated from the cooled glass recondensed on the glass at room temperature and could he liberated thence by baking. If the gas that disappeared was pure hydrogen, the gas restored was also pure hydrogen. This important result could be established with great certainty by measurements of the glow potential. It was pointed out in §6 that the glow potential of hydrogen differs so greatly from that of, Disappearance of Gas in the Hlectric Discharge. 687 -any other gas, and is so sensitive to impurities, that there is no “possibility of a mistake. It is clear then that, in accordance with the conclusion of Tangmuir, the hydrogen that disappears adheres to the glass, especially to the cooler parts, and that it can be liberated therefrom, at least in part, by rise of temperature. In the first few moments after liberation 1t is still in a con- dition to adhere to the glass, but it generally loses that power and reverts to the normal condition of hydrogen. There appears to be a definite quantity of hydrogen that can adhere to a definite area of glass at a definite temperature ; the rate of disappearance of the gas, while the filament is hot, falls continuously and approaches zero asymptotically. This quantity increases as the temperature of the glass dlecreases, but it also varies with the exact condition of the glass. Sometimes no eas will adhere to the glass until it is ‘well below room temperatures ; sometimes at room temper- atures as much will adhere as corresponds to a monomole- cular layer on the glass, and perhaps even more. At the temperature of liquid air much more than a molecular layer will adhere; but even here the amount that will adhere is definitely limited, and a state can be reached in which no more hydrogen can be caused to disappear. According to Langmuir, the abnormal hydrogen, produced by the contact with hot tungs ten, which will adhere to glass, is made up of monatomic molecules. This conclusion is based partly on the abnormal thermal conductivity of the hydrogen, partly on its abnormal chemical activity. Lang- muir found that the active hydrogen, liberated by warming the cooled glass, would combine with nee in the cold. We have not been able to confirm certainly this abnormal activity, but we do not pretend that, our experiments are conclusive. It appears to us that any theory of the dis- appearance must be framed in the light of the observations on other gases that will be described below. 13. Disappearance of hydrogen in the discharge.—I{ the temperature of the filament and the potential between cathode and anode are raised so that a discharge passes through the hydrogen, the phenomena are somewhat differ ent. “First, the gus floceinen wenain pure hydrogen. It becomes Ceeromnaied with a substance which has a lower glow potential and is condensible in liquid air. This sub- stance is almost certainly water. Its identity is Judged partly by the temperature at which it will condense, partly PAW Ao, 688 Research Staff of the G. E.C., London, on the by its action upon sodium, which results in the evolution of hydrogen. It is not definitely known whether the water is produced by the oxidation of the hydrogen * or evolved as. water from the glass walls (see § 17). Second, the quantity that can be made to Feappear is less than that which will disappear if there is no discharge. If the walls are saturated without the discharge, and the discharge is then started, hydrogen is evolved and not absorbed. How ever,, in order that this effect may be observed, the hydrogen must be pertectly pure; if traces of impurity are present, the discharge may cause the disappearance of more hydrogen. Third, until the limit of absorption is reached, disappearance is more rapid with the discharge than without it. As before, part of the hydrogen can be liberated again by rise of temperature of the walls, whether they have been cooled in liquid air or have been at room temperature. In some cases all the hydrogen absorbed during the discharge has been liberated in this manner. Any loss may be ex- plained by the oxidation of the gas to water, and this expla- nation receives some support from experiments in which the original pressure was restored by allowing the gas access to sodium or heated magnesium. But since it was not possible in all cases so to restore the original pressure, possibilities indicated by later experiments must be taken into account ; the hydrogen may adhere to the wallsin a way which cannot be reversed by heating to the softening point of the glass. 14. The disappearance of carbon monoxide.—Iin the first paper (§ 10) it was said that, in the absence of a cooled tube to remove the carbon dioxide formed, the absorption of this. gas reached a limit when a pressure of 0°009 mm. was. reached, although the discharge stili continued. It has been found since that this observation was vitiated by the presence of a trace of mercury vapour { (cf. § 7). Jf mercury vapour is completely removed +, the disappearance continues until the discharge stops, owing to the rise of the glow potential— at any rate, if a potential greater than 300 volts is not employed. If fresh CO is admitted, it disappears again under the discharge at a rate apparently unchanged ; a limit to the * Ifthere are reducible substances present, e.g. copper oxide on the: wires, they are certainly reduced by the discharge in hydrogen. But the lamps used here were thought to be free from such substances. + The belief, apparently current, that mercury can be ee aeaeid by a trap cooled in solid CO. is erroneous (cf. M. Knudsen, Ann. d. Phys. 1. p. 472 (1916)). The glow potential can certainly detect the vapour of mercury at that temperature. On the other hand, no test has been found which will show the vapour at the temperature of liquid. alr. Disappearance of Gas in the Hlectric Discharge. 689 absorption has not been found, although a quantity has been absorbed equal to at least five times that representing a monocular layer on the walls. ; If the vessel is now baked (the temperature of baking, so long as it is over 600° K. is apparently immaterial), the greater part of the gas is restored. Most of it is restored as CO,,and not as CO. The amount of CO liberated from the walls was never found to be greater than 4 a monomolecular layer. But if it is supposed, as indicated in $10, that the CO, is formed from the CO by abstraction of carbon, so that a given volume of CO, represents double that volume of CO, then at least 90 per cent. of the gas can be restored by baking. It was not investigated whether there was also water vapour in the restored gas, but it now appears probable there was. Water vapour seems always to be produced when a discharge is passed in a vessel made of glass (lead-soda) . used in these experiments. 15. The disappearance of nitrogen.—Nitrogen disappears from the vessel under the action of the discharge, exactly as does carbon monoxide, in apparently unlimited quantities. But at the same time the filament wastes rapidly and the walls become covered with a black deposit. Some blackening always occured after prolonged operations in hydrogen and CO, but it was altogether of a lesser order. In nitrogen, absorption of gas and blackening of the glass (measured by the wastage of the filament) are roughly proportional. When the walls are heated up to the softening point of the glass, only a very small proportion of the gas absorbed is evolved ; and much of this gas is not nitrogen. ‘The nitrogen cannot be liberated in any quantity by such baking. (Again no inquiries for water vapour were made.) This fact was taken at first to indicate the formation of Lang- muir’s nitride WN, by reaction between the tungsten and the gas; indeed, Langmuir asserts that the compound is formed in the discharge. On the other hand, he insists that the compound is brown; while the deposit on the walls was grey or black, and indistinguishable in colour from that formed in other gases. An alternative (or, perhaps, addi- tional) explanation of the disappearance of nitrogen will be indicated later ; it is that the liberation of the nitrogen from the walls is prevented by the deposition over it of tungsten spluttered from the cathode. 16. The disappearance of argon.—The gas employed always contained about 5 per cent. of nitrogen, but was free from oxygen. ay 696 Research Staff of the G. EH. C., London, on the Under the discharge the gas disappeared like other gases, but at a very much slower rate. The rate of disappearance: increased very suddenly when the pressure fell below 005 mm., but even at these low pressures was not more than one-fifth of the rate at which nitrogen would disappear. (In nitrogen there was a similar rapid increase in the rate of disappearance at about ‘012 mm.) At the same time, rapid blackening of the bulb oceurred. In both nitrogen and argon there is the intimate connexion between cathode spluttering and absorption of gas which Vegard* has noted in the discharge without an incandescent cathode. An interesting fact may be mentioned, though it has probably no direct bearing on the matter under discussion. The blackening which oceured with argon was not, like that with nitrogen, distributed almost uniformly over the lamp. Round the central portion of the lamp where the electrons. projected from the cathode would be expected to hit the walls, there was absolutely clear glass with no sign of blackening; on this clear glass were the sharp shadows of the anode wire and the wire supporting the Alnnlonies and elsewhere the division between the blackened portions and the clear glass was absolutely sharp. The black material is. not deposited on the portion of the walls struck by electrons projected at right angles to the cathode surface. Traces of a similarly unequal blackening were seen in nitrogen, but they were only traces of a “distribution that is perfectly definite with argon. When the lamp in which argon had disappeared was. heated, more gas was evolved (non-condensible in liquid air) than had been absorbed. It appeared probable that all the absorbed argon was contained in the evolved gas, together with hydrogen, doubtless derived from the glass, although, as always, the olass had been baked until no more gas was: evolved before the experiment started. 17. The discharge in mercury vapour.—If the vessel is. baked just below the softening point of the glass until no more gas is evolved, and completely exhaused of ail gases other than mercury vapour, and if the discharge is then passed through this vapour in the cool lamp, the pressure in the lamp increases rapidly. The gas evolved is hydrogen, perhaps mixed with a little water. As Jong as there is liquid mercury present from which a continual supply of vapour can be drawn, the evolution continues without * L. Vegard, Ann. d. Phys. 1. p. 769 (1916). Disappearance of Gas in the Electric Discharge. 691 apparent diminution while the discharge lasts. In 48 hours enough gas to fill the lamp to a pressure of 1 cm. has been aired without any signs of fatigue, so long as the gas formed is continu: uly pumped away. Meanwhile the filament wastes rapidly, and the experiment is brought to a close by its breakage. Other experiments conducted inthis laboratory have shown that it is probable that the gas is evolved from the walls in the first instance as water, which is reduced to hydregen by the tungsten filament. If the supply of mercury vapour is not olen es the action seems to come to an end after many hours ; the mércury has disappeared. It is not known what becomes of the mercury, and indeed it is not absolutely certain that it disappears. The process is interesting as showing definitely another action which must be taken into account in considering the disappearance of gases under the discharge. The discharge here evolves gas trom the walls. It has been already noted that this action oceurs al-o in argon, and probably i in hydrogen ; it may also occur in nitrogen am CO. And it should be noted that the gas thus liberated cannot be liberated by mere heating of the walls to their softening point; gas can be attached to the walls in some such way that it can be liberated by the discharge, but not by heating. Of course, the attachment may consist of chemical combination; it is possible that glass contains hydrogen chemically combined, probably as water. But it should be observed that the hydrogen liberated, if piled up on the glass, would form a layer at least 25 molecules thick ; some of it must therefore have come from a layer at least 25 molecules deep. Since the potential driving the discharge in these experiments was often as low as 30 volts, it is hardly to be expected that the electrons or ions could penetrate so far into the glass simply in virtue of the energy which they receive from the discharge. It seems easier to believe that a layer on the surface, subject to the action of these particles, is constantly renewed by diffusion from within. 18. Conelusions.—The following conclusions are based on these experiments, though some of the evidence for them is derived from the next section :—— (1) All gases can be made to adhere to glass by the dis- charge in such a way that part, at least, can be restored by heating the glass. (2) \ The amount of eas that can be made so to adhere depends on the nature of the gas and on the state of the glass. 692 Research Staff of the G. H.C., London, on the (3) The adhesion is not due primarily to chemical reaction, although such reaction (as in the conversion of CO to GO.) may aid adhesion by converting the gas into another which adheres more readily. (4) The discharge can also liberate gas from the walls, doubtless by bombardment of the char ged particles, and - some of the gas so liberated cannot be liboratenl by heating the glass to the softening point. (5) The limit in the “disappearance of the gas is reached, when the rate at which gas is caused to adhere to the elass by the discharge becomes equal to the rate at which it is liberated by the bombardment. Tor ACTION OF PHOSPHORUS. © The main object of the research was to elucidate the well-known action of phosphorus in promoting the dis- appearance of gas. The experiments so far described were undertaken in the hope that they would throw light on that action. We must now proceed to discuss what happens when phosphorus is present in the discharge vessel. (19. The physical properties of phosphorus.—It is well known that phosphorus exists in at least two forms at room temperatures—the white and the red ; the red is stable, the white metastable. The transition point is usually given as 520° K., and heating to just below this temperature is necessary to convert the white rapidly into red. On the other hand, the conversion of red into white does not proceed rapidly at tempertures below 720° K., which is just above the softening point of the glass used. It was not possible to convert red into white rapidly without passing the softening point of the glass; but a slow conversion doubtless takes place at lower temperatures. At room temperature the red phosphorus has a vapour- pressure too low to be determined by any of the ordinary methods. In the course of the experiments a rough determi- nation of the vapour-pressure of white phosphorus at 290° K was made by a modification of Victor Meyer’s method. For this purpose an evacuated vessel was filled with the vapour, and then placed in communication for a few seconds through a narrow tap witha vessel containing a neutral gas at a known greater pressure. If itis assumed that no phosphorus diffuses out against the entering stream of gas, the vapour-pressure can be calculated from the amount of that gas entering. It was found that at 290° K. the vapour-pressure was 0:014 mm. No previous measurements at such low temperatures could Disappearance of Gas in the Electric Discharge. 693 be found, but the vapour-pressure has been determined recently in the chemical department of this laboratory by Walker’s method of drawing a neutral gas over the phos- phorus and determining the weight of the material subse- uently frozen out of the stream. The results will be published elsewhere, but they were .sufficiently concordant with the value just given, if the molecule of the vapour is taken (in accordance with the accepted view) to be P,. 20. The discharge in phosphorus vapour.—A side tube containing white phosphorus was attached to the lamp (fig. 1) and separated from it bya tap A*. The lamp was also separated by a tap B from the remainder of the apparatus. By exhausting with both taps open, closing B and then A, the lamp could be filled with phosphorus vapour at the pressure corresponding to room temperature ; or, by cooling the side tube, at the pressure corresponding to any lower temperature (T). There was no evidence that heating a tungsten filament in the vapour to 2500° K. produced any chanve either in the filament or in the vapour. But if the potential of the anode was gradually raised, a definite glow potential, similar in all respects to that discussed in § 5, was reavhed at 46°5 volts a= 290), At T—273, V, was 98 volts. Thus the glow otential, as in other gases, is greater at the lower pressure. The glow had a characteristic blue colour, but observations with a pocket spectroscope did not disclose any recognizable dines other than those of CO, which were doubtless derived from the tap grease. The glow, and the increased current which accompanies it, only lasts for a second or two. It very soon ceases, and is not renewed unless the potential is raised further. The phosphorus vapour disappears in the discharge, like most other gases, but much more rapidly than any of the other gases investigated. And it is easy to determine what has happened to the vapour that has disappeared (or at least to a great part of it). At the moment that the glow discharge passes, the walls of the lamp assume a faint yellow tint, precisely that of the ‘* yellow bulb,” which is so familiar to lamp makers. If further charges of vapour are admitted and the process repeated, the vapour disappears as before, and the coloration of the wall gradually deepens to brown. * Of course, taps always mean the presence of grease-vapour; and it is impossible to be see that this vapour has no effect on the result. But all the evidence that could be obtained by varying the conditions in all possible ways convinced us that the grease vapour had no effect what- -ever on the experiments about to be described. 694 Research Staff of the G. E.C., London, on the The yellow substance does not disappear from the walls if the lamp is exhausted for many hours toa very high vacuum through a liquid-air trap. If oxygen is admitted to the vessel, it is not absorbed at room temperature. The yellow substance is not therefore, as seems to have sometimes been believed, white or “yellow” phosphorus. It is almost certainly red phosphorus or some modification with similar properties. If the lamp is baked to 650° K. or more, the yellow coloration disappears gradually ; some of it merely distils to the cooler connecting tubes, but some collects in the liquid-air trap as white phosphorus, which will absorb oxygen at room temperatures. Ifa little oxygen is admitted during baking, it is absorbed when the temperature reaches. 650° K. These observations are completely accordant with the view that the yellow substance is red phosphorus, deposited on the walls in an extremely fine film. The discharge converts the white vapour into the red solid ; the change is one of those chemical actions (see § 11) we should expect to be brought about by the discharge. Accordingly, the disappearance of phosphorus vapour under the discharge does not seem to differ essentially from that of the other gases that have been noticed. In all cases the gas that has disappeared is deposited on the walls ; in phosphorus, as in CO, this deposition is aided by a chemical change in the gas which makes it adhere more readily to the walls. Nevertheless, these are important peculiarities about: phosphorus. The first is that,the product of the chemical change is a stable solid body which has no appreciable vapour-pressure even when it is not adhering to glass*; no evidence of the formation of such stable solids has been found in the other gases. * An attempt was made to determine the vapour-pressure of red phos-- phorus by the method just mentioned; but the only result that could be obtained was that the vapour-pressure at room temperature is less, and probably much less, than 0001 mm. The glow potential of red phospho- rus vapour, whether introduced as a powder or deposited on the walls by the discharge through white phosphorus vapour, is certainly greater than 600 volts—-a result which indicates again that the vapour-pressure must be very small, In the earlier work some very puzzling indications were obtained of a residual vapour-pressure of the substance deposited on the walls. amounting to 00009 mm., giving a glow potential as low as 32° volts. It is now believed that this vapour was that of oxides (or possibly other compounds) of phosphorus, formed by the reaction of the phosphorus vapour with carbon monoxide under the influence of the discharge. But as it has been found so far impossible to reproduce these early experi-. ments, the matter has not been cleared up satisfactorily. However, no doubt is now entertained that the vapour-pressure of red phosphorus is: ‘inappreciable for these experiments. Disappearance Pe Gas in the Electric Discharge. 695> The second concerns the values of the glow potential. If the molecular weight of phosphorus vapour is 124, the glow-potential curve corresponding to fig. 4 should lie between those of argon and mercury (ef. § 6). Only one point on the curve is definitely known (p=0°014 mm., V,= 46°5), but this point (and probably that for T=273 K.) lies well above the argon curve and close to that for CO. This. discrepancy sugeests interesting considerations which will be raised when the general theory of the glow potential is discussed in a later communication ; but it may be pointed out that an explanation would be obtained if it could be established that the ionization of the phosphorus molecule splits it into its constituent atoms, so that the weight of the phosphorus ion is 31. At present, however, this suggestion is merely speculative* The third peculi iarity lies in the great difference between V, and V,’, the rising and falling glow potentials. It is. impossible to stute the difference quantitatively without measuring fully the glow-potential curves; such measure- ment is difficu It, because it involves a rapid method of determining the pressure of the phosphorus vapour present in the vessel. But an indication is obtained by observing by what amounts the potential must be raised, in order to start the discharge once more when it has ceased, owing to the disappearance of the vapour (7.e. by tracing out the line NABCDE in fig. 3); this amount is the difference between V, and V, at the same pressure. If in argon (fig. 3) we start with a glow potential of 38 volts, the successive values of V requisite to restart the discharge when it has stopped will be 38, 54, 76, 105, 150,.... In phosphorus vapour, the successive values are approximately 46°5, 78, 350. They are difficult to determine because, at the lower pressures, the glow is so transient that it may be missed; moreover, unavoidable impurities become important; but there is no doubt that, after the glow has been started at a potential less than 100 volts, the restarting of the glow after it has stopped needs a potential very much higher “than in similar conditions in other gases. No ex- planation is attempted at present of this difference ; but it will appear that the fact is of great practical importance. * Note added Feb. 9, 1921.--It has been discovered that these con- clusions are doubtful. The vapour-pressure of white phosphorus in the presence of hot tungsten is considerably lower than its normal vapour- pressure. The hot ‘tungsten converts some, but not all, of the vapour into the red modification ; equilibrium is reached when the concentration of the vapour present is less than the normal, but still finite. The high glow potentials are doubtless due to this lowering of the equilibrium vyapour-pressure. 696 Research Staff of the G. E. C., London, on the As in other gases, no evidence could be obtained that a discharge through phosphorus vapour under a potential less than the glow potential would cause any disappearance of the vapour, even if the (/, V) characteristic indicated con- siderable ionization. 21. The disappearance of gases in the presence of phos- phorus.—F or this investigation the lamp was filled with the gas in question to a known pressure; tap B was then closed, A opened, and sufficient time allowed for the phosphorus vapour to attain by diffusion its equilibrium vapour-pressure throughout the lamp. The filament was then heated, and the potential V raised till the glow discharge occurred. The glow potential was never higher then that of the phosphorus vapour ; if there was much gas present it was less. The presence of phosphorus did not seem to alter considerably the current passing in the glow discharge; since this current represents the’ saturated thermionic current, the thermionic emission was unaltered. The gas disappeared initially in the discharge much more rapidly 4 than if phosphorus were not present, except possibly if the gas were argon. If the gas were hydrogen, the gas would not disappear ceed unless the initial pressure of the gas were below a certain limit (about 0°04 mm. in the lamp of fig.1). Ifit were CO or nitrogen, it would disappear however ‘ureat the initial pressure *: in nitrogen the dis- appearance was accompanied as before by a rapid blackening -of the walls and wastage of the filament. In argon, so far as could be ascertained, the presence of the phosphorus made little difference, but the disappearance of this gas is so slow and irregular that it is difficult to decide anything definitely about it. Mercury noalll not be examined, because the metal acts directly with the vapour, and one or other disappears completely without any discharge at all. If, instead of a definite charge of p hosphorus vapour being mixed with the gas initially, the tap A were lett open so that a continual supply of the vapour was provided, then the disappearance continued rapidly with hydrogen, nitrogen, and CO as long as the discharge lasted and fresh gas was supplied; now the amount of hydrogen that could be made to disappear was, like that of the rte two gases, unlimited. * This statement is not strictly true. If the initial pressure is too high, greater than 0°15 mm., the disappearance of gas begins almost indefinitely slowly. But if, while the pressure of the gas is always kept below (say) 0°05 mm., fresh gas is admitted as the disappearance proceeds, the amount of CO and nitrogen that can be made to disappear seems almost unlimited. a Disappearance of Gas in the Electric Discharge. 697 These statements apply to the total quantity of gas which can be made to disappear. What is far more important for some purposes is the lowest pressure that can be reached, even if the amount of gas initially present is very small. In $9 it was pointed out that the lowest pressure that can be reached in the absence of phosphorus is that at which the failing glow potential becomes equal to the potential which can be applied between the electrodes. This pressure depends largely on the form of the electrodes, and is in general lower the more uniform the field between them. In the lamp of fig. 1, with an applied potential of 150 volts, the limits were for H, 0:01 mm., for N, and CO 0-0012. (To reach this hmit in hydrogen it is necessary to remove the water formed.) On the other hand, in the presence of phosphorus, with the same potential, the discharge and the absorption of gas do not cease until the pressure has fallen to 0:°0002 mm., and when the discharge ceases at this. pressure it is not started again by raising the potential to 300 volts. An explanation of this effect of phosphorus in lowering the limit of pressure that can be reached may be based on the facts narrated already. ‘The absorption of gas ceases when the glow discharge ceases; and the glow discharge ceases when the falling glow potential becomes equal to the applied potential. The admixture of phosphorus vapour with the gas increases the pressure corresponding to any given partial pressure of the gas, and thus decreases the glow potential corresponding to that partial pressure. It therefore enables the discharge and the absorption of gas to continue when the partial pressure has fallen so low that, if the gas were present alone and its partial pressure were the total pressure, the potential applied could no longer be sufficient to maintain the discharge. When the partial pressure of the gas has fallen sufficiently, the phosphorus vapour, and not the gas, begins to disappear ; and the disappearance of this vapour proceeds until the pressure of the phosphorus has fallen so low that the discharge can proceed no longer. And when the discharge ceases because the falling glow potential has become equal to the applied potential, it is not started again except by a very great increase of potential, owing to the wide difference between the falling and rising glow potentials in this vapour (cf. $9). But why does the gas disappear before the phosphorus vapour (or before the greater part of it), as this explanation demands? Since pure phosphorus vapour disappears much more rapidly than pure gas, it might be expected that the ‘698 Research Staff of the G. E. C., London, on the order of disappearance would be reversed. It is not because the current is carried by the gas rather than the phosphorus vapour, for there is evidence that, even in the presence of gas, part of the discharge is carried by the phosphorus. Such evidence cannot be based on spectroscopic observations, for the spectrum of phosphorus is not readily discernible ; but it is obtained from the observation that, if sufficient phosphorus vapour is present, a “yellow bulb” is formed ‘before all the gas has disappeared. The reason for the per- sistence of the phosphorus is, we believe, to be found in the reversibility under the discharge of tne reaction which leads to the conversion of the white phosphorus vapour into the red solid. Even when nothing but phosphorus is present, the reaction is probably reversible, in accordance with our general principles. But the equilibrium hes so far on the side of red phosphorus that the residual pressure of white is barely detectable*. But when the red phosphorus on the walls is bombarded, not only by phosphorus molecules which may be themselves converted into solid, but also by gas molecules which cannot be so converted, the equilibrium is pushed towards the vapour phase. So long as there is gas present in considerable quantity, the conversion of white into red is never complete; there is always enough white phosphorus re-evaporating to maintain the discharge; and it is only when the gas has been greatly reduced in quantity ‘that the equilibrium moves once more towards the solid phase, and a complete disappearance of all gaseous molecules is -obtained. Is tHE ACTION OF PHOSPHORUS CHEMICAL ? 22. We have no doubt ourselves that this explanation of the lower limit of pressure attainable in the presence of phos- ‘phorus vapour is right in esseutials; such uncertainty as affects its detail will probably be removed when the exact mechanism by which the glow discharge causes the adhesion -of gas to the walls is better known. But it appears to be believed very generally that the action of phosphorus in aiding absorption by the discharge is chemical, and arises from the formation of definite chemical compounds of phos- phorus with the gas; such a theory seems to pervade all the * But it zs detectable by that very sensitive test, the glow when air or oxygen is admitted. If air is admitted to a bulb in which phosphorus vapour has been absorbed by the discharge, whether with or without gas, the flash can always be seen. On tke other hand, it is not seen if powdered red phosphorus is placed in the bulb and freed from white by long-continued exhaustion in contact with the vessel cooled in liquid air. Disappearance of Gas in the Electric Discharge. 699 literature of the subject that we have discovered. It is therefore necessary to consider that theory. It is doubtless based on the belief that a greater amount of gas can be absorbed in the presence of phosphorus than in its absence. This belief is doubtless well-founded, but it must be insisted strongly that the facts described so far afferd no warrant for it. The important action which we have just discussed depends only on a reduction of the lower limit of pressure attainable and on a consequent increase of the potential that can be applied without a discharge passing. The attainment of this lower limitis not necessarily associated _with the absorption of more gas, for the limit is set, not by the cessation of absorption if the discharge continues, but by the cessation of the discharge. So far, then, there is no evidence for the chemical theory, We will now proceed to give the evidence that has been obtained against it. This evidence may be summarized in three statements :— (1) if the gas that has disappeared is restored, it is found to be in the same chemical state as it would have been if it has disappeared in the absence of phosphorus. (2) ‘There is no simple relation bet. een the quantity of gas that can be made to disappear and the quantity of phos- phorus necessary for its disappearance. ‘here is nothing approaching to a ‘‘ law of constant proportions ”. (3) The amount of gas that will disappear depends very greatly on the surface condition of the walls ot the discharge vessel. These three statements will be expanded in order. 23. Restoration of absorbed gas.—(1) When the gas has disappeared in the presence of phosphorus, part of it at least ean usually be restored by baking the vessel as described in §12. If sufficient phosphorus has been used to give a “yellow bulb,” the evolution of gas appears to accompany closely the disappearance of the yellow colour. If the gas that has disappeared is CO, the gas can be restored almost completely as a mixture of CO and COs, the proportion of the latter being the greater as the total amount increases. That is exactly the result obtained without pho-phorus. If it is argon, the gas restored is argon with some hydrogen— as before. If it is hydrogen, the gas restored is hydrogen (partly in the “‘active”’ condition) together with water vapour. Sometimes the whole of the gas can be restored ; sometimes only part; the proportion restored varies consider- ably from experiment to experiment in much the same way 700 Research Staff of the G. E. C., London, on the and within much the same limits as if there were no phos- phorus present. The question arises whether any phosphine is restored. Langmuir found that his active hydrogen would react with phosphorus to form phosphine, and, in accorddénce with the principle of § 11 that any chemical compound will be formed to some extent that can be formed, some phosphine is to be expected. It is very difficult to detect in the presence of hydrogen, phosphorus vapour, and water; the only method seems to be to condense as much as possible of the gases restored (7.e. P, H,O, PH) in liquid air, and to note the increase of pressure as the temperature of the cooled tube is allowed to rise. Phosphine boils at 153° K. Some observations made in this manner seemed to indicate the presence of a small amount of gas which volatilized between 120° and 180° K.; since water does not volatilize appreciably till 200° K., and phosphorus until still higher temperatures, the gas may have been phosphine. But its amount was variable and never more than 25 per cent. of the restored hydrogen; accordingly, even if phosphine is formed, its formation cannot be the main cause of the increased amount of gas that will disappear. Moreover, since phosphine is a gas, its formation will only result directly in the reduction of the volume of hydrogen by one- third. But since it is possible that umlimited quantities of this gas, as of CO, can be absorbed in the discharge, its for- mation might aid in the reduction of the hydrogen. If the gas that has disappeared is nitrogen, then no gas is restored by baking the vessel*. But in this case the yellow layer does not disappear on baking. This fact, taken in conjunction with the marked blackening of the walls, appears to us to suggest that the nitrogen and the phosphorus are not liberated on baking, simply because they are held to the walls by a layer of spluttered tungsten deposited over them. When the general nature of the results is considered, this appears to us a more plausible view than that the nitrogen disappears by the formation of a nitride. The loss in weight of the filament was always decidedly less than that indicated by Langmuir’s formula * More accurately, by baking it for 5 minutes, the.time sufficient to restore all the CO and, in that case, remove all the yellow. If the vessel is baked for some hours, the yellow disappears and possibly gas is restored; but since it seems impossible by previous baking to exhaust entirely the gas from the walls, the gas coming off may not be the absorbed nitrogen. The previous baking in all cases had been so pro- longed that the pressure would not rise by more than 0-0001 mm. during 4 minutes’ subsequent baking, unless gas had been absorbed in the interval. Disappearance of Gas in the Electric Discharge. TOL WN,; it approached more nearly that corresponding to WN;. But once more, according to § 11, it is probable that some nitride is formed. 24. The absence of “constant proportions.” —A very large number of observations have been made with the object of determining ue the amount of gas that disappears under the influence of phosphorus is proportional to the amount of phosphorus present. It is clear that, of the gases mentioned, hydrogen alone is suitable for the investigation, because It alone shows a definite limit to the disappearance. ‘The following experiments appear conclusive :— (a) It the law of constant proportions were true, the pressure of hydrogen which would disappear when mixed with the vapour of phosphorus at a definite temperature pale be inde; pendent of the size or shape of the vessel. On the contrary, fume Foemil ko depend considerably, and in a very complex manner, on both the size and shape. (>) In a lamp of a certain type the pressure of hydrogen which would disappear when mixed with one charge of phosphorus vapour was 0-039 min. If hydrogen was filled to a greater pressure than this, together with a charge of phosphorus, and the discharge passed, the pressure could be reduced by about 0°039 mm. If, now, a second charge of phosphorus was mixed with the hydrogen left, the amount that could be made to disappear was much less than 0°039 mm.; and if further charges of phosphorus were added, the decrease of pressure due to each a ge appeared to decréase still further. The subsequent charges do not eause the absorption of as much hydrogen as the first. (c) The same experiment can be repeated in rather a different form. If, instead of mixing the phosphorus vapour with the hydrogen, we first introduce a charge of the vapour, pass the discharge through this vapour, depositing it on the walls as a yellow film, and then introduce hydrogen ; the amount of hydrogen that can be made to disappear is the same as that which would aisappear if the phosphorus had been in the .form of vapour, 7. ¢. 0:039 mm. If, how- ever, we introduce initially and deposit on the walls, not one charge of the vapour, but many charges, the amount of hydrogen subsequently introduced that will disappear does not seem to change. When the phosphorus is introduced in this manner, the : ‘amount of ice that can be made to disappear seems independent of the amount of phosphorus present, so long as it is above a certain limit. Phil. Mag. 8. 6. Vol. 41. No. 245. May 1921. 3A 702. ~—Research Staff of the G. E. C., London, on the (d) If not pure hydrogen, but hydrogen mixed with CO, is admitted, the amount of hydrogen that can be absorbed is almost or quite independent of the amount of CO that is absorbed at the same time. ‘Thus, in one experiment, the limit for pure hydrogen was 0°042 mm.; if 0:04 mm. of hydrogen together with 0:06 mm. of CO, making a total pressure of 0-10, all this mixture was absorbed. This appears to be the strongest proof that the action is not chemical, at least in the case of CO. On the other hand, direct experiments showed that the amount of hydrogen that can be absorbed in the discharge does increase in some measure and in some conditions with the amount of phosphorus vapour mixed with it. It was difficult to obtain quantitative results, but such measure- ments as could be made did not indicate proportionality. between phosphorus added and hydrogen absorbed. Some of these observations are noted below. 25. The influence of the walls of the vessel—These experi- ments prove conclusively that the disappearance of the gas is not due simply to chemical combination with the phos- phorus. The next experiments show that the state of the walis of the vessel influences profoundly the disappearance of the gas. (a) The most striking experiment was made for the first time by accident. If the lamp is filled with phosphorus vapour, and then air admitted, the phosphorus is oxidized and deposited on the walls, and alters somewhat their lustre. If the lamp is now exhausted, and either hydrogen or CO admitted (the other gases were not investigated), then none of the gas can be made to disappear by the discharge, whether or no phosphorus is present. If phosphorus is present, it is deposited on the walls as usual as a yellow film, but no gas disappears. The film of phosphorus oxides on the walls entirely prevents the absorption of gas, (b) Other experiments directed to changing the state of the walls so that the amount of gas absorbed should be altered were not definitely successful, although indications which it is hoped to discuss in a later communication have been obtained. But it could be proved that this state affected the condition of the gas after disappearance. For this purpose hydrogen was introduced into the lamp at various pressures, and the discharge passed with the tap A open, so that an unlimited supply of phosphorus vapour was available. In those circumstances any quantity could be made to disappear. The lamp was now baked in conditions known to liberate in a few minutes all the gas that could be Disappearance of Gas in the Klectric Discharge. 703: liberated by baking at all (unless it is prolonged for many hours), and the pressure of the restored gas measured. Since the gas was in contact with liquid air, only the restored hy drogen, and not water or phosphine, was measured in this way. “It was found that, in any given state of the vessel, the ratio of hydrogen re-evolved to that absorbed was remark- ably constant and independent of the amount absorbed ; in the lamps used the normal ratio was 0°6. This ratio could be repeated many times by absorbing and liberating hydrogen in the same vessel, but if the experiment was sufficiently prolonged, the ratio would at some stage always fall suddenly to a lower value, and again remain constant at that value. Later it would fall again until usually a ratio as low as 4 could be reached. It was noticed that any signs of blackening of the walls were immediately accompanied by a fall in the ratio, 7. e. by the re-evolution of less gas on baking. An attempt was made to determine whether the. hydrogen which did not reappear on baking could be accounted for as water and phosphine (and ‘‘ active hydrogen”) condensed in liquid air. In a few experiments, which were always the first experiment made on a new lamp, it seemed that, by allowing the products of baking to act upon sodium, prac- tically all the hydrogen could be restored. But when the ratio of re-evolved to absorbed hydrogen fell, there was no rise in the quantity that could ‘be restored ‘by action on sodium. ‘The fall is not due to the production of a larger proportion of water ; the hydrogen disappears in a way that cannot be reversed by baking. 26. Asagainst all these facts tending to show that chemical action is not the cause of the absorption of hydrogen in the presence of phosphorus, we can only set one which supports that view directly. Ifthe lamp in fig. lis filled with phospho- rus vapour without other gas, the discharge causes the walls to become vellow. If an amount of gas is present which is near the limit that can be absorbed, then the yellow coloration does not appear. Further, if the walls are first made yellow by the dischargein pure phost yhorus and then gas admitted and absorbed, the yellow colour vanishes. These facts might indicate that a colourless compound of phosphorus and the gas isformed. But the general nature of our results leads us rather to believe that the admixture of gas with the very thin layer of phosphorus on the walls * changes its optical properties. * How thick is the thinnest layer which will give a yellow colour, it is impossible to say without a better knowledge of the cross-section of a phosphorus molecule. It may be only one molecule thick, but it may be 2 or 8 thick—the latter alternative fits in best with the views suggested in the following section. OA 704 Research Staff of the G. E, C., London, on the 27. How does phosphorus increase absorption ?—But if the action of phosphorus is not chemical, why is the amount of gas that can be absorbed with phosphorus greater than that which can be absorbed without? For it is undoubtedly greater—at least in hydrogen. If the supply of phosphorus vapour is unlimited, then there is no limit to the amount of hydrogen that can be absorbed. We suggest that the answer is this. Under the discharge a limit is set to the absorption of hydrogen, because absorption i is balanced by evolution of gas alr eady absorbed in the walls by the bombardment of the changed particles. When phosphorus is mixed with the hydrogen, it is deposited on the walls as a stable solid ; this ro) solid chaslde from bombardment the absorbed hydr ogen and prevents it from being detached. If sufficient phosphorus is present to form a solid layer over the hydrogen absorbed on the glass, this layer provides a new surface on which gas may once more be absorbed. If the supply is unlimited, an unlimited number of layers can be found, consisting alter- nately of hydrogen and phosphorus, and there is no limit to the amount of gus that can be made to disappear. (Of course, the layers of gas and phosphorus are not likely to be completely distinct.) On this view, if conditions can be obtained in which absorption of hydrogen takes place without evolution, then the amount absorbed should not be increased by phosphorus unless it is present in such quantity that it can form a layer - covering the hydrogen completely and providing a new sur- face for absorption. For in such conditions the limit to absorption will be fixed by the amount of gas which can baliore, all at one time, to the surface of the walls. Now, such conditions can be obtained in hydrogen, for the gas can be made to adhere to the walls in the presence of hot tungsten without any discharge at all. It is strong support for our view that we can discover no evidence whatever that more hydrogen can be absorbed by the discharge in the presence of pho sphorus than can be absorbed by Langmuir’s method without any phosphorus at all—unless so much phosphorus is supplied that a distinct yellow coloration appears on the walls. In order that phosphorus may increase absorption above the maximum quantity that can be absorbed in its absence, so much must be used that there is definite evidence of the formation of a complete layer of phosphorus on the Bee Tt is difficult experimentally to establish that proposition with perfect certainty, but we have observed nothing that cannot be reconciled with it. Disappearance of Gas in the Electric Discharge. 705 On the other hand, it has been noted that the amount of hydrogen that can be absorbed when the glow-discharge passes is definitely less than that absorbed without the discharge. This difference is due to evolution by bombard- ment. The difference is certainly decreased by the addition of phosphorus vapour, even when there is not enough to form a yellow bulb; but there is no evidence that, while the bulb does not turn yellow, the sign of the difference can be reversed by phosphorus and more absorbed with than with- out it. The marked increase in the absorption of hydrogen which is obtained with an unlimited supply of phosphorus vapour is invariably associated with the formation of a “yellow” bulb, the depth of the coloration of which increases with the amount of hydrogen absorbed. Summary. The paper is a continuation of that in Phil. Mag. xl. p- 985, Nov. 1920. It discusses the disappearance under tbe electric discharge of hydrogen, carbon monoxide, nitrogen, argon, both in the absence and the presence of phosphorus vapour, § 12. In the presence of incandescent tungsten, hydrogen disappears without the passage of the discharge, as discovered by Langmuir. The gas disappearing adheres without change to the surface of the glass, especially to the cooler parts of it. § 13. When the discharge passes, the rate of disappearance of hydrogen is not greatly altered ; but the phenomenon is complicated by the simultaneous liberation of absorbed gas by bombardment and by the appearance of water vapour. The gas that has disappeared is again adhering to the glass walls either as hydrogen or as water. § 14. In the discharge, carbon monoxide is converted into dioxide (see § 10); the dioxide, as well as the original monoxide, adheres to the glass walls and can be liberated thence by heating. The quantity of CO, which will adhere to the glass is very much greater than that of hydrogen which will so adhere. § 15. Nitrogen disappears under the discharge, but cannot be liberated again by baking the evacuated vessel. At the same time the filament wastes and the walls are blackened. It is probable that Langmuir’s nitride WN, is formed, but it 706 Disappearance of Gas in the Electric Discharge. is thought that part of the gas is held to the walls in its original form by a covering layer of tungsten. § 16. Argon disappears like nitrogen with much blacken- ing of the bulb; but the gas can be restored by baking together with hydrogen produced by the bombardment of the glass. § 17. Mercury has not been proved to disappear. The discharge through the vapour liberates large quantities of gas from the glass, even if they have been previously baked in a high vacuum. § 18. The conclusions based on this part of the work are given on p. 691. § 19. Some of the relevant properties of phosphorus vapour are stated. § 20. Phosphorus vapour disappears rapidly in the dis- charge, being converted into red phosphorus which is de- posited on the walls. The glow potential of phosphorus vapour is anomalous. § 21. Hydrogen, carbon monoxide, and nitrogen when mixed with phosphorus vapour disappear in the discharge together with the phosphorus. A lower final pressure of gas can be reached with a given applied potential in the presence of phosphorus than in its absence. ‘This fact is correlated with the abnormal glow potential of phosphorus vapour. § 22. It is asked whether there is any evidence for the prevalent view that the action ot phosphorus in removing gases in the discharge is chemical. Evidence against this view is adduced thus :-— § 23. The state of these gases when they have disappeared in the presence of phosphorus seems not to differ from the corresponding state in the absence of phosphorus. § 24. There is no indication of a “law of constant proportions.” § 25, The state of the glass walls modifies profoundly the absorption of gas. § 27. An alternative theory of the effect of phosphorus is proposed. Itis suggested that the deposited red phosphorus covers the deposited gas and prevents it from being liberated again by bombardment. At the same time it provides a new surface on which gas can be absorbed. ae LXIITI. The Physical Significance of the Least Common Mul- tiple. By NorMAN CAMPBELL, Sc.D., and E. C. C. Baty, C251., MiSc., Poss | _Novre.—This paper is the result of a long private discussion between its authors. They have resolved some of their differences and misunderstandings, but since a residuum of disagreement remains they think it well to express their views in the form of a criticism and a reply. L (By N. RB. GC) 1. § N interpreting his work on absorption spectra Prof. me) Baly (2. ¢. bie Mag. xl. ip. 1; 1920) has: based some of his conclusions on the observation that one of the frequencies (or wave-numbers) characteristic of the spectrum is the least common multiple of other characteristic fre- quencies. So far as I can discover, this is the first time that physical significance has been attributed to the L.C.M. of measured magnitudes ; the occasion seems appropriate to examine what that significance may be. 2. The conception of an L.C.M. is usually applied only to integers; Prof. Baly applies it to fractions, still meaning, of course, by the L.C.M. the least common integral multiple. No objection can be taken to this extension in general ; any fractions, so long as they are commensurable, have an L.C.M. which is given by es VW L.C.M. of numerators of the fractions ~"F.C.F. of denominators of the fractions ’ (1) where H.C.F. means highest common integral factor. But it may be observed at once that the conception is applicable only if the fractions are commensurable ; there cannot be any L.C.M. of the length of the side of a square and the Jength of its diagonal. In speaking of an L.C.M., Prof. Baly is assuming that the real frequencies characteristic of an absorption spectrum are commensurable. Though no arguments can be raised against this assumption, I believe that none can be raised for it; there can be no arguments until relations are found for absorption spectra such as are known for the “ hydrogen-like”” emission spectra. The tre- quencies of the Balmer series are doubtless commensurable ; but it is still doubtful whether the frequencies of the similar spectra in other elements are so. However, Prof. Baly might well claim that the success of his interpretation * Communicated by the Authors. 708 Dr. N. Campbell and Prof. E. C. C. Baly on the justified his assumption, and therefore no further objection en this score can be taken. 3. The point to which I wish to draw attention is that the L.C.M. given by (1) is determinate only if the fractions are known ah perfect exactitude ; it is quite indeterminate if there is any experimental error whatever. In this the L.C.M. differs from the functions which are most frequently used in physics. Thus, if we are concerned with the product of two magnitudes A and B, an uncertainty of (say) 1 per cent. in their values produces an uncertainty of only about 1 per cent. in their product. But the same uncertainty, or any uncertainty whatsoever, produces an infinite uncertainty in their L.C.M. If we do not know the values exactly, then there is an-infinite number of values covering an infinite range, any of which may be the L.C.M. For there is an infinite number of ratios plq (where p and g are prime to each other) which differ by less than any assigned amount from each other ; whatever our experi- mental accuracy, there is an infinite number of pairs p, @¢ the ratio of which can be used with equal right to represent the measured value. Let these ratios for one of the measured values be 7 Lee ; for the other aus Me Voit Then, by Up Sy So i (1), the ee of and = will not be equal, or even 71 1 approximately equalin general, to the L.C.M. of ft and — : Tf (py, G15 715 81) are all prime to each other then he L.C. M. is pyr}, and will differ enormously according to the approxi- mation adopted. Thus, if we cannot measure a magnitude to 1 in 1000, we cannot have any reason to adopt the approximation 22/7 rather than the approximation 355/113 ; but if we take the L.C.M. of this magnitude with (say) 31/10, the two results we shall. get will be 682 and 11005, which are in the ratio of 22 to 355. ‘Their ratio is not that of any small integers; they are magnitudes as completely different as any magnitudes can be. 4, This is so obvious that, if we had heen in the habit of expressing fractions in the vulgar form, the indeterminate- ness of the L.C.M. of any fractions of which the value is not known with complete mathematical accuracy would have been immediately apparent. It is only if we use, as we always do, decimal notation (a term which will be employed to denote also similar notation based on a radix other than 10) that there is any appearance of determinateness. Tor when we adopt such notation, we fix the denominator of all Physical Significance of the Least Common Multiple. 109 our fractions and therefore fix also the numerator and the L.C.M. Then (1) leads immediately to the following rule, which is that actually employed by Prof. Baly :—Hxpress the values with the same number of decimal places ; remove the decimal point ; take the product of the resulting integers and replace the deeuned place in the product. Thus the .C.M. of 7-7 and 5°6 is 431:°2.. It.is-to be observed: that we must take the product ond not the L.C.M. of the integers. For the number of decimal places expressed are those which represent amounts greater than the experi- mental error; we do not know what the remaining places are. If we took the L.C.M. of 77 and 56 (viz. 616) In place of the product 4312, we sienle be assuming that the poserossed places were all zeros, and for such an assump- tion there is not, of course, the slightest justification. This rule, which is imposed on us by decimal notation. implies the choice of one out of the infinite number of alternatives for the ratio p/q which is to represent the value. Moreover, the choice is determined whollv by the radix of the econ and the physical unt. But the L.U.M. is determined by the choice; and the L.C.M. arrived at is thus a quite arbitrary selection from an infinite number of pos- sible alternatives. 5. In place of a general discussion it will perhaps be better to take a single example. Suppose that the real value of the two magnitudes is 7:69438... and ,9°62936... inches, and that the possible experimental error is not less than 1 per cent. Then we shall choose the numbers already given, 7:7 and 5°6, and find 431-2 as the L.C.M. Butif we had been measuring in centimetres and had never heard of inches (obviously a permissible supposition) the real values moulconave been! 19-5438 ...and 14:2986... ;. remem- bering our 1 per cent. error, He shall choose 19°5 and 14:3, of which the L.C.M. is 2788°5 cm. or 1976°6 inches——a result quite different from the value 431:2 at which we arrived before. (It is, of course, 10/2°54 times as great.) Accor- dingly a change of unit healt by the rule to which we are forced, to the identification of a perfectly different length to represent the L.C.M. of the original lengths. Now let us change our radix to 8, and express number in the scale of 3initalics. Then 7°69438... =27°20020202., and 95°62936... =72:12722227.... With an accuracy of 1 in 243 we shall choose 27°200 and 72°721. Of these the L.C.M. is 1720212:200=1157:-7—again quite a different result from our original 431°2. Indeed it is not difficult to see that by an appropriate 710 Dr. N. Campbell and Prof. E. C. C. Baly on the choice of unit or of radix, and employing always the rule which Prof. Baly employs, it is possible to make out any number whatsoever to be the L.C.M. of any other numbers whatsoever. And this indefiniteness, it must be insisted, does not result from Prof. Baly’s use on the wrong rule; so long as there can be a right rule to determine something that does not exist, it is the right rule. The plain fact of the matter is that, if the numbers of which the L.C.M. is to be taken are not known with perfect and complete mathematical accuracy, there is no such thing as the WOM. SEhserule merely ace one out of an infinite number of alternatives; and the alternative which it adopts depends wholly on the unit of measurement and the scale of notation. 6. Are we then to conclude that the numerical agreements which he finds are pure coincidences ? The answer to that question I leave to those familiar with spectroscopic work, for they alone can determine whether the accuracy of the measurements is such as to make the probability of the necessary coincidence sufficiently great for that explanation to be plausible. But it seems that if the answer is negative, there is only one conclusion to be drawn. lf the L.C.M. is to be significant the values must be known with complete accuracy. There is only one kind of magnitude which has no experimental error and can be known with complete accuracy; that magnitude is the number of something, which is necessarily an integer *. If we were measuring electric charges and found values 9:6 x 1057, 14°3.« 10°, 28:6 x 10° @s.¢.9:5) ie should doubtless conclude that we were measuring 2, 3, 6 electronic charges. and might justifiably attribute sig nificance to the fact that the third number is the L.C.M. of the first two. And the conclusion would not be invalid because there was experimental error; the error arises in deter- mining what is the charge of which the measured charges are integral multiples; we know without any error at ‘all that the ‘charges are integral, and not fractional, multiples of that charge, and since our measurements are accurate enough to distinguish between successive integers, we know exactly what multiples they are of the unknown ae We know that the real charges are px, ga, rx where p, q, 7 are integers; and we can draw the conclusion that qv is the L.C.M. of pa and rv without knowing x. The conclusion, moreover, would be independent of any change of unit (which would merely change «) or of change of radix. * See ‘Physics: The Elements,’ Chap. xvi. Physical Significance of the Least Common Multiple. 711 This represents, in my opinion, the only case in which the L.C.M. of magnitudes affected by experimental error can have an L.C.M. which is physically significant. Its charac- teristic is that all the magnitudes are integral multiples of some “natural unit” of which fractional values cannot occur. If Prof. Baly’s conclusions cannot be explained as numerical coincidences, they must prove that there is an indivisible unit of frequency in the absorption spectra which he considers. But since there is, I believe, no general belief at present in such a unit, it will be well to point out that the measurements must fulfil certain conditions to be consistent with that hypothesis. If they do not fulfil them, then, so far as I can see, the agreements he finds must be numerical coincidences, however small the apparent probability of such a coincidence may be. If measured values a, }, ¢ are to be integral multiples of a natural unit and, at the same time, the statement is to be significant that cis the L.C.M. of a and 6, then it must be possible to find an # such that, within experimental accuracy, (a, b, c)=(p, g, 7). x, where p, g, r are integers cul ris the CM. of TDS This condition raises some difficulties when, as in Prof. Baly’s examples, the values obey his rule, being expressed in units such that they are not integral. Tor consider a=7°7, b=5°6, c=431°2. The only possible value of x is near to Waa be oa if e=0°1, the numbers of nataral units in each of the meaeaced magnitudes is 77, 56, 4312. And since the only way of proving the existence of a natural unit with any satisfaction is by showing that measured values are in the ratios of small integers, the mere fact that all Prof. Baly’s magnitudes must be represented, in terms of natural units, by integers that are by no means small, makes it difficult for him to establish from these experiments (I say nothing of others) the only proposition that will make his results sig- nificant. But once more, the only alternatives are (1) that his agreements are mere ‘coincidences, (2) that all experi- mental physics 1 is founded on a delusion and scales of notation have physical importance. i (ey HC. CB.) Whilst minor objections might be raised to some of Dr. Campbell’s statements the main criticism he makes against the least common multiple is obviously sound. I naturelle am in complete agreement with him in his conclusion that the least common “multiple of two different energy quanta 712. Dr. N. Campbell and Prof..E.C.C. Baly on the can have no physical significance unless these quanta are integral multiples of a fundamental unit. Dr. Campbell speaks throughout of frequencies and the least common multiple of these frequencies, and in conclusion he states his opinion that there is no general belief in a fundamental unit.of frequency. I would point out that my theory is based on the assumption that each elementary atom is characterized not by a frequency but by its energy quantum, the frequency being due to the time factor in the process. involving the absorption or emission of one quantum, namely the shift of an electron from one stationary orbit to another. To my mind the difficulty in postulating a fundamental unit of energy is not so great as it might be in postulating a natural indivisible unit of frequency, ‘especially when it is remembered that this fundamental unit is ea hypo- thesi associated with matter. In my papers I made the assumption that two atoms in combining together contribute each an equal share towards. the total energy loss and therefore emit an equal amount of energy as whole numbers of. their characteristic quanta. This very simple hypothesis naturally means that each atom evolves an amount of energy which is the L.C.M. of the quanta characteristic of the two atoms, and in putting it forward J was tacitly assuming that each elementary quantum is an integral multiple of some fundamental unit. I am very indebted - ‘to Dr. Campbell for pointing this out. The existence of this fundamental unit of energy asso- ciated with matter does not seem to me to be difficult of acceptance. Very possibly the fundamental unit is the elementary quantum characteristic of the hydrogen atom, the quanta characteristic of other atoms being integral multiples of this. The very interesting possibility that every atomic nucleus is built up of hydrogen nuclei certainly does not increase the difficulty of belief in this, and in addition J may refer to the statement in my paper that simple arithmetical relations do seem to exist between the ele- mentary quanta | have been able to calculate. Apart from the question of the correctness of any particular theory, the non-existence of a fundamental unit of energy would seem to lead to utterly chaotic and unco- ordinated relations between the frequencies and energy contents of substances. It would I think stultify all hopes. of quantitatively coordinating energy and chemical reaction, and as is well-known many most promising results have been obtained by Bodenstein, W. C. McC. Lewis, and others in this direction. Physical Significance of the Least Common Multiple. 713 Dr. Campbell says that all the numerical agreements I have obtained must be coincidences if the fundamental unit does not exist, but after all the multiplication of coincidences must sooner or later engender a belief in their reality. The “* coincidences *’ I have obtained are very numerous indeed. The calculation of all the 600 individual lines in the absorp- tion band of sulphur dioxide from the infra-red absorption spectrum of hydrogen sulphide, the calculation of the lines of water from sulphur dioxide are only two isolated cases. I laid more stress upon these because of their very great accuracy. There are many others equally striking *, and | do not think I am unduly biassed in saying that it surely is easier to believe in the existence of a fundamental unit of energy associated with matter than to disbelieve in the reality of these results. There have also been obtained in these laboratories some results which very strongly support the L.C.M. principle and hence the existence of the fundamental unit of energy. In my paper on molecular phases t I suggested that the observed deviations from Hinstein’s law of the photo- chemical equivalent are due to the re-absorption by the surrounding reactant molecules of the energy radiated during the reaction. This will clearly result in more than one molecule reacting for every quantum of energy absorbed. In any photochemical reaction it is obvious that the energy evolved must be radiated by the products of the reaction and at frequencies characteristic of them, since if the reactant molecules radiate the energy they have absorbed they will no longer be in a reactive condition. It must be emphasised that, if the energy radiated by the products is absorbed by the reactant molecules, the quanta charac- teristic of the former must be exact multiples of the latter, which is the fundamental basis of my theory and indeed is the point at issue between Dr. Campbell and myself. If, therefore, it can be proved thatin any reaction the energy radiated by the resultant molecules is absorbed by the re- actant molecules, I venture to think that the integral relations between the quanta characteristic of the two will be very strongly supported, if not absolutely proved. In the paper referred to it was pointed out that the re-absorption by the reactant molecules of the energy radiated by the resultant molecules will obviously depend upon two factors. In the case of a photochemical reaction it will depend firstly on the concentration of the reactant molecules, * Astrophys. Journ. xlii. p. 4 (1915). pele bil Migot scp. 1) (1920). 714 Dr. N, Campbell and Prof. E.C. C. Baly on the and secondly on the density of the radiated energy and there- fore on the intensity of the activating light. In other words, the number of molecules reacting above and beyond that demanded by Hinstein’s law will vary, firstly with the con- centration of the reactant molecules when the intensity of the activating light is kept constant, and secondly with the intensity of the activating light when the concentration of the reactant molecules is kept constant. The fact that the deviation from Hinstein’s law varies with the concentration of the reactant molecules has already been proved, and this was pointed out in my paper. Further, Bodenstein * has found that the amount.of hydrogen chloride formed from a mixture of hydrogen and chlorine in unit time under constant illumination varies as the square of the concentration of the chlorine. Clearly, therefore, the first condition as regards the re-absorption is established. The second condition, namely the variation in the intensity of the light with a constant concentration of the reactant molecules, has been investigated by Mr. W. Barker and myself. The photochemical union of hydrogen and chlorine under constant pressure in the presence of water has been studied, the concentration of the chlorine and hydrogen being kept constant by the dissolution of the hydrogen chloride in water as tast as it is formed. The detailed results will be published elsewhere, but a brief statement as to their nature may now be made. In the first place, it is obvious that, if re-absorption by the chlorine of the energy radiated by the hydrogen chloride takes place, ihe amount of hydrogen chloride formed in unit time with constant intensity of the activating light will at first conform with Hinstein’s law and then steadily increase up to a constant maximum. This increase in the rate of the reaction up toa constant maximum was first noted by Bunsen and Roscoe + and now has been amply confirmed by Mr. Barker and myself. In the second place, we have found that these maximum rates are not proportional to the intensity of the incident light but that they increase far more rapidly than the in- tensity. In other words, the deviation from Hinstein’s law increases with the intensity of the incident light exactly as foretold from the theory. It may also be pointed out that the velocity curves indicate that at a finite intensity the number of molecules reacting for every quantum absorbed * Zeit. Phys. Chem. \xxxv. p. 297 (1915). + Poge. Ann. c. p. 481 (1865). Physical Significance of the Least Common Multiple. 715. becomes infinite. With this critical intensity the reaction passes as an explosion wave through the mixed gases. Reference may also be made to Slade and Higson’s results * with a photographic plate, which show that the amount of silver obtained with a given exposure is not proportional to the intensity of the light but increases at a greater rate than the intensity. All the phenomena foretold from my theory have therefore been experimentally proved. I lay great stress on the fact that this re-absorption by the reactant molecules of the energy radiated by the resultant molecules is only possible if the quanta characteristic of the former are exact integral multiples of those characteristic of the latter. Moreover, these observations lie outside the purview of Dr, Campbell’s. criticism and the question of coincidence does not arise. These results would seem to have considerable importance apart from the present discussion, for they clearly establish the possibility of a new type of photocatalysis. A reaction,, for instance, which requires light of extremely short wave- length (Schumann region) should be induced by light of a longer wave-length in the presence of a suitable cataly st. The criterion of this catalyst will be that it contains the same atoms as the reactant molecules. This substance when absorbing light rays of its own characteristic irequency will radiate this energy in the infra-red at frequencies which are exactly equal to those of the reactant molecules with the result that, provided the radiation density is sufficient, these molecules will become reactivated and will react. Experi- ments on these lines. were commenced some time ago in these laboratories and are still in progress. During the writing of this note an interesting paper has been published by Daniels and Johnston f in which is de- scribed a typical instance of this very phenomenon, These authors have proved that hght of wave-length 400-460 pu has no action on pure nitrogen pentoxide, but that this sub- stance is decomposed by light of this wave-length in the presence of nitrogen dioxide which is known to absorb these rays. They put forward the same explanation as given above, but state that the energy radiated by the dioxide can be absorbed by the pentoxide if the absorption bands of the two substances in the infra-red overlap. This process is im- possible on the quantum theory unless the infra-red quanta characteristic of the two are exactly equal, that is to say are integral multiples of a fundamental unit. * Proc. Roy. Soe. xeviii. p. 154 (1920). + J. Amer. Chem. Soc. xliii. p. 78 (1921), 716 * Dr. J.8. G. Thomas on a Null-Deflexion Tn conclusion it may be said that the evidence for the exact integral relationship between the quanta characteristic of atoms and the quanta characteristic of the molecules formed by the combination of these atoms is well-nigh overwhelming. The hesitation to accept the existence of a fundamental atomic quantum of energy, possibly that associated with the atom of hydrogen, must surely give way before the experi- mental evidence now adduced. Whilst admitting that my previous results may by some freak of nature have been coincidences in spite of their number, I feel that the more recent work gives extraordinary support to my original hypothesis of the least common multiple principle. LXIV. A Null-Deflexion Constant Current Type of Hot- Wire Anemometer, for use in the Determination of Slow Rates of Flow of Ga together with an Investigation of the Effect of the Free Convection Current upon such Determi- mations. By J. S. G. Taomas, Sc. bonds ease: (Wales), A.R.C.S., AT.C., Senior Physicist, South Metro- politan Gas Company, London *. Introduction. T was pointed out in a recent communication f, that the laws governing the convection of heat from fine heated wires are such as to indicate the hot-wire anemometer as pre-eminently the type of instrument to be employed in the investigation of slow rates of flow of gases. Employing the Morris | type of hot-wire anemometer in such investigations, difficulties are encountered owing to the existence of the iree convection current arising from the heated wire. In recent papers f, the author has discussed a type of directional hot-wire anemometer, in which these difficulties are largely obviated, and by the use of which the range of application of the hot-wire anemometer may be extended to the investi- gation of very slow-moving streams of gas. The sole co) uncompensated effect arising from the existence of the free convection current in the directional type of instrument referred to, is due to the difference in the magnitudes of the * Communicated by the Author. + Phil. Mag. vol. xii. p. 240 (1921). ie Lela. Mag. vol. xxxix. pp. 525-527 (1920); vol. xl. pp. 640-655 (1920) ; Proc. Phys. Soc. vol. xxxzi. Part 3, pp. 196-207 (1920). Constant Current Type of Hot-Wire Anemometer. 717 respective free convection currents arising from the two exposed wires owing to their small difference of temperature when exposed to the convective effect of an impressed stream of gas. In the ty pe of hot-wire anemometer inv estigated in detail by King ™*, the heat-loss from the wire due to forced con- vection was ascertained by adjusting a measured current through the wire so as to bring its resistance to a value corresponding to a predetermined temperature. A Kelvin double bridge was employed, and the arm opposite the sensitive exposed platinum arm was of manganin of negli- gible temperature coefficient. In the Morris type of instru- ment, the arm of the bridge opposite to the exposed arm is constituted of a wire similar in all respects to the latter, but shielded by means of a surrounding tube from the cooling effect of any impressed gas stream. It is clear that, with this latter arrangement, the balanced condition of the bridge is, except in so “far as a difference exists in the respective free convection currents from the two wires owing to their different respective dispositions to their immediate sur- roundings, independent of the actual temperature of the fluid medium in their neighbourhood, so long as this temper- ature is the same in each case. It has, however, been shown by the author J, that the balanced condition of the bridge is, owing to the difference in the cooling effects experienced by the exposed and shielded wires, due to their respective free convection currents, dependent to some extent upon the heating current employed in the bridge. It appeared, there- fore, desirable to investigate the possibility of constructing a null-deflexion type of hot-wire anemometer in which the bridge current was maintained constant, the deflexion produced by an impressed gas stream being annulled other- wise than by increasing the heating current through the arm of the bridge exposed to the stream. The present paper details some of the results obtained in the course of such an _ Investigation. In a previous paper f, attention has been directed to the fact that the resistance of a fine heated wire varies con- siderably when the wire, through which a constant current passes, is inclined at various inclinations to the horizontal. * Phil. Trans. A. 520, vol. ccxiv. p. 385 (1914). + Phil. Mag. vol. xxxix. pp. 511, 528, fig. 16 (1920). ee TOC: Phys. Soey vOluxxxus part 5, pp. 291-314 (1920). Also Phil. Mae. vol. xl. doc. ctt. Phil. Mag. 8. 6. Vol. 41. No. 245. May 1921. 2B 718 Dr. J.8. G. Thomas on a Null-Dejflexion With the wire horizontally disposed, the free convection current passes immediately away from the wire, whereas with the wire set vertically, the wire is laved by its free convection current whereby the heat-loss, owing to this current, is materially reduced. With variations in the ~ inclination of the wire between the two positions specified, it is clear that the cooling the wire experiences also varies. It follows, therefore, that 1 up to a limit of the velocity of the impressed stream of. gas, possibly dependent upon the value of the heating current employed, the loss of heat from a horizontally disposed wire exposed to the cooling action of an impressed stream may be compensated for, maintaining the heating current in the wire constant, by rotation of the wire from its original horizontal position, whereby the heat- loss due to free convection is reduced so as to compensate for the thermal loss due to the impressed stream. The following experimental arrangement was employed :— Apparatus. Fig. 1 shows the anemometer-tube employed which was eG in the flow tube of equal bore by means of the spigot unions 8,8. A represents the exposed platinum wire, and B the shielded wire, cnt from the same specimen as A, and surrounded by the shielding tube T. The ends of the respec- tive wires are connected by means of short lengths of thick copper wires to the screw terminals U, C. The mode of insertion of the wires A and B in the anemometer-tube is similar to that described in previous papers. The anemo- meter-tube was made in two separate sections, DE and Constant Current Type of Hot-Wire Anemometer. 719 FG, the ends of which were carefuliy turned so as to be accurately ai right angles to the axis of the tube. The ends of these sections were likewise carefully turned so as to afford a good sliding fit into the sockets K, K, K shown, and the tubes could be separately rotated axially. No difter- ence in the anemometer readings corresponding to a given flow could be detected whether the outer portions of these sockets were sealed with wax or not. The small screws M, the ends of which moved in triangular grooves cut at right angles to the axis of the tube as shown, served as guides during rotation of the tubes. After rotation, the positions of the tubes were secured by means of the screws H. (Although the diagram shows only a single screw M and a single screw H at each of the sockets, it may be mentioned that “actually in every case three screws disposed radially at angles of 120° were used.) The inclination of either wire A or B to its initial horizontal position was read by means of the pointers P, P which were attached to the respective sections of the anemometer-tube as shown, and which moved over the circular plate S of 8 inches diameter similariy affixed to the central socket-tube K, and divided into degrees. VY, V were ebonite blocks affixed to the anemometer-tube in order to facilitate the rotation of the respective sections of the tube. The portions of the anemometer-tube in the neighbourhood of the wires A and B for a distance of inches on either side of the wires were wrapped round with soft felt and the wires were inserted in a Wheatstone bridge in the manner detailed in the papers previously referred to. The constant ratio arm was throughout adjusted to 1000 ohms. The bridge current was adjusted to any desired value and maintained constant by means of a rheostat. The drop of potential across either wire was ascertained by means ofa Weston voltmeter of resistance about 200 ohms, the indi- cations of which were correct to within + per cent. The resistances of the wires were deduced therefrom, using the value of the current employed in the bridge. The temper- atures of the wires in any case were ascer tained therefrom, employing the values of their respective resistances at atmospheric temperature, determined by means of a Gallen: dar and Griffiths bridge, and the value of the temperature coefficient of the portion of wire from which they were cut. Precautions were taken to age the wires before measure- ments were made. 3B 2 720 Dr. J. 8. G. Thomas on a Null-Deflexion Heperimental Results and Discussion. The following particulars refer to anemometer-tube R 3, used in the present investigation :— Ratio cimmneeerree so fe emo 1000 ohms. Interna! Diameter of Flow Tube. ... 2°039 em. External Diameter of Flow Tube. ... 2°238 em. Internal Diameter of Shielding Tube. 0°242 em. External Diameter of Shielding Tube. 0°419 em. Mean Diameter of unprotected and PRGUECHEM WINE... 0.3: 2. aeene cake 0°101 mm. Temperature coefficient of wire. ...... 0:003588. Ro. of exposed wire (C=0°02 amp.) = 0°2337 amp. Ro. of shielded wire (C=0°'02 amp.) 0°2445 amp. The whole of the flow system, including the anemo- meter-tube, was tested for leakage by closing the outlet and establishing a pressure of 10°5 inches of water within the tube. At this pressure the leak was ascertained to be 0:06 cubie feet per hour. As the pressure in the flow tube during a series of calibrations never exceeded 0:1 inch of water, the leakage in the system is obviously extremely small and negligible. A series of determinations was made of the variation occurring in the values of the resistances of the respective wires, heated by various currents (1°5 to 0°9 amp.) with vari- ations in their inclinations to the horizontal. No detectable variation with inclination occurred in the resistance of the shielded wire, for all values of the inclination from the horizontal to the vertical position. Subsequent investigation showed that any such variation as occurred was certainly less than 0:2 per cent. of the initial resistance of the wire. The enclosure shielding this wire appears, therefore, to have been of such dimensions that no appreciable con- vection current from the wire employed is set up therein when the heating current employed has any value up to 15 amp. ‘The determination of the dimensions of a chamber wherein the free convection effect experienced by a fine heated wire is negligible is of importance in connexion with the design of ‘the katharometer * , and the method of rotation of the chamber as described affords a ready means of ascertaining the degree of elimination of such free convection current in any given case. In the case of the exposed wire, considerable alteration of resistance * Daynes, Proc. Roy. Soc. A. vol. xcvii. p. 276 (1920). Constant Current Type of Hot-Wire Anemometer. 721 accompanied its rotation through successive angles from the horizontal to the vertical position. The magnitude of these variations when heating currents of poe 0:'9°to 1:5 amp. were employed will be seen from fig. 2 wherein are given the results for successive angles of Faclination to the horizontal increasing by 10°, readings in each case being taken with the pointer indicating the same inclination to the right and left of its initial vertical position. The initial horizontal position of the exposed wire can be very accurately ISTANEE er WIRE (O14). RES INCLINATION OF WIRE TO HORIZONTAL (SECREE 3) determined by employing the bridge as previously described. It is clear that if the bridge is baloneod with the wire either horizontal or vertical, equal galvanometer deflexions are obtained on rotation of the exposed wire through the same angle in a clockwise or counter-clockwise direction. The temperature of thé surroundings varied from 16°C. to 20°:8 C. while the results shown in fig. 2 were being obtained. For the purposes of comparison it appeared desirable to reduce the experimental results to a uniform basis in whieh q2 Dr. J. S. G. Thomas on a Null-Defleaion the temperature of the surroundings of the enclosure con- taining the wire was 0°C. An investigation of the depen- dence of the temperature of a fine platinum wire heated by a constant current within an enclosure, upon the surrounding temperature, showed that, in the present case, no appreciable error would be introduced by assuming that an increase in the temperature of the medium surrounding the enclosure containing the wire, is accompanied by an increase in the temperature of the heated wire equal to such increase of temperature. The results plotted in fig. 2 have thus been deduced from the experimental results, assuming that the enclosure is surrounded by a medium at 0°C. ‘The values of the resistance and temperature of the wire in the horizontal and vertical positions when heated by various currents, the surroundings being at 0°C., are set out in Table I. herewith :— TABLE I. | Resistance of Wire | Mean temperature of | ih (ohm). i Wire Oe a , ; eating : r ss | «| Ratio = “mae a nee WV ovtieal IHoveparet| Werreal ea epee o> Rn. Rv. | On. ea dee: | O95) 04189 || (0438905225 Fain 250 1-048) ea 10 | 04680 | 04950; 288 | 322 1058 34 11 | 05973 | 05687| 364 | 414 1069 «50 1:2 | 05958 | 06350 | 456° | 510 1-066 54 13) |) 0 6692) VeOio4) | osama 22 1069 65 ee | 10:7428)) | 07086) Goss) e ces 1068 || 7% 15 | 08260 | 08680) 786 | 852 1-050 66 It will be seen that on rotation of the wire frem the horizontal to the vertical position, an increase of from 5 to 6 per cent. occurs in the resistance of the wire. The accompanying increase in the temperature of the wire is shown in the 7th column. The temperatures 0, and @, are probably correct to within 0°5 per-cent. Employing the unshielded wire in the manner already described, a series of determinations was made of the angular rotation of the wire about the axis of the flow tube necessary in order to maintain a balanced condition of the bridge when the wire was subjected to the cooling action of a current of air moving with a determinable mean velocity in the tube, Constant Current Type of Hot-Wire Anemometer. 123 the bridge current being maintained constant meanwhile. The air stream was derived from a 5 cubic feet gas holder, and was controlled and its velocity determined in the manner detailed in previous papers. The results obtained employ- ing values of the bridge current ranging from 0°9 amp. to 15 amp. are shown in fig. 3. The main features of the calibration curves are briefly as follows :—With gradual increase of the impressed stream from zero, the initial comparatively large rotation necessary to restore balance of the bridge is succeeded by a region of velocities for which the necessary rotation increases but slowly with increase in ANGULAR ROTATION (DEGREES) a VELOCITY (CMS. PER SEC, VOLUMES REDUCED To O'C aND 760MM) the magnitude of the impressed velocity. With subsequent increase of the impressed velocity the necessary rotation increases extremely rapidly. These characteristics of the calibration curves are readily interpreted by reference to the inclination-resistance curves of fig. 2. It is seen that with continuous increase of the inclination of the wire to the horizontal, the resistance of the wire initially increases extremely slowly. With further rotation, the resistance increases comparatively rapidly until inclinations approaching the vertical are reached, when the rate of variation of resistance with inclination again becomes very small. The velocity-inclination curves (fig. 3) for inclinations of the 724 Dr. J.S. G. Thomas on a Null-Deflexion wire approaching verticality are seen to be practically perpendicular to the axis of velocities. These extremely steep portions of the calibration curves represent a condition of affairs where the velocity of the impressed stream has attained such a value that the temperature of the resultant convection current from the wire differs very little from atmospheric, so that very little temperature change occurs in the wire on rotation. Moreover the fact that the con- vection current from the wire is of approximately uniform temperature over a region considerably wider than the diameter of the wire*, is a factor likewise operative in accentuating the steepness of the calibration curves in this region compared with their smaller inclination in the region of the origin. . The curves in fig. 3. show that, employing a current of’ 1°5 amp., the type of anemometer described may be usefully employed for the determination of velocities up to about 4:5 em. per sec. The range of application diminishes with decrease in the bridge current employed. For a current of 0°9 amp. the maximum velocity measurable with accuracy is about 2 cm. per sec. Determination of the effective velocity of the free convection current arising from the wire when inclined to the hori- zontal. As already remarked, the cooling effect, due to the free convection current, experienced by the heated unshielded wire in a position inclined to the horizontal is less than when the wire is horizontal. Such diminished cooling effect arises from the greater thermal shielding influence afforded by the free convection current in the case of the inclined wire, and may for purposes of calculation be ascribed to a diminution in the velocity of the free convection current. The approximate values of such effective velocities under the conditions of the present experiments can be readily determined from the results represented in fig. 3. In the case of any one of the curves shown therein, the total heat-loss from the wire is the same for any point on the curve, as the points represent a balanced condition of the bridge. If, then, v»=effective velocity of the free convection current with the wire horizontal, 7,=the effective velocity of the same when the wire is inclined at an angle a to the * Proc. Phys. Soc. vol. xxxii. Part 5, p. 801 (1920). Constant Current Lype of Hot- Wire Anemometer. 725 horizontal, and if, moreover, V,=velocity of the impressed stream corresponding to the rotation a, we evidently have since the total cooling effect experienced by the wire is the same in all cases, m=r/ 0 Va vg — Va's asswning that the cooling effect due to the walls of the tube etc. is the same in all cases. This assumption would be most strictly justified in the case of experiments carried out in a channel of large dimensions compared with the length of the heated wire. Values of v) appropriate to various temperatures of the heated wires in the present case were determined in the manner described in a previous paper ”*. The present results are not strictly comparable with the results obtained in the previous paper owing to the differ- ences in the mounting of the wire, the diameter and lagging of the tube, also the difference in the diameter and temper- ature coefficient of the wires in the two cases. In each case, the velocity of the free convection stream, assuming the stream to be at the temperature of the wire, was found to be linearly related to the temperature of the wire, the relation being of the form V=«(O—20) where V is the velocity of the free convection current, and 6° C. the temperature of the wire. While tlie determinations were being made, the mean atmospheric temperature was 20°C. lHvidence has pre- viously been given { that the free convection stream is not raised to the temperature of the wire owing to the existence of a stagnant gas film surrounding the wire. The effective velocity of the free convection stream, assuming its temper- ature to be T, the wire being horizontal and at temperature , is, in the present case, given by the relation T+ 273 0 +273 For purposes of comparison with results previously given in various calibration curves, in which velocities have eee recorded, assuming the impressed air stream to be at 0°C., the “ane of the effective velocity of the free convection cur- rent for various inclinations of the wire have been calculated as already explained, taking the temperature of the stream as V =0:026(0— 20)( aa * Phil. Mag. vol. xxxix. pp. 518-523 (1920). + Ibid. pp. 5381-534, See also Langmuir, Proc. Amer. Inst, Elec. Eng. xxxi. pp. 1011-1022. Trans. Amer. Electrochem. Soc. xxiii. p. 298 (1913). 726 Dr. J. 8S. G. Thomas on a Null-Deflexion 0°C. The results obtained are set out in Table II. here- with :—— TABLE II. Diameter of Wire 0°101 mm. Effective Velocity of Free Convection Current when Wire is inclined to Hori- Current (amp.). Temperature zontal at Angle Specified (em. per sec.). of Wire (° C.) Temperature of Stream taken as 0° C. 02... 10° 20° — \30°%4eonaome 0:9 212 2:8 DTT EES Li alles 0 1-0 279 Sidi 3-2) 0) 2:9) a0 2 0 aes 1-0 Hel Bd4 - 38 1 oe Os M2 Oe meas 1°5 1-2 445 Ae?) al) FO. LO peel 15 1:3 548 46 45 41 3°0) 25. es 1-4 651 ag) aor 4-4 31 2:6 1-8 15 766 1 aon MGNT 26 pal GA: It is clear that the numbers in the above table are only of significance for comparative purposes, as the values of v,. given in the third column, were obtained by employing the tube in a vertical position, whereas the values of the respec- tive velocities given in the remaining columns are deduced therefrom, employing results obtained with the axis of the flow tube horizontal. The various results would be strictly comparable only in the case where the experiments were carried out in a channel whose dimensions were large com- pared with the length of the wire employed. Further, since dvx= —dvo, it is clear that the effect of an error in v upon Vn the value of v, deduced therefrom will be greater, the greater the inclination of the wire to the horizontal. Thus, an error of 0°1 cm. per sec. in the determination of vp corre-- sponding to a heating current of 1°5 amp. introduces an error of about 0°4 cm. per sec. in the deduced value of v, at 90°. For this reason, therefore, no great accuracy can be attributed to the absolute values of the velocities, relating more parti- cularly to large inclinations of the wire to the horizontal,. contained in ‘lable If. The curves yielded by the results in Table II., plotting velocities as ordinates against the re- spective inclinations as abscisse, resemble that obtained by Constant Current Type of Hot-Wire Anemometer. 727 King * for the relation of the convection constant P, to the inclination of the wire. The results clearly indicate that very considerable reduction in the magnitude of the effective velocity of the free convection current occurs when the wire is rotated from the horizontal to the vertical position. Such considerable reduction is to be anticipated from a com- parison of the results contained in a previous paper ts showing the variation of the resistance of a heated wire, when subjected to an impressed downwardly-directed cur rent of air, and the variation of the resistance of the wire when rotated from the horizontal to the vertical position given in Table I. The latter variation is of the same order of magni- tude as that in the previous work corresponding to complete elimination of the free convection effect in the case of a horizontal wire. The effective velocity of the free convection current corresponding to any inclination of the wire could be very accurately determined by experiments along these lines carried out in a wind channel. lt has previously been pointed out f that an inversion ia the respective sensitivities of an anemometer of the Morris type with horizontal wire, employing two different values of the heating current, occurs as the velocity of the impressed stream is gradually increased from zero,and that the velocity of the impressed stream corresponding to this point of inver- sion is larger, the larger the free convection current corre- sponding to the greater of the two heating currents concerned. A similar effect due to a decrease in the magnitude of thie effective velocity of the free convection current, when the wire is rotated from the horizontal to the vertical position, is shown in fig. 4, wherein are given the forms of the cali- bration curves obtained employing currents of 1°5 and 1:0 amp. in the anemometer bridge, the bridge in each cuase being balanced in the absence of an impressed flow of air with the unshielded wire (1) horizontal and (2) vertical, corresponding to the respective inclinations at which the wire was subsequently used. The galvanometer shunt was throughout equal to 27 ohms. It is seen that the vertical disposition of the wire affords the greater sensitivity (c/. eurves A and ©, or B and D), and that whereas in the case of the horizontal wire, the larger heating current affords the greater sensitivity only for velocities oreater onl rains: Avocet, p42. Pehle Mast vol. xxxix. plixi: fig. 13 (1920), ‘ Phil. Mae. vol. xxxix. p. 51! 5 (198 20). 728 Dr. J.8. G. Thomas on a Null-Dejlexion than 5°6 cm. (point P in diagram), with the vertical dis- position of the wire the inversion in the respective sensi- tivities corresponds to an impressed velocity of about 1°9 cm. per sec. These velocities are of the same order of magnitude as those given in Table II. for the respective effective velo- cities of the free convection current for the horizontal and vertical disposition of the wire when heated by a current of 1°5 amp. | A.B. WIRE HORIZONTAL VELOCITY (GMS.PER SEC, VOLUMES REDUCED To O'C AND 760 1M) C.D. WIRE VERTICAL CURRENT X15, @I-0 MM eOO TL re 2O0n” "300" ) a, © 600 (iia DEFLECTION The alteration in the value of the effective velocity of the free convection current occurring on rotation of the wire was utilized in the following manner to determine the magnitude of the velocity of the impressed stream of air for which the effect of the free convection current became negligible in the case of astream of air flowing horizontally. This matter has been considered theoretically by King*. In the present * Phil. Trans. A. loc. cit. p. 426. i i fA 4 a eS ee eee Constant Current Type of Hot-Wire Anemometer. 729 instance the bridge was balanced in the absence of an impressed flow with the wire disposed horizontally, and there- after calibratiun curves were determined with the wire disposed either horizontally or vertically, readings corre- sponding to the respective horizontal and vertieal dispositions being taken alternately for the same value of the impressed velocity of the air stream. Similar determinations were then made with the bridge balanced in the absence of an impressed flow with the wire disposed vertically. The temperature of the wire corresponding to the various flows 8 = 5 z @ Ss 5 nr CURRENT |AMP BALANCED CALIBRATED _ ~~ HORIZONTALLY. T woRZONTALLY. BALANCEO CALIBRATED “"" "VERTICALLY. O VERTICALLY: VELOCITY{CMS.PER SECVOLUMES REDUCED TO O'C ano 760i") 602 DEFLECTION was determined in the manner previously described. Cali- bration curves were determined employing values of the heating current ranging from 0:9 to 15 amp. They all showed the characteristic features of the curves drawn in fig. 5, which gives the results obtained when a heating current of 1:000 amp. was used in the bridge. A difference of 1 scale division in the respective deflexions when the wire was disposed vertically or horizontally could be detected with ease. With the exception of the velocity corresponding to the point Q, the points P obviously correspond to a minimum value of the impressed velocity for which the 730 Dr. J.8. G. Thomas on a Null-Dejlexion deflexion is independent of the inclination of the wire, and consequently, in accordance with what has preceded, of the magnitude of the velocity of the free convection current. Velocities between those represented by the points P and Q ean be determined from the calibration curve, irrespec- tive of the inclination of the wire, and hence of the free convection current, with a possible percentage error not exceeding 6 per cent. ‘he results obtained, using the respective heating currents mentioned, are summarized in Table. III. The respective temperatures of the wire in the horizontal and vertical positions in the absence ef flow are contained in ‘Table I. (fourth and fifth columns respectively). iUNe re UOE Temperature of Wire corre- sponding to point Heating P (fig. 5), for Impressed ve- Temperature Impressed ve- Current which the de- locity corre- of wire corre- locity corre- (amp.). flexion is inde- sponding to P. sponding to sponding to Q. pendent of ineli- point Q (fig. 5). nation of wire. On (° C.). (cm. per sec.). Og CC:): (cm. per sec.). 09 179 a 215 37 1°0 233 9 278 4-0 11 281 11 352 4°5 12 331 13 439 50 13 390 15°5 538 5°6 1-4 470 18°5 633 6°3 15 547 22-0 7a7 6°8 The results obtained in this table and in columns 4 and*5 of Table I. indicate that the velocity V, corresponding to the point P, for which the deflexion is independent of the inclination of the wire, is related to the corresponding temperatures Oy, 0, and @, of the wire by the empirical relations, | V,=—0 0100, =0:029 0, =0-0267.. The corresponding empirical relation in the case of the points represented by Q is V,=0:0060 6, + 2°4=0°0057 6, +2°4 =0°0052 0,+ 2:4. % Constant-Current Type of Hot-Wire Anemometer. 731 Finally, in fig. 6 are’given the forms of the calibration curves obtained employing a heating current of 13 amp. in the bridge for various inclinations (specified in the diagram) of the wire to the horizontal. The bridge was in every case initially balanced in the absence of an impresed flow and the galvanometer shunt adjusted throughout to 4 ohms. . The diminished influence of the free convection current upon TO HORIZONTAL AT RESPECTIVE ANGLES SHEWN . VELocITIES (cMS. PER SEC. VOLUMES REDUCED To O°c AND 760m.) ° Ook um eco DEFLECTION. the form of the calibration curve as the wire is more inclined to the horizontal is clearly brought out by the curves, the initial steep portion of the curves becoming continually less pronounced as the inclination of the wire to the horizontal is increased. Excluding these initial portions, the curves are seen to be practically parallel to one another. 732 Constant Current Type of Hot-Wire Anemometer. The research detailed herein was. carried out in the Physical Laboratory of the South Metropolitan Gas Co. Mr. W. H. B. Hall assisted in the experimental part of the work. The author desires to express to Dr. Charles Carpenter, C.B.E., M.I.C.E, his sincerest thanks for his unfailing readiness to provide all facilities for carrying out the work, and for his inspiring interest in the investigation. Summary. A form of hot-wire anemometer is described in which the heating current is maintained constant, and the cooling effect experienced by the exposed wire due to a small impressed velocity of the gas stream, compensated by alter- ation of the inclination of the wire from its initial horizontal position. The limits.of application of such a device are shown by a series of calibration curves, in which the neces- sary alteration is plotted against the value of the impressed velocity, for values of the heating current ranging from 0°9 to 15 amp. The results are ‘employed to deduce the value of the effective free convection current corresponding to any given inelination of the wire when heated by the respective electric currents. The alteration in the effective velocity of the free convection current accompanying alter- ation in the inclination of the wire is utilized to determine the inferior limit of the value of the impressed velocity of the stream for which the effect of the free convection current may be safely neglected. Empirical linear formulz are obtained relating such limiting velocities to the temperature of the wire. The effect of the free convection current upon the form of the calibration curves is illustrated by reference to the forms of the curves obtained employing two values of the heating current for both the horizontal and vertical dispo- sitions of the wire, and for various inclinations of the wire employing a constant heating current. Physical Laboratory, South Metropolitan Gas Company, 709 Old Kent Koad, 8.E. 17 Jan., 1921- Percy LXV. Scattering and Absorption of Hard X-Rays in the Lightest Elements. By 'Tycuo H:son Avren, Dr. phil.* Y the aid of a compensation method in adjusting the thickness of a water layer so that its absorption for a pencil of X-rays is equal to that of a Haid layer of the thickness of 1 cm. or of a sheet of metal of known thickness, I have made determinations of the relative atomic absor ption coefficients of a number of elements, and I have given an account of the results of these researches in a previous paper f. Barkla and other authors have shown that, for many elements and at different wave-lengths, the relation of the absorption coefficients is independent of the wave-length. In calcu- lating the coefficients, I have assumed that such was the case for all elements, As ‘the influence of tle effect of scattering in the lightest elements is so great compared with true absorption that it cannot be neglected, it was supposed that the deviations from the constant relation of the absorption coefficients in different wave-lengths was entirely due to this effect. By means of the determination of the absorption in H, which may be considered, with no appreciable error, to be exclusively dependent on scattering, a measure of the scattering effect caused by one electron was obtained, and aided by this measure I then tried to state the number of electrons, 7. e. outer electrons, which in different elements produce scattering. As far as I know, there has not yet been performed any investigation into the question whether the relation of the values of absorption coefficients of the hghtest elements corrected for scattering is constant, which lacuna has certainly been left owing to the difficulty there is in deter- mining accurately the extent of the scattering effect. My estimation just mentioned of the number of the outer electrons being essentially dependent on the determination of absorption in the lightest elements, I have found it necessary, in recent experiments, to try to determine more accurate values of the magnitude of absorption in these elements. Particularly it has appeared of importance to me to bring about more accurate measures regarding absorption eof hydrogen. The method of research has been identical a that used earlier, with the exception of using a Coolidge ‘bulb and considerably shorter wave-lengths than before. * From the Meddelanden fran K. Vetenskapsakadenuens Nodelinstitut, ‘Bd. iv. No. 5. Communicated by the Author. _ + Medd. fr. K. Vet.-Akad. Nobelinstitut, Bd. iv. No, 3 (1919), and Phil. Mag. xxxvii. p. 165 (1919). Phil. Mag. §. 6. Vol. 41, No. 245. May 1921. 30 (34. Dr. Tycho H:son Aurén on Scattering and The tension, applied at the bulb from a high-tension trans- former, could be read off on a kilowattmeter attached to the primary circuit of the transformer, and could also be accurately regulated. In the course of my experiments tension has been continuously observed, and a continuous regulation has been made. The variations in the tension have only exceptionally reached 0°5 kv. The composition of radiation has been varied not only by variation of tension, but also by aluminium filters of the thicknesses of 0°5-3°6 em. -For each determination, as a rule, five different measure- ments have been made, which have very nearly agreed with one another, and have seldom diverged more than 0°5 per cent. from the mean. Repeated determinations of the absorption of the same substance have shown that the agree- ment of the values found is always remarkably close. In Table J. there are noted down the tensions used in the different compositions of radiation, the thickness of the respective aluminium filters, the values observed in these radiations for the atomic absorption coefficient of Cu relative to water (KcwxH,0), the corresponding values of the mass absorption coefficient of Cu (Few) and the effective wave- length (A,) in Angstrom units, calculated by the aid of the table of Barkla and White*. UN oop Ie Ne kv. filter in em. En sei 70 0°5 65:4 6°30 0°359 TELS are 9) 1:0 49:4 36-4 0°S02: 1 LEN DEE aoe 80 peso aS 40:3 2°62 0°264 TVET SU ARS 85 2:0 308 2-02 0:237 VRE: se3: 95 2°5 28°6 1:69 0°215 Walon 100 36 23°4 £28 0°194 In the determination of the atomic absorption coefficient of H relative to water in my experiments published pre- viously, I have started from the point of view that the law of additivity is strictly valid, and, consequently, that this. * Barkla and White, Phil. Mag. xxxiy. p. 272 (1917). Se F Absorption of Hard A-fays in the Lightest Elements. 735 coefticient can be calculated, if the corresponding coefficient of oxygen relative to water is known. As for the validity of the law of additivity, I have not, in a considerable cana of my experiments in which widely ditterent substances have been investigated, found anything in any case indicating that the law should not be strictly in force. On the other hand, it must be said that the determination of xojq,0 is liable to an error of great importance in the calculation of «yn,0. This error is owing to the fact that this coefficient could only have been calculated from the difference of the molecular absorp- tion coefficients of substances, which, in regard to their chemical composition, differ from each other by one atom of oxygen, because the experimental errors of both coefficients appear in this difference. Since the value of «gjn,o 1s calculated yy the subtraction of the coefficient of kojH,o from H,0/H,0 OF 1, its value will be liable to error, because of the inaccuracy of kojn,0, and also because «yy,0 is in itself a small quantity compared to xom,o- However, I have pre- viously shown that absorption can be determined with great accuracy in organic compounds, and it may, therefore, be of more advantage to calculate the absorption of hydrogen directly from these determinations. So I have made a new series of absorption determinations in various organic sub- stances at the effective wave- lengths recorded in Table [. In these researches, only pure preparations free from water have been used. In Table II. are recorded absorption coefficients ( Soh 0096 0046 0019 0002 —0-004 —0-013 KN/H.O °C 0663 0675 0674 0683 0684 0-685 TkKH/H,O CC O47Gn 0:67, 9 0616, 0651) 7 .0:672) 0-686 a Nia 0187 0108 0-058 0032 0-012 —0-001 KO/H,O 0874 0838 0824 0814 0808 0804 en gece 0544 0648 0704 0744 0768 0784 jh Se eee 0:330 0190 0120 0070 0040 0-020 8 «H/H,0 is considerably greater, which shows that true absorption is proportionaliy greater in O than in lighter elements. If we compare the values of «,7,0 for the three elements with one another, we find that while the values for Cand N are continually increasing towards a maximum, they are, on the contrary, decreasing for O and approaching to a minimum. Indeed, according to formula 3 this must be the case, if the term fad? is proportionally much greater in the case of O than for the other two elements. As I have _ previously found, it also results from these experiments that true absorption is increasing much more rapidly at O than with the preceding elements. I have assumed, as a possible explanation of this increase of absorption, the suggestion that there is a re-grouping taking place in the electrons, in so far as four electrons are concentrating nearer to the nucleus and are forming an inner region, whereas the remaining: four electrons form a group of outer electrons (valence electrons). The suggestion that the number of the outer electrons is likely to be four at O is supported by the fact that of all chemical compounds hitherto known of which this element is a constituent, oxygen does not appear to be capable of a greater number of valences than four. Nobel Institution for Physical Chemistry. Stockholm, February 1920. MuiAe. «| LXVI. On the Correction for Shear of the Differential Equa- tion for Transverse Vibrations of Prismatic Bars. By Prof. S. P. TrmosHENKo *. N studying the transverse vibrations of prismatic bars, we usually start from the differential equation et J 2() 0 relay in which ET denotes the flexural rigidity of the bar, Q the area of the cross-section, and : the density of the material. When the “rotatory inertia” is taken into consideration, the equation takes the form 4 4 2 myo le Oy Oey ae Bet g BaF g OF I now propose to show how the effect of the shear may be taken into account in investigating transverse vibrations, and I shall deduce the general equation of vibration, from which equations (1) and (2) may be obtained as special cases. Let a bed (fig. 1) be an element bounded by two adjacent cross-sections of a prismatic bar. M and Q denote respectively the bending moment and the shearing force. * Communicated by Mr. R. V. Southwell, M.A. Translated from the Russian by Prof. M. G. Yatsevitch. On the Transverse Vibrations of Prismatic Bars. 745 The position of the element during vibration will be deter- mined by the displacement of its centre of gravity and by the angular rotation ¢ in the (a, y) plane: the axis Ov may be taken as coinciding with the initial position of the axis of the bar. The angle at which the tangent to the curve into which the axis of the bar is bent (the curve of deflexion) is inclined to the axis Ow will differ trom the angie ¢ by the angle of shear y. Hence, for very small deflexions, we may write OY For determining M and Q we have the familiar expressions ig CON ca OY M=—EIS®, Q=ACOy=200 (S¥—4), “1 @) where C denotes the modulus of rigidity, for the material of the bar, and 2X is a constant which depends upon the shape of the cross-section. The equations of motion will now be :— for the rotation— ee ee Qda=! es 26 dx, Ow toler o¢ 19 (OY eK or BIS 2 +ACO(5i eae OSE = (9) if we substitute from equations (A) ; for translation in the direction of Oy— 28 jy PDH ae D 0? AL NENTS oly oY Op) a Bliminating from (5) and (6), we obtain the required equation in the form Oo 8 Oiy))\\p1 ORL) Oy HI — Se ee = Orion oe. Vg ' ts POC i GENO Ol a =. 4@) Introducing the notation | Klg Aaa eye mame pre we may write oe (7) in the form 1 0'Y ie Oe) Le) “ = 8 a oh 4 SH B(1+ 3) so 20 t? F OO ot oS). 746 = Onthe Transverse Vibrations of Prismatic Bars. In order to estimate the influence of the shear upon the frequency of the vibrations, let us consider the case of a prismatic bar with supported ends. The type of the vibrations may be assumed to be given by | MNT Vv y=Y sin 7 C08 Pails) +) ae (9) where / represents the length of the bar, and p,, is the re- quired frequency. By substitution from (9) in equation (8), we obtain the following equation for the frequency : _g mim eS a BE ) hy kK? Sik a [t Pin [? Lr AC Pun J gaGlm =(0). (10) If only the first two terms on the left side of this equation are retained (this will correspond to the equation (1)), we have 2 1 a naa [2 ea L? ? l where L=— represents the length of a wave. m By retaining the first three terms of equation (10) (2. e. by neglecting the terms which involve ), we find an” 1 wk? Mas nee 2 Tr) . ° ° ahd es (12) approximately : this result corresponds to equation (2), where the rotatory inertia is taken into consideration. By using the complete equation (10), and neglecting small quantities of the second order, we find an? { L727? EN aa ee [l= 9 ae + a) . e . (13) approximately. | Assuming the values ae ee i we have 1G i and hence we see that the correction for shear is four times greater than the correction for rotatory inertia. The value of the correction of course increases with a decrease in the wave-length L, 2. e., with an increase in m. Yougoslayia, Videm. Summer 1920. port LXVII. On the Einstein Spectral Line Effect. By H. J.. PRIESTLEY, Professor of Mathematics, University of Queensland *. . | a prediction of the LHinstein displacement of the spectral lines is based on two assumptions, namely : (a) the atom behaves as a natural clock, giving a value of ds which is the same for each vibration ; (6) the time period dé of a vibration is transmitted by the radiation from the source to the observer. If we assume, as an alternative to (4), that the Hinstein interval ds is transmitted by the radiation, the effect will arise on the transference, not of the source, but of the observer to a different gravitational field. The object of the present paper is to discuss the reasonableness of rejecting (b) and adopting the alternative. The usual method of discussing the deflexion of a beam of light by a gravitating field, by applying the principle of Least Time to determine the course of a ray propagated in three- dimensional space with velocity 1--2mr71, leads one towards the adoption of (b). The use of the principle of Least Time appears to imply an underlying constancy of period. Conse- quently it is advisable to investigate the deflexion by a method which makes no appeal to Pre-Relativity Physics. The equations obtained from $\ds=0, d= —-ydr—r dg? +y dt’, are dy (dd * 2) Mie ay > || CAP 2a Dern (=) +1 (s) = f?—1+ +2m-—, (aL) oth ‘ ope = . (2) Gil lea f Ue =h, (3) where / and & are constant. Blimination of s from (1) and (2) by means of (3) leads to By .- at 63) =o? [(1—h-? + 2mk-?2/r + 2mhPk-*/23], (4) ( 2D ohh. By pes ti) fie st 1a ak SoS teem * Communicated by the Author. ‘748 On the Einstein Spectral Line Effect. If the last term in (4) be transformed by means of (5), the equation (4) becomes (Si) tr (Zz) =? il ey (6) Further, by the Principle of Equivalence, dr dd 5 | a) tv (3) =y «Gr | age eae along the ray. It follows from (6) and (7) that the constant k must be infinite for a light pulse. In that case h is also infinite and (5) can be written Uae Os . aie eee aes = (8) -where a is the ratio of the two infinite constants. EKlimination of ¢ from (7) and (8) leads to i ee dd +yu = 9 where u=7—1, -and, on differentiation, this equation becomes d?u dg’ Since m= 1°47 km.and wu cannot be greater than (697000) -! the term on the right-hand side of (9) is small compared with u. Hence (9) can be solved by approximate methods. If u=Acos®@ is a first approximation, the second is -given by pus Smuts). ee u=Acos@+3mA?|1—cos2g]. . . (10) “The directions of the asymptotes are given by uw=0, or 2mA? cos’ d — 2A cos 6—4mA?=0. T£ second powers of mA be neglected, the solution of this equation is coso=—2mA, ae iB - ama |. IIence the angle between the asymptotes of the light ao is approximately 4mA. It follows from (10) that A=ho -where R is the distance of perihelion. The Degradation of Gamma-Ray Energy. 749 The deflexion cf the ray is therefore 4m/R, the value verified by the Solar Eclipse Expeditions. Since the deflexion can be thus established without any appeal to the Eon aple of Least Time or any other ideas of pre-relativity Physics, there is no @ priori reason for adopting assumption () above. The following argument suggests that the proposed alter- native is more in accordance with the ide eas of Relativity. Consider two light pulses leaving A at times tg, t, + dt, and arriving at B a times fp and a Since, by. the Principle of Equivalence, ds=0 along the world line of each pulse, it seems that the Einstein interval ds, between the two departures from A is equal to the interval dsg between the two arrivals at B. hat is, the in- terval ds is transmitted by the radiation. ‘University of Queensland, Brisbane. EXVIT. Vhe Degradation of Gamma-Ray Knergy. By Arraur H. Compron, PA.D., Wayman Crow Professor of Physics, Washington University * qe has long been known that when matter is traversed by gamma rays, it becomes a source of secondary tT gamma ee The relation between the primary and_ the secondary gamma rays, however, has not been definitely established. Although the secondary radiation is very appreciably less penetrating than the primary rays, it has usually been considered to be due principally to true scattering t. It is the purpose of the pre esent paper to investigate the nature and the general characteristics of secondary gamina rays, and to study the mechanism whereby * Communicated by Prof. Sir EH. Rutherford, F.RS. t In this paper the term ‘‘secondary” gamma radiation is used to denote any radiation of the gamina type excited either directly or indirectly by the passage through maiter of primary gamma rays. By “scattered” radiation is meant the radiation emitted by the electrons in matter (that due to the positive nuclei is theoretically negligible in comparison) due to the accelerations to which they are directly subjected by the primary rays. The term “fluorescent ” radiation signifies as usual radiation of the energy absorbed from the primary bexm and stored temporarily in the kinetic and potential energies of the electrons. Its frequency theretore depends jointly upon the frequency of the primary rays and the nature of the radiator. t Cf. e.g. E. Rutherford, ‘Radioactive Substances, ete., p. 282. ws A Gray, Phil. Mac: xxyi. p. tee (LE 118). D.C. H. lorance, Phil. Mac. xxvil. p. 225 (1914). K. W. F. Kohlrausch, Phys. Zectschr. xxi. ‘p. 193 (1920). Phil. Mag. 8. 6. Vol. 41. No. 245. May 1921. 3D 750 Prof. A. H. Compton on the comparatively soft secondary radiation is excited by relatively hard primary radiation. From theoretical considerations, both scattered and fluorescent radiation should undoubtedly be present in secondary gamma rays. According to J. J. Thomson’s. well-known theory *, when the wave-length is so short that there is no appreciable co-operation in the scattering by the different electrons, the mass scattering coefficient should be about 0-2 for all elements and all wave-lengths (if the number of electrons per atom effective in the scattering is equal to the atomic number, as seems to be the case for hard X-rays). The magnitude of the scattering to be. expected is considerably reduced if the wave-length ap— proaches the size of the electron, and may, indeed, become very small if the ratio of the wave-length to the diameter of the electron approaches unity t+. There is, however, on the basis of the classical electrodynamics, no means of eliminating completely the scattered radiation. Fluorescent radiations of a comparatively soft type, presumably the characteristic K radiations, have been detected in the secondary gamma rays from elements of high atomic weights {. But, in addition to this, there should be excited in all elements a harder flucrescent radiation by the impact of the high-speed beta particles liberated by the primary rays. The number of such electrons expelled by gamma rays is known to be much the same per unit mass for all elements, andit has been shown experimentally § that gamma rays in not greatly different amounts per unit mass are- excited when beta rays fall upon different substances. Thus. one would expect to find in the secondary gamma rays an appreciable amount of fluorescent radiation, which, like the scattered radiation, does not differ greatly according to» the element used as radiator. ‘The usual method of distinguishing between scattered and fluorescent radiation is by comparing the absorption co- efficients of the primary and secondary radiations. It is assumed that the scattered rays are of the same hardness as the primary rays, whereas all known high - frequency fluorescent radiations are of a less penetrating type. Gray he and Florance§, however, have rejected this criterion, for although they find that the secondary radiation excited by hard gamma rays is of a distinctly softer type than the- : J . J. Thomson, ‘Conduction of Electricity through Gases,’ 2nd ed: p. ¢ ZO. ‘Peale . H. Compton, Phys. Rev. xiv. p. 23 (1919). A. Gray, loc. ert. C. H. Florance, loc. cvt. Je D. tr t+ Degradation of Gamma-Ray Energy. om primary radiation, they conclude that the primary rays are truly scattered, but in the process of scattering are so modified as to become less penetrating. It is therefore important to determine under what circumstances, if any, the hardness of the scattered rays may differ from that of the primary rays. If the scattering is due to electrons of negligible dimen- sions which are separated far enough to act independently of each other, there is no question but that the scattered ray will be exactly similar to the primary ray in every respect except intensity ; for since the accelerations to which each electron is subject are strictly proportional to the electric intensity of the primary wave which traverses it, and since the electric intensity of the scattered ray (ata great distance) due to each electron is proportional to its acceleration, the electric vector of the scattered wave is strictly proportional to the electric vector of the primary wave. ‘Thus the fre- quency, the wave-form, the damping, etc., will be the same in both beams. Radiations scattered by such electrons should, therefore, be identical in character with the primary waves. _ Whatever type of scattering unit be assumed, it is also clear that, if the primary wave is pertectly homogeneous— 2. é., if it is an indefinitely long train of simple harmonic waves of constant frequency,—the scattered waves must also be homogeneous and of the same frequency. If, however, the scattering unit—whether a group of electrons or the individual electron—is of dimensions comparable with the wave-length of the incident radiation, theory demands that the scattering, especially at large angles, shall be less for short than for longer waves. This prediction i is confirmed by measurements of the scattering of X-rays and gamma rays over a wide range of frequencies. If the primary beam consists of very short, highly-damped pulses, or of waves of some irregular form, it may, of course, be considered as the Fourier integral of a large number of long trains of waves of different wave-lengths. Thus, unless the primary beam consists of long trains of monochromatic waves, the scattered radiation will, in general, be softer than the primary rays, and the hardness will be ereater at small than at large angles withthe incident beam. This corresponds qualitatiy ely with the properties of the secondary gamma rays *. * This explanation of the difference in hardness, as well as an ex- planation of the distribution of the intensity of the scattered gamma radiation, has been discussed in detail, for the special case of scattering by a ring electron of comparatively laree size, by the writer (Phys. Rey. xx. p. 30 (1919)). Sie WG) Prof. A. H. Compton on the While it is possible to account in this manner for the difference in penetrating power of the pr imary and secondary radiation if a sufficiently heterogeneous primary beam is postulated, it is clear that, as a result of scattering, there can be no transformation of radiation of one frequeney into radiation of another frequency. That is, the scattered rays can be no softer than the softest components of the primary rays, and removal by filtering of the softer components of the primary radiation must “harden also the secondary beam. An experimental method of determining the relative amount of scattering and fiuorescence has been applied to a study of the secondary radiation excited by the hard gamma rays from radium C. In figure 1 is shown diagrammatically the arrangement of the experiment. A source of hard Hig. 1. gamma rays § excites secondary radiation in a block R, and the intensity of this radiation is measured by an ionization chamber I, which is screened by heavy lead blocks from the direct beam of gamma rays, and which in the final experi- ments is surrounded on fcur sides by about 4 em. of Jead to keep out secondary radiation from the walls of the room. The ionization current when the radiator R is removed, is approximately balanced by the ionization current produced by an adjustable source of gamma rays S in a second chamber I'. The intensity of the secondary radiation is then measured by the difference in the readings of the electro- meter E when the radiator R is in place and when removed. The test for the presence of fluorescent radiation was made by comparing the intensity of the secondary radiation when an absorption screen was placed alternately in position A, in front of the source of gamma rays, and position B, in Oo Degradation of Gamma-Ray Energy. 75 front of the ionization chamber. Supplementary tests showed that when the primary rays entered directly into the ionization chamber, the absorption was nearly the same whether the lead screen was at A or at B*. Let us suppose that the primary beam consists of any number of components of different wave-lengths Aj, A», ..., that L,, Lb, ... are the intensities of these components in the primary beam, ¢), Co, ... the fractions of each component scattered into the ionization chamber, *,, k:. ... the fractions of the respective energies transmitted through the absorption screen when placed at A, and ky',k,’, ... the corresponding fractions when placed at B. Then it is clear that, when the absorption screen is placed at A, the intensity of the beam scattered into the ionization chamber is Tek), + ekol, +... = de,k,1,, and, when placed at B, the intensity is ae =¢k; ~- Ghee ee = deh, I. But for all wave-lengths, 4, is very nearly equal to 4,’. Hence I is nearly equa] to I’: that is, for truly scattered radiation, the observed intensity of the secondary radiation should be approximately the same whether the absorbing plate is in the positicn of A or B. If, on the other hand, the primary radiation excites in the radiator R a fluorescent radiation which is more readily absorbed than the primary rays, the observed intensity of the secondary radiation will be less when the absorption sereen is in the position B; for if £4, is the fraction of the primary radiation transmitted through the absorption screen, while &, is the corresponding transmission factor for the fluorescent radiation, the ratio I'/I of the intensity of the fluorescent radiation when the screen is at B to that when the screen is at A is obviously k,/kp. Thus the effect of any fluorescent radiation will be to make the fraction I'/I less than unity. If all the secondary radiation is of the fluorescent type, the ratio k/kp, and hence also of I'/I, should become in- definitely small as the thickness of the absorption screen is * The supplementary experiment referred to showed that for the gammia rays from radium C'filtered through 2 mm. of lead, and using an absorption screen of 1 em. of lead, the value of % was 0°57 and of 4’ was 0°52. The difference is doubtless due to the difference in the amount of secondary radiation reaching the ionization chamber in the two cases. This difterence will be relatively less important for softer radiation, but will be relatively somewhat more prominent for greater thicknesses of the screen. 754 Prof. A. H. Compton on the increased. Ifor wholly scattered radiation, as we have just seen, the value of this ratio should remain approximately unity for all thicknesses of the absorption screen. If the secondary radiation is a mixture of the two types, it will be seen that the ratio I'/I should approach, for large thicknesses of the absorption screen, the constant value jl=cligia, . where c, is the fraction of the primary beam scattered into the ionization chamber witl no absorption screen at B, and cz is the corresponding fraction for the fluorescent radiation. Thus by measuring the ratio of the intensity of the secondary radiation when suitable absorption screens are placed altern- ately in front of the ionization chamber and the source, it is possible to determine the relative magnitude of the scattered and the fluorescent radiation. The results of measurements of this ratio Alsiees different angles with the primary beam are shown in Table I. TABLE I. Thickness of Ratios I'/I for iron radiator. lead screen. 45°. 90°. foe 0 1 1 1 0°15 cm. i Res 0°45 0:5 eee 0°30 —0-02 1-0 0:52 0:13 — 0:02 2:0 0:39 0:02 30 0°26 4-1 0:20 t each angle the measurements were continued until the intensity was too low for accurate determinations of it The probable error of the final measurements of the ratio at 135° was about 0:02, at 90° about 0 03, and at 45° about 0°04. On the basis of the above discussion, we may therefore conclude that for gamma rays which have traversed several centimetres of lead the secondary radiation at angles greater than 90° is, except for the small probable error, all of the fluorescent type. At 45° it appears that the value of the ratio I’/I is approaching a constant value for large thicknesses of the absorption screen. The limiting value of this ratio would seem to be of the order of 5 or 10 per cent., which, according to expression (L), would represent appr oximately the fraction of the secondary radiation at this angle which is due to true scattering. It is clear, in view of the magnitude of the Degradation of Gamma-Ray Lnergy. (ys probable error, that such extrapolation fer large thicknesses of the screen is precarious, and the evidence for any true scattering cannot be considered conclusive. Probably at least 90 per cent. of the secondary radiation at this angle is of the fluorescent type. Though we thus find that there is very little of the pene- trating primary gamma radiation present in the secondary rays, 1t is not impossible that some of the softer components of the unfiltered primary beam may be appreciably scattered, but be so strongly absorbed that they are not detected through the lead screens employed. An upper limit to the amount of such soft scattered radiation that may be present ean be assigned in thefollowing manner. If at any specified _angle a certain screen suffices to make the ratio I’/I sensibly zero, it is clear that with this screen in position A there is no appreciable scattered radiation entering the ionization chamber, and practically all of the secondary radiation is fluorescent. The absorption coefficient for the primary rays which excite the fluorescent radiation may then be determined by measuring the decrease in ionization when additional screens are placed at A. Assuming that this absorption coefficient remains constant for -all thicknesses of the absorption screen, which experiments on the primary beam show is very nearly the case, the intensity of the fluorescent radiation when no absorption screen is employed is io) ° Or where J is the observed intensity for a screen of thickness x, and pw is the linear absorption coefficient. Of the ioe secondary radiation I) observed when no screen is employed, the traction Jo/I, at least consists of fluorescent radiation. Of the remainder, (Iyp—Jdo) )/To, a part may be fluorescent, Price the true value of fe is presumably greater for small thicknesses of the absorption screen, and the rest will xepresent the truly scattered radiation. Thus the fraction (Ip>—Jo)/Ip is an upper limit to the amount of scattered radiation which may be present in the secondary gamma rays when no absorption screens are employed. Since theoretical considerations would lead one to expect the scattering to be greater at the smaller angles, an experi- menial determination of the value of this fraction was made for the secondary radiation at 45°. In this experiment the window of the ionization chamber consisted of 0°15 em. of lead, and the same thickness of lead surrounded the source of gamma rays. This was necessary in order to cut 796 Prof. A. H. Compton on the out the beta rays. The results are shown in the following table :-— BLE [f. . iL. J/Ip- Ijin 3'l em. 0:7 O17 0-98 0:02 The values in the fourth and fifth columns are calculated from the experimental data in the first three columns. After the primary gamma rays have passed through 3:1 em. of lead, we have seen that probably not more than 10 per cent. of the secondary radiation is of the scattered type, but the intensity is so weak that the probable error of these ‘measurements is necessarily rather large. It may be con- cluded, however, that at 45° probably not as much as 15 per cent. of the whole secondary radiation consists of scattered. primary rays. At the larger angles the results given in Table I. show that if there is any appreciable scattered radiation it must be of a very soft type, and since it is unable to penetrate a centimetre of lead, it cannot be identified with the hard gamma rays from radium CU. It will be seen from this. table also that the absorption coefficient of the fluorescent radiation is rapidly approaching that of the primary rays at the smaller angles. This fact, together with geometric difficulties which prevent securing intense secondary radiation at tbe smaller angles, makes very difficult any effort to separate the scattered and fluorescent radiation at angles much smaller than 45°. The question of the presence of scattered radiation at the smaller angles will be discussed. in another paper on the basis of some experiments of a different type*. In this paper some positive evidence wilk be presented for the existence of true scattering at angles. less than 15°. ; In order to find out-how far the actual scattering falls short of that to be expected from theoretical reasoning, a measurement was made of the relative intensity of thes primary and secondary radiation at several angles. If the dimensions of the electron are negligible compared with the wave-length of the gamma rays, and if the electrons all scatter independently of each other, the usual theory T gives for the intensity of the scattered radiation Te e{(f + cas? 0) 2) aera 2G |.” Sa ee ( * Infra, p. 770. +.J. J. Thomson, oc. cit. Degradation of Gamma-Ray Energy. (ya where I is the intensity of the primary beam at the ionization: chamber when the radiator is replaced by the source of gamma rays, N is the number of electrons which are effective: a scattering, e and mare the charge and mass respectively of the electron, 0 is the angle with the primary beam at which the Po iicred beam is observed, / is the distance of the radiator from the source of gamma rays, and C is the velocity of light. Taking the number of electrons per atom as equal to the atemic number, and using the experimental values 10°3 cm. for J and 234 g. for the mass of the iron radiator, this expression gives for the ratio I)/I at 90° the value 0: 023. The experunental value of this ratio was 0°0017. But of this we have seen that less than 3 per cent. probably represents true scattering. The value of 4he ratio I,/I, where I, is the observed true scattering, is therefore less than 0-00005, only 2 per cent. of that required by theory. Similar results for the scattering at 45° and 135° are given in Table III. It will be seen from this table that at large angles, if there is any true scattering, itis probably less than a thousandth part of the amount predicted on the basis of the usual electron theory. TABLE III. i : Ts obs. Angle. aa ae Pia. Ts eale. 45° 0-015 c. 0-001 0035 c. 0:03 90° 0:0017 <0-00005 0-023 < 0-002 135° 0:0008 <0-00002 0:035 <0:0005 It is not impossible to account for this very low value of the scattering on the basis of the classical electrodynamics, if suitable assumptions are made with regard to the wave- ‘length of the primary gamma rays and the properties of the electron. ‘Thus, for example, the writer has shown else- mere that af ihe clcctnon ia’a rigid sphere which is not subject to rotational displacements by. the primary beam, the scattering at all angles becomes negligible when the ratio of the wave-length to the fone of the electron is less than about 2°4. Certain other ypes of electron give a similar result for different values fe this ratio. If this explanation is the correct one, the wave-length of these gamma rays must be considerably shorter than that of the hardest X-rays which have yet been studied, since for these rays the scattering, though somewhat smaller than that predicted by the usual theory, is apparently of the proper order of A, H. Compton, Phys. Rev. xx, p. 25 (1919). 758 - . Prof. A. H. Compton on the magnitude *. It seems premature to attempt any detailed explanation of the failure of the usual electron theory until more definite information is available with regard to the wave-length of the hard gamma rays. The Characteristics of the Fluorescent Radiation. Let us now consider the properties of the fuorescent radiation excited by the hard gamma rays. The observed relative intensities of the secondary radiation from aluminium, iron, and lead at different angles: with the primary beam, when gamma rays from radium C filtered through a centi- metre of lead are employed, are shown in Table LV. & | Tasne LV. Secondary Angle : LACIALOR I eo eee Doe eGo. lego 90°. 120°) 13522 ote NN eeee eee ae 6:2 4:0 ie7/ (1:0) O-7 0-4 0:3 Wee: 10 76 4°6 2:0 (1:0) 0-6 0-5 0-4 Phinces 11 6°8 4:3 2°2 (1:0) 0-8 ree 07 In comparing the intensity of the scattered beam at any two angles, 0, and @., the effect of absorption was eliminated as completely as possible by the well-known method of placing the radiating plate with its normal at an angle (0,+6,)/2 with the primary beam. - In this case the absorption is the same at the two angles at which the secondary radiation is compared. Since the window of the ionization chamber consisted of 0:15 cm. of lead, any soft fluorescent radiation was strongly absor bed. Data similar to those given in this table have been published by Florance f and Kohlrausch f, except that in the present case the effect of absorption by the radiator has been largely eliminated, and the seh beam was rendered homogeneous by filtering through a suitable lead sereen §. * Cf. e.g. Hull & Rice, Phys. Rev. viii. p. 326 (1916). Barkdaw& White, Phil. Mas. xxxiv. p. 277 (1917). + D. C. H: Florance, Phil: Mae. xx. p. 921 (1910). t K. W. F. Kohlrausch, doe, cit. § It should be noted that on account of the differing hardness of the secondary radiation at different angles, the relative intensity observed at a given angle depends upon the fraction of the radiation absorbed by the ionization chamber. The ionization chamber used in the present experiments was so designed that it absorbed a large part of even the hard primary rays. This probably, accounts for the fact that the writer's experiments show relatively more intense radiation at the smaller angles where the secondary rays are hard, than do the experiments of Florance and Kohlrausch, whose ionization chambers presumably absorbed only a small fraction of the incident radiation. 4 Degradation of Gamma-Ray Energy. 4799 In common with these experimenters, it is found that the secondary radiation, which in the writer’s work censisted almost wholly of fluorescent radiation, is very much more intense at small angles than at large angles with the incident gamma rays. The relative amount of fluorescent radiation excited in different substances per unit mass is shown in Table V. ANB Ey Vo ‘Awol Relative fluorescence per unit mass: 8 Paraffin. Al. Fe. Sn. Pb. 135° iLcylDs 1-04 (1:00) 0-78 O74 45° lei 0:9 (1:6) 0-8 0:9 gs at 135° were taken for primary rays filtered through 0°5 cm. of lead, and those at 45° were with a 4°1 cm. lead filter. Thus it was made certain that practically all of the secondary radiation was of the fluorescent type. Suticiently thin plates of the radiating materials were employed that the necessary corrections for the absorption of the primary and secondary radiation in the radiator were not large. The values here given therefore represent the amount of the fluorescent radiation excited in unit mass of the different radiators, which penetrates the 0-15 em. lead window of the ionization chamber. It will be seen that the values do not differ greatly over a wide range of atomie weights. The constancy is even more marked when the fluorescence per electron is calculated by multiplying each of the above values by the ratio (atomic weight)/(atomie number), as is done in the following table. Atthe angle 135° the constancy The reading Tas.e VI. MAE Relative fluorescence per electron : Paraffin, Al. Fe. Sn. Eb: 135° 0-91 101 (1:00) 0-87 0°87 45° 1-4 0:9 GLO) yee 0:9 Le of these values is somewhat accidental, since, as we shall see, the absorption coefficient of the fluorescent radiation at this angle is considerably greater for the radiation from the light than for that from the heavy elements. At 45°, how- ever, the hardness of the fluorescent radiation 1s practically the same for the different radiators, so the constaney of the values at this angle is of real significance. The result expressed by this table is confirmed ‘by the more quantitative experiments of Ishino *, who found that the magnitude of * M. Ishino, loc. cit. 760 Prof, A. H. Compton on the the total secondary radiation (which includes any scattered rays that may be present) per atom is more nearly pro- portional to the atomic number than to the atomic weight. The amount of the fluorescent radiation excited is therefore approximately proportional to the number of electrons. traversed by the primary gamma rays. It will be seen on examining Table I. that the Auk escent radiation at small angles with the primary beam is con- siderably harder than that at right angles. This matter was. examined in greater detail for a number of different elements, with the results shown in Table VII. Care was taken in these experiments also to eliminate any possible soft scattered. Tasue VII. Mass absorption coefficients in Lead of the eae fluorescent radiation excited in different Bes materials by hard gamma rays: Paraflin. Al. Fe. Sn. Pb. 45° 0-10 0°10 O1l 0:09 0:05 (?) 90° 4 ADE 0-21 ies 422 135° 0-78 0-50 0°50 0:32 0-15 - radiation by interposing o suitable absorption screens between the source of gamma rays and the radiating material. Asa result, the intensity of “the secondary radiation was so low that the values of the absorption coefficients obtained can be considered only approximate. The data suffice to show, liowever, that while the radiation from all substances is harder a small angles, the difference is less for elements of high atomic weight, so that whereas at large angles the radiation from the heavier elements is considerably more: penetrating, at 45° the hardness differs but little from element to element *. Interesting information is obtained on examining the absorption coefficients of this penetrating fluorescent radiation in various materials. This was done for the secondary radiation from iron at 135° after the primary gamma rays. had been filtered through 0°5 em. of lead, aon the results shown in the following table. With a “inate geometric TasLe VIII. Mass absorption coefficients in different elements of the fluorescemt radiation at 135° excited in iron by hard gamma rays from radium C: Fb. Sn. Fe. ; Al. 0125) 0 0-18 0:08 0:07 * The constancy observed for different elements at 45° is confirmed by” the measurements of Florance, Phil. Mag. xx. p. 935 (1910). Degradation of Gamma-Ray Energy. 161 arrangement, the mass absorption coefiicient of the primary rays in lead was 0° 062, which presumably means that this fluorescent radiation is of very appreciably longer wave- length than the primary g gamma rays. On the other hand, the. experiments of Hull “and Rice * show that X- rays of wave-leneth 0:122x 10-§ em. have a mass absorption coefficient in lead of about 3°0, which indicates that even the softest part of this fluorescent radiation is of shorter wave-length than the critical wave-length 0'-147 x 107° em. required to excite the characteristic K radiation in lead. This conclusion is confirmed by the fact that the mass absorption of this fluorescent radiation is greater in lead than in tin, which is the reverse of the case for wave-lengths between ihe IG radiamont teomelead and the radiation from tin. ‘There can thus be no question but that the fluorescent rays under examination are of a distinctly harder type than the characteristic K radiation from even the heaviest elements. In an experiment with a Coolidge tube operated by an induction coilat a maximum potential of 196,000 volts, Rutherford has obtained X-rays whose mass absorption co- efficient in lead is as lowas 0:75 7. This is practically the same as the value observed for the fluorescent gamma adiation from paraftin at 135° (Table VII.). According to the quantum relation, hy=eV, the wave-length in Rutherford’s ee must have been greater than 0:063 A.U. ‘The ave-length of the softest part of this penetrating fluorescent eeation must therefore lie between 0:06 and 0°12 A.U. Tt is interesting to note that these secondary g oamima rays bridge the gap which has existed between the hardest X- rays and the ver y penetrating gamma rays; for as we have just seen, the softest part of “this seeondar y radiation falls within the wave-length of the hardest X-rays, while Table VII. shows that at small angles it is nearly as penetrating as the oO hard gamma rays from radium C. The Origin of the Fluorescent Radiation. Although the secondary gamma radiation under exami- nation seems, without doubt, to be fluorescent in nature, it differs in several important respects from the characteristic fluorescent K and L radiations excited in matter when traversed by hard X-rays. In the first place, whereas these characteristic radiations differ greatly in hardness from * Hull & Rice, loc, cit. } E. Rutherford, Phil, Mag. xxxiv. p. 153 (1917). 762 Prof. A, H. Compton on the element to element, the secondary gamma rays, especially at small angles with the incident beam, are of nearly the same hardness over a wide range of atomic numbers. And in the second place, while the characteristic radiations are found to be distributed uniformly with regard to intensity and quality at all angles with the primary beam, the fluorescent gamma rays show marked asymmetry in both quantity and quality in the forward and reverse directions. There is therefore good reason to suppose that the oscil- lators which give rise to this fluorescent radiation are radically different in character from those which are responsible for the K, L, and M characteristic radiations *. An explanation of the origin of the fluorescent radiation which appears to be satisfactory is that the high-speed secondary beta particles liberated in the radiator by the primary gamma rays excite the secondary gamma rays as they traverse the matter of the radiator. On this view the fluorescent gamma rays should be identical in character with the so-called ‘* white’’ radiation excited in the target of an X-ray tube by the impact of the cathode particles. Hxperi- ments have shown that when cathode rays or beta rays strike a target which is so thin that the particles are not greatly scattered and in which no considerable amount of characteristic radiation is excited, the X-rays emitted are more intense and harder in the general direction of the cathode ray beam than in the reverse directiont. This asymmetry is of the same kind as that observed for the secondary gamma rays, and though not so marked, is found to increase with the speed of the impinging electrons. For speeds comparable with those of fast beta rays the asymmetry may well become as great as that observed in the present experiments, But it is also known that the beta rays liberated by gamma rays are much more intense in the direction of the gamma ray beam than in the reverse * The idea suggested itself that the secondary radiation which was being studied was a fluorescent radiation excited in the lead screens which surrounded the source of gamma rays, this fluorescent radiation being in turn scattered by the radiator into the ionization chamber. It is obvious that such a radiation would not be eliminated by placing additional lead screens over the source, while the ionization would he considerably reduced by placing screens over the ionization chamber. Considerations of the energy involved and of the characteristics of the secondary radiation rendered this suggestion improbable, but the possibility was definitely eliminated by removing all the lead screens and replacing them with iron. The phenomenon in this case was identical with that when lead screens were employed. + G. W. C. Kaye, Proc. Camb. Phil. Soc. xv. p. 269 (1909). J. A. Gray, loc. cit. . Pa) ee a Degradation of Gamma-Ray Energy. 763: direction. Indeed, “‘the results indicate that the beta particles initially escape in the direction of the gamma rays,. and with the same speed for all kinds of matter” *. Thus the hypothesis that the fluorescent gamma radiation is due to the impact of the secondary beta particles accounts quali- tatively for the observed asymmetry in the hardness and intensity of the secondary gamma rays. With regard to the relative intensity of the fluorescence excited in different materials, attention may be called to the fact that the number of the beta particles excited by hard gamma rays is approximately the same per electron in different elements t. There is a somewhat larger number produced in the very heavy elements such as mercury and lead, which appears to be connected with the excitation of the characteristic K radiation in these elements. But since. such radiation is too soft to have an appreciable effect in the present investigation, it seems probable that the number of effective beta particles excited in these elements does not differ greatly from that for the lighter elements. If, there- fore, the simple assumption is made that the amount of secondary gamma rays excited depends only upon the’ number of electrons traversed by the secondary beta particles, our hypothesis gives a satisfactory account of the fact that the amount of secondary gamma radiation per electron is practically the same for all elements. The greater scattering of beta particles by elements of high atomic weight means that in these elements a relatively larger number of the beta rays move in a direction opposed to the primary beam. Tor this reason we should expect, as Table VII. snows is actually the case, that at large angles with the incident beam the fluorescent radiation from the heavier elements will be more penetrating than that from the light ones. An estimate of the relative energy in the secondary rays can be obtained by integrating over the surface of a sphere the observed relative intensity of the scattered beam «at various angles. A rough summation of this kind, using the data of Tables IT]. and IV. and extrapolating by the help of Kohlrausch’s data for the very small angles, shows that the ratio of the energy (as measured by the ionization) of the secondary radiation from iron to the total energy absorbed from the primary beam is about 0°69. This result is in good agreement with Ishino’s estimate that the “ scattering ” by iron accounts for 62 per cent. of the total absorption. A * EK. Rutherford, ‘ Radioactive Substances, ete.,’ p. 276, + Eve, Phil. Mag. xvii. p. 275 (1909). 764 Prof, A. H. Compton on the correction must be applied to this value to make allowance for the fact that while the greater part of the secondary radiation which enters the fouiaacion is absorbed, in the present experiment only about half of the primary beam was thus absorbed. Taking this correction factor to be about -O'7, we find that approximately 50 per cent. of the absorbed prareny rays is transtormed into radiation of sufficiently high frequency to penetrate 0°15 cm. of lead”. The efficiency of transformation of the energy 1s therefore of a much, higher order than that observed in an X-ray tube operating at usual potentials, in which case not as much as 1 per cent. of the energy of the cathode rays appears as X-rays. It is possibie that this difference is to be eccounted for by an excitation of gamma rays when the secondary beta particles are liberated in addition to that produced when ‘they collide with other electrons. A question of great theoretical importance is—What kind -of oscillator can give rise to radiation which not only is more intense in one direction than in another, but also differs ‘in wave-length in different directions ? Since the secondary ‘radiation differs in frequency from the primary rays, it would seem impossible to invoke any interference between the radiation from the different oscillators to account for ‘this phenomenon. Such an explanation is rendered the more difficult, by the fact that to explain the different hard- ‘ness of the rays in different directions, oscillators of different frequencies would have to be present, between which there -.gould be no fixed phase relations. An obvious means of accounting for the observed phenomenon is to suppose that the radiator which gives rise to the secondary rays is moving at high speed in the direction of the primary beam. ik this case, both the intensity and the frequency of the fluorescent radiation will be greater in the forward than in the reverse direction, as is demanded by the experiments. A rigid calculation of the relative intensity of the fluores- cent radiation at different angles, according to this hypothesis, is not at present possible, hecaues the scattering of the beta particles results in an irregular distribution of their velocities. It will nevertheless be fate uctive to consider the relative energy radiated in different directions by an_ oscillator moving at a speed comparable with that of light. It can be * The estimate here made of the efficiency of transformation is, of course, based upon the assumption that unit energy of one frequency produces the same total number of ions as unit energy of another frequency. Though this assumption has not been tested over the ran: ze of frequencies mere considered, it does not appear probable that any _ error thus introduced can change the order of magnitude of the result. Degradation of Gamma-Ray Energy. 765 shown that for an electron whose acceleration is unpolarized relative to the observer*, and which is travelling at a velocity BC, the mean square of the electric vector at a great distance 7 is T 2p. aon, e a 167?C*r? : (1— 8cos@)?—1(1—?) sin?6 + 1 cos A(28-- cos 0 — B’ cos @) (1—8 cos 6)§ where I” is the mean square of the acceleration relative to the observer at the moment the pulse under observation le!t the electron, e is the charge of the oscillator, C is the velocity of light, and @ is the angle between the direction of motion of the particle and the observed beam. Thus the ratio of the intensity of the radiation at an angle 6, to that at an angle @, is given by the expression eae I, me { 1—Bcos0, \ *(1—B8eos0,)?—4(1—8’)sin70, Is 1—8cos6,) (1—8cos@,)?—4(1— ?) sin? d, +4 co0s0,(28 — cos6,— 8? cos 0; ) + 4+ cos 0,(28— cos@,— B? cos 6.) (4) Assuming that all the radiating particles are moving in the same direction, the ratio of the intensity of the fluores- cent radiation at the angle 45° to that at 135° has been calculated from this expression, with the results shown in figure 2. It will be seen that it is possible on this view to account for any reasonable degree of asymmetry of the secondary radiation. In the case of paraffin, in which the least scattering of the beta particles occurs, the observed ratio of the intensities at 45° and 135° was about 20. In addition to the effect of the scattering of the secondary beta rays, experimental errors arise because much of the soft radiation at 135° is absorbed before it enters the ionization chamber, while a considerable part of the hard radiation at 45° traverses the ionization chamber without being absorbed. The rapid increase of R with @, however, makes it reasonably certain, on the present view, that the average speed of the oscillators which emit the secondary gamma radiation does not differ greatly from half the speed of light. * Of course such an oscillator will not be unpolarized relative to an observer moving with it. A slight polarization will not, however, make any great difference in the value of the ratio (4). t+ The values of the three components of the electric vector from which this expression is derived may be found, e,g. in O, W. Richardson's ‘Electron Theory,’ p. 256. Phil. Mag. 8. 6. Vol. 41. No. 245. May 1921. 3 E 766 | Prof. A. H. Compton on the Since the speed of even the swiftest alpha particles is only about one-tenth that of light, it is clear that the radiating particles cannot have mass comparable with atoms, but must 45 (— 4500 40 | . i) 3500 30 3000 L45° I) 35° 25) 2500 20 2000 15+ 1500 10 1/000 5 500 0 0.2 OfP 0.6 0.8 1.0 B be individual electrons. Weare thus led to the idea that it is the vibrations of the secondary beta particles themselves which give rise to the fluorescent gamma rays”. * The corresponding hypothesis that the “ white radiation ” from an X-ray tube is due to vibrations of the cathode particles has been sug- gested on the basis of similar considerations by D. L. Webster, Phys. Rey. xiii. pp. 308-305 (1919). Degradation of Gamma-Ray Energy. 767 Substituting the value 8=0°5 in equation (4), we can ealculate the relative scattering to be expected at various angles. The result is shown in the solid curve of figure 3. Fig. 3. nae ies ~ Ss Curves showing intensity at different angles with motion of oscillatcr whose velocity is BC. Solid circles writer’s, open circles Kohlrausch’s values of relative intensity of gamma rays. The solid circles in this figure represent the writer’s obser- vations on aluminium, as given in Table IV. The open circles at the small angles show the results of Kohlrausch referred to the writer’s value at 45°. It is not impossible 3 EK 2 768 Prof. A. H. Compton on the that the large amount of secondary radiation at these small angles is due in part to the presence of some true scattering. However this may be, the generally satisfactory form of the theoretical curve suggests that we are working along the right line. If this view of the origin of the fluorescent radiation is the correct one, we are supplied with a means of estimating roughly the wave-length of the primary gamma rays. It has been shown above that the wave-length of the softest part of the fluorescent radiation lies between 0-06 and 0-12 A.U., and probably nearer the former. But according to the Doppler principle, if the oscillators producing the fluorescent radiation are moving in the direction of the primary rays, the ratio of the wave-length at an angle @, to that at an angle @, is dm 1—B cos, 1, 1 uee8,) (5) Thus, if we take A, to be about 0:08 A.U. at 6,=135°, and the value of B to be 0°5, the wave-length of the penetrating fluorescent radiation at 45° is about 0-04 A.U. ‘This result does not vary greatly with different values of 8 ; but the extreme hardness of the fluorescent radiation at 45° indicates that it is more nearly similar to the primary rays than to the soft secondary radiation which appears at the larger angles. Thus we shall probably not be far wrong in assigning a value 0:02 to 0:03 A.U. as the wave-length of the most effective part of the hard gamma rays from radium C. This result is not in disaccord with the calcu- lations of Rutherford * based upon the quantum hypothesis. Summary. The principal experimental results of this investigation may be summarized as follows :— | By far the greater part of the secondary gamma radiation from matter traversed by the hard gamma rays from radium C is fluorescent in nature. If any truly scattered radiation is present, at 45° it probably amounts to less than 15 per cent., and for angles greater than 90° to less than 3 per cent. of the secondary rays. * E. Rutherford, Phil. Mag. xxxiv. p. 153 (1917). According to the quantum relation, an electron must have a velocity 8=0'8 in order to excite radiation of wave-length 0:04 A.U. On the present view, there- fore, the radiating beta particle must already have lost a large part of its energy of translation. Degradation of Gamma-Ray Energy. 769 At large angles with the primary beam the scattered energy is probably less than 0-001 of that required by the usual electron theory. The secondary fluorescent radiation is found, in accord with observations by others on the whole secondary radiation, to be harder and more intense at small angles with the incident beam than at large angles, and tables are given showing the manner of this ‘variation. While at large angles the radiation from heavy elements is somewhat more penetrating than that from the light elements, at small angles both the hardness and the intensity of the fluorescent radiation are approximately the same from elements covering a wide range of atomic numbers. A study of the absorption coefficients of this radiation in various elements shows that the softest parts of it, though of shorter wave-length than the K radiation from lead, are not harder than the most penetrating X-rays. The hardest parts approach in penetrating power the primary gamma rays from radium It is pointed out that the very small scattering observed is not incompatible with the classical electrodynamics, if the wave-length of the gamma rays and the diameter of the electron are of the same order of magnitude. A satisfactory qualitative explanation of the observed fluorescent radiation is found in the gamma rays produced by the impact of the secondary beta particles liberated in the radiator by the primary gamma rays. The observed asymmetry in the intensity and hardness may be accounted for if the oscillators which give rise to the fluorescent radiation are electrons moving in the direction of the primary beam with about half the speed of light. The wave-length of the softest part of the observed fluorescent radiation is shown to lie between 0°06 and 0:12 A.U., probably nearer the former value, while the wave- length iof the hardest part is probably about half as great. _ By a comparison of absorption coefficients, the effective wave-length of the hard gamma rays from radium is estimated as about 2 or 3x 107° em. The writer performed these experiments at the Cavendish Laboratory as National Research Fellow in Physics. He desires to express his appreciation of the interest. which Professor Rutherford has shown in the work. Washington University, St. Louis, September 24th, 1920, Cre Oa UXIX. The Wave-Length of Hard Gamma Rays. By ARTHUR H. Compton, Ph.D., Physics Laboratory, ee thgton University *. ae only recorded attempt to measure directly the wave- length of hard gamma’ rays is apparently that of Rutherford and Andrade+, using the method of reflexion from a crystal of rock-salt. In these experiments spectrum lines were observed at angles as small as about 44 minutes, _ corresponding to a wave-length of about 0:07 A.U. It was thought that this line, as well as one of wave-length 0°10 A.U., could be detected through a 6-millimetre screen of lead, which would make it appear that these lines represent the hard gamma rays from radium ©. Professor Rutherford informs me, however, that the appearance of these lines through the lead screen was doubtful. His more recent measurements of the absorption of X-rays of very high frequency { have indicated rather that radiation, whose wave- length is about 0:08 A.U., has an absorption coefficient i in lead that.is very much gr eater than that of the hard gamma raysfrom radium. ‘Thus, while the crv stal reflexion measure- ments show that radium gives. off gamma rays of wave- lengths 0°07 A.U. and longer, the very penetrating radiation which it emits probably has a much shorter wave-length. Various lines of theoretical reasoning suggest. that there are in hard gamma rays components ranging in wave-length from 0:01 to 0-04 A. U. Rutherford has pointed out § that radium C gives off beta rays with an energy corresponding to a fall through from 5 to 20x10° volts. According to the quantum relation, hv=cV, the limiting wave-length produced by the slower of these electrons would be about 0:03 A.U., while that due to the fastest ones would be as dha ag DOO. Te he second place, using an absorption formula which is satisfactory for hard X-rays of known wave-length, itis found || by extrapolation that the absorption coefficient of hard gamma rays corresponds to a wave-length of about 0:04A.U. And finally, knowing approximately the * Communicated by Prof. Sir E. Rutherford. + Rutherford and Andrade, Phil. Mag. xxviii. p. 263 (1914). J EH. Rutherford, Phil. Mag. xxxiv. p. 153 (1917). § Ibid. || A. H. Compton, Washington University Studies,’ Scientific Series, Jan. 1921. The Wave-Length of Hard Gamma Rays. rial wave-length of the “incident” secondary gamma radiation, and calculating from this the wave-length of the “emergent” secondary radiation on the hypothesis that the difference in wave-length is a Doppler effect due to motion of the particles emitting the secondary radiation, the wave-length of the primary gamma rays can be estimated, since the absorption coefficient of the primary and the “emergent” secondary radiation is nearly the same. This method leads to a value of between 0:02 and 0:03 A.U. for the effective wave-length of the hard gamma rays from radium *. In the present paper anew method of measuring the wave- length of high frequency radiation will be proposed, and the method will be applied to the determination of the wave- length of gamma rays. Instead of studying the spectrum lines reflected by a grating composed of regdlarly arranged atoms in a crystal, this method consists in observing the diffraction pattern due to the individual atoms. To consider an optical analogy, if the reflexion of X-rays from a crystal is compared with the spectrum from a ruled grating, the method of atomic diffraction corresponds to a study of the diffraction pattern due to a large number of parallel lines ruled at random distances. The distance between the different order lines in the spectrum is determined by the grating space between the lines ruled on the grating, while the distance between the bands of the diffraction pattern is determined by the breadth of the individual lines. The advantage of the method as applied to gamma-ray measure- ments lies in the fact that the effective diameter of the atom is much smaller than the distance between two atoms in a crystal, go that the effective width of the diffraction band is much greater than the distance between two spectrum lines. Thus, whereas the spectrum of hard gamma rays from a erystal grating would have to be studied at angles less than 1/2 degree, atomic diffraction measurements may be made at angles in the neighbourhood of 10 degrees. In order to use the method quantitatively, it is of course necessary to know the effective diameter of the atom. This may be determined, in a manner that will be described below, by measurements with X-rays of known wave-length. Debye has shown that if an atom is composed of N electrons, and if at any instant the distance between the mth and the nth electron 1s syn, the probable intensity of the X-rays * A. H. Compton, Phil. Mag. supra, p. 749. 12 Prof. A. H. Compton on the scattered at an angle @ with the primary beam whose intensity 1s) hast. Hee a sin gi NON sin ine Onn ») 1 ee ae zon Sm SU Xu 2 where X is the wave-length and J, is the intensity of the rays scattered by a single electron. This expression supposes that the forces holding the electronsin position are negligible in comparison with the forces due to the traversing radiation— an hypothesis supported by experiments on the seattering of X-rays. If py,.dsis the probability that the distance spn will lie between s ands +ds, the average value of the intensity for all possible arrangements of the electrons in the atoms is, ? Sy sume) 2 x Np sin} we AT Sma = Sma ORNGET 7 ILS ALMA Ty BOLO y 2 tk > =I sin 6/2 met hia X : mn or 2 =F (N, p, sin 6/2/r). 1 Since for any particular atom the quantities N and p remain constant, for an atom of atomic number N this ratio may be written, I, NG If the quantity sin (6/2)/d is sufficiently large, it will be seen that co-operation in the scattering by different electrons will be almost wholly a matter of chance, and the “ excess scattering ” function x will become practically unity. On the other hand, for very small values of this quantity co- operation between the electrons will be almost complete and the value of the function W will approach N. For inter- mediate values of sin (6/2)/X the function will have a different value for every atom, since for no two atoms will the probabilities pn, be identical. The experimental values of ar=I,/I,.N for different materials, as measured by Barkla and Dunlop +, together with the values calculated for certain = bx (sin aN * P. Debye, Ann. d. Phys. xlvi. p. 809 (1915). + Barkia and Dunlop, Phil. Mag. xxxvii. p. 222 (1916). Wave-Length of Hard Gamma Rays. 173 arbitrary arrangements of the electrons in the atoms of the different elements *, are shown infig.1. The measurements for the different wave-lengths were all made at an angle g— 90°. Let us suppose that according to these experiments the ratio of the value of wy for lead to its value for copper is R when sin (6/2)/A=c. Then, if for some unknown wave. length 2’ the value of this ratio becomes R at an angle 6’, it * Cf. A. H. Compton, Phil. Mag. (soon to be published), 774. Prof. A. H. Compton on the is clear that sin (6'/2)/A'=c=sin (6/2)/d, whence ese) 2 (2) SANE Vk Thus, if it is possible to find an angle at which gamma rays scattered from lead and copper are in the same ratio as the X-rays scattered at 90° in Barkla and Dunlop’s experiments, we have the data necessary to calculate the effective wave- length of the gamma rays. Scattering of Hard Gamma Rays at Small Angles.—An element of uncertainty is introduced into the application of this method of determining the wave-length of gamma rays by the fact that recent measurements have shown* that only a very small part, if any, of the secondary gamma’ rays observed at large angles with the primary beam is truly scattered radiation. At an angle of 45° these measurements indicated that perhaps 5 or 10 per cent. of the secondary radiation consisted of scattered primary rays, though the absorption coefficient of the fluorescent secondary radiation was so nearly the same as that of the primary rays that it was not possible to establish with certainty the existence of any scattered rays. With the hope of placing the present wave-length experiments on a more certain footing, careful examination of the character of the secondary radiation at 22°°5 was made, using, with some refinements, the same general method as that employed in the earlier experiments. It was found that, though for small thicknesses of the ab- sorption screen the absorption coefficient of the secondary radiation differed by only about 8 per cent. from that of the primary beam, even after traversing 5°6-cm. of lead the two absorption coefficients were still measurably different. Thus at least 50 per cent. of the radiation at this angle is certainly fluorescent in nature, but the fluorescent radiation is absorbed at so nearly the same rate as the primary rays that it was impossible to decide what part of the remaining 50 per cent. was fluorescent and what part might be scattered radiation. The experiments are, however, consistent with the view that for angles smaller than 30° with the primary beam a con- siderable portion of the secondary gamma radiation consists of truly scattered radiation. Several investigators have found that at angles greater than 30° the ratio of the secondary radiation from one element * A, H. Compton, supra, p. 749. Wave-Length of Hard Gamma Rays. 175 to that from another is practically independent of the angle * An examination was therefore made of the relative intensity of the secondary radiation from copper and lead at angles less than 30°. The experimental arrangement is shown in fig. 2. A strong source of hard gamma raysS (usually about Fig. 2. 100 millicuries) is placed at a point on the axis of the ionization chamber I, anda lead cylinder C is placed between to cut off the primary gamma rays. The sample of copper or lead under acu innate | isin the form of a ring R supported coaxially on the lead cylinder bya piece of aerclanel It is clear that all parts of the ring will scatter gamma rays into the ionization chamber at approximately the same angle. In order to secure the greatest possible intensity, the dimen- sions of the apparatus were so adjusted that the maximum differences in the angle @ made with the primary beam by the secondary rays entering the ionization chamber were between 6/2 and 30/2. By this arrangement ionization due to the secondary radiation from the ring R ean be obtained which is very considerably greater than that due to stray rays from.the walls of the room, ete. Jt was found con- venient, however, to balance ine stray radiation against econ pr aaticed by a small source 8’ of 2 gamma rays In a second chamber I’. The secondary Tahation was then measured by the difference in the readings of the electro- meter when the radiator R was in place and when removed. The results of the experiments are shown in the followi ing table. The values of the observed ratios of the intensity of eECy. nage Nia ishing. Phil, Maey xxxii. p, 140 (1914); I. Wok. FeoMliauech: Phys. Zeitschr. xxi. p. 193 (1920); A. H. Compton, sepra, p. 749. 776 Prof. A. H. Compton on the the secondary rays from lead to that from copper, shown in the second column, are the averages of large numbers of readings :— TABLE I. Observed Ratio. Relative Angle. ___ Intensity for Pb. | Intensity Intensity for Cu. per electron. BOS Pe eT, -605+-012 86+ 02 FAN epee i ih ‘601 -- 009 85+ 01 DD ae ee ee ete sn 638+ 008 89+ 01 TO ecto eee es 67742 023 "95 +:03 In the third column the observed intensities are corrected for the difference in absorption of the rays in traversing the lead and copper rings, and the ratio of the relative intensity per electron is calculated, taking the number of electrons per atom as equal to the atomic number. lt will be seen that while at all angles at which measurements were made the value of this ratio is slightly less than the theoretical value unity, the ratio shows a tendency to increase at the smaller angles. It is unfortunate that at still smaller angles the energy of the secondary radiation was so low that no satis- factory measurements could be made. It may be mentioned that Kohlrausch * has recently made measurements similar to these at angles as low as 10°, and that his measurements do not show this tendency for the intensity of the secondary rays ?from lead to inerease at the small angles more rapidly than that from the lighter elements. While Kohlrausch used several times as strong a source of gamma rays as that employed in these experiments, his apparatus was not specially adapted to taking measure- ments at small angles, and it would appear that his probable error at these angles was greater than that of the present measurements. In view of the consistency of the experiments here recorded, it appears probable that the observed increase in the ratio at 10° is not the result of chance. : The Wave-Length of the Gamma Rays.—A comparison of the values given in Table I. with the data in fig. 1 shows that the present experiments have not been carried to angles sufficiently small to give values in Ip,/Ica overlapping Barkla * KX. W. PF. Kohlrausch. loc. cit. : = Wave-Length of Hard Gamma Rays. (ae and Dunlop’s experimental values for the scattering of X-rays at 90°. Since their measurements included wave- fo} lengths as short as 03 A.U., we may conclude from equa- tion (2) that the effective wave-length of the gamma rays here employed is less than sin 5° sim 45° 00a 7 A UE If Barkla and Dunlop’s results are extrapolated according to the theoretical curve for lead shown in fig. 1, it will be seen that the ratio Ip»/Icu at 90° begins to increase appreciably for wave-lengths in the neighbourhood of +2 to °25 PANY from Table I. we take 10° as the angle at which the ratio begins to increase for gamma rays, the effective wave-length of these waves is by equation (2) between 0:025.and 0:030A.U. According to the present experiment: this may therefore be taken as the approximate wave-length of hard gamma rays which have traversed about 8 mm. of lead. The three principal elements of uncertainty which enter into this calculation are: (1) the wave-length 0:2 A.U. at 90° is an extrapolated value, (2) a possible error in the experi- ments, and (3) the lack of positive evidence that the radiation measured in these experiments contains an appreciablefraction of truly scattered rays. None of these difficulties seem sufficiently serious to render improbable the correctness of the result as to order of magnitude. Indeed this methed of estimating the wave-length of hard gamma rays is perhaps the most direct one that has been employed, and in as far as its results are In agreement with the predicted values con- sidered at the beginning ot the paper, they may be taken as a support of the theoretical bases of these predictions. The writer performed this experiment at Cavendish Laboratory as National Research Fellow. He desires to thank Professor Sir H. Rutherford for the free use of the laboratory facilities and for valuable suggestions with regard to the experimental procedure. Washington University, Saint Louis, U.S.A. December 1, 1920. hs | LXX. Dissociation of Hydrogen and Nitrogen by Electron Impacts. By A. Lu. Hucues, D.Sc., Research Professor of Physics, Queen’s University, Kingston, Canada*. HYDROGEN. ANGMUIR f found that hydrogen could be dissociated by contact with a tungsten (or other metallic) wire when its temperature was raised above 1300°K. He cal- culated from his experiments that the work necessary to dissociate a gram molecule of hydrogen was 84,000 calories. This may be expressed in terms of the energy necessary to dissociate a single molecule. It is the energy which an electron would acquire in falling through a potential dif- ference of 3°6 volts. It occurred to the writer that possibly the molecules could be dissociated by direct impact of electrons possessing energy in excess of that corresponding to 3°6 volts. The experiment failed to show any appreciable dissociation by electrons whose energy corresponded to 3°6 volts. It, how- ever, led to an investigation of the disappearance of hydrogen when subjected to bombardment by electrons possessing higher energies. A number of investigations have from time to time been carried out on the reduction in pressure when an electric discharge is passed through a gas. Insome respects, the central idea of the present investigation is new. The experimental conditions are simplified so as to give as direct information as possible as to the ratio of the number of molecules disappearing from the gas in terms of the number of collisions between electrons and molecules, for different values of the electron velocities. A stream of electrons is passed through hydrogen and the progressive decrease in pressure noted. It is assumed, for reasons discussed later, that the hydrogen which disappears, does so, because it is dissociated into atoms which condense on the walls when they strike them. Apparatus.—The final form of apparatus used is shown in fig. 1. The earlier experiments showed the necessity for trapping the atomic hydrogen as completely as possible. This was secured (as in some of Langmuir’s experiments) by keeping the experimental tube in liquid air throughout a set of observations. This called for a source of electrons * Communicated by the Author. + Langmuir, Journ. Amer. Chem, Soc. xxxvil. p. 451 (1915), Dissociation of Hydrogen and Nitrogen. 7179 which would give out as little heat as possible in order to conserve liquid air. The experimental tube EH was a glass tube 2 cm. in diameter internally. The source of electrons was a platinum strip (IF) 1 cm. long and 1 mm. wide, coated Fig. 1. To Mcleod gauge Fd tube | On 5; Sez (PH — To UI diffusion pump Acce/. potential Tm . TT 77 C with a mixture of BaO and SrO. The anode was a nickel gauze (40 mesh to the inch) which fitted snugly inside the glass tube. The advantage of this type of anode was that there were no large areas of glass surface to become charged up and give erratic results, as so often occurs when electrons and ions have an opportunity to lodge on such areas. The number of electrons available was controlled by a rheostat, while the energy of the electrons was determined by the accelerating potential applied between the gauze and the filament. The experimental tube was connected to a mer- cury trap and a diffusion pump through a U-tube. The 780 Prof. A. Ll. Hughes on D%ssociation of ressure was measured by a Mcleod gauge. Hydrogen was admitted through a palladium tube. ‘The apparatus was thoroughly outgassed by prolonged pumping and heating to 400° C. When systematic observations were carried out, the apparatus was subjected to an outgassing for 30 minutes at 400° C. between each run, the filament being heated to a white heat, and the nickel gauze being bombarded by elec- trons. The runstabulated below were obtained after several weeks’ preliminary testing, during which there had been many short heat treatments with the pump in action, and during which no gas had been admitted except hydrogen up to pressures of from ‘001 to *1 mm. It is extremely unlikely therefore that there could be any impurity in the hydrogen. To keep mereury out of the experimental tube EH, it was customary, after heating both K and U, to surround the U-tube with liquid air and to continue the heating and pumping for some minutes to ensure that no appreciable amount of mercury was left in KE. Method of Experiment.—Before admitting the hydrogen, a test was always made (1) on the amount of gas given out when the filament was heated at the temperature to be used in experiment on the hydrogen, and (2) on the amount of gas given out when electrons were driven across the tube. (1) was in general not measurable, and (2) was usually very small compared with the pressure changes observed when the hydrogen was in the apparatus. Then hydrogen was admitted to approximately the pressure desired. A test was now made for the amount of hydrogen cleaned up by the hot filament—-the purely thermal effect discovered by Langmuir. It was found that, at any temperature above a dull red heat, there was a very appreciable clean up. The experiments on the clean up due to the electron stream had therefore to be carried out witha lower filament temperature. Fortunately, the filament gave out an ample supply of electrons when heated to a temperature at which it was barely visible in the dark. The electrons were driven across by the accelerating potential chosen, the electron stream being held constant, if necessary, by slight adjustment of the heating current. Pressures were read every four minutes. It was impracticable to take gauge readings oftener than this. At the end of 40 minutes the electron current was stopped and the constancy of the pressure with the filament still hot, but without an electron current, was checked. The residual hydrogen was pumped out to a pressure less than ‘00001 mm. Then the evolution of gas FAlydrogen and Nitrogen by Electron Impacts. 781 due to electron bombardment of the gauze was measured, using the same accelerating potential as before. This was appreciable, especially with the higher accelerating potentials and after large amounts of hydrogen had been cleaned up. The liquid air had to be replenished from time to time, to keep its surface at the same level around the tube H. Results. The clean up with different initial pressures is shown in Table I. The accelerating potential was 71 volts in every run. TABLE TT. Hlectron current =650 microamps. Accelerating potential=71 volts. Pressures taken at intervals of 4 minutes. A | B C D _ 5080 x 10-5 mm.) 2160 10-5 mm. | 94210-5 mm. | 200x10-5 mm. — 4880 1602 642 122 | 4060 | 1262 482 a — 8880 _ 1070 398 52°2 | 3550 926 338 35°6 _ 3380 800 292 27-4 | 3200 i oe. 252 19:2 3060 | 656 230 16°8 2890 | Bee 204 12:4 | 2740 | 546 184 9°6 | 2600 | 512 164 88 The absolute amount of hydrogen cleaned up decreases as the initial pressure is made smaller, but the ratio of the initial pressure to the final pressure increases. Table II. vives the results obtained on varying the acceler- ating potential, but keeping the other conditions constant. It was impossible to do more than to get the initial pressure approximately constant. A slight but definite clean up was observed at 13°3 volts, a pressure decrease from 432 x 10-?> mm. to 408 xX 107° mm. being observed in 60 minutes with an electron current of 185 microamperes. (This will be referred to as run L.) No evidence of any clean up at all could be obtained when the electrons were accelerated by 89 volts. A careful test was made® over a period of 90 minutes, and the very slight Phil. Mag. 8. 6, Vol. 41, No, 245. May 1921. 3F 782 Prof. A. Ll. Hughes on Dissociation of TaBLeE II. Electron current 650 microamperes. Initial pressures approximately constant. Unit of pressure 10-5 mm. Pressure taken at intervals of 4 minutes. Pressure. | | E Bee aay H | 1 ee 29] 141 71 49: 208 Sea Oar 17°9 volts. volts. | ‘volts. volts. volts. volts. volts. 918 1 3970 942 970 900 892 884 | 656 | 636 642 730 724 738 796 | EBA || eee) 482 586 612 662 744 | 462 398 S98) fia) OL 52 592 ~=——=—s- 708 398) wilmoas 338 450 508 572 692 SOP Zoe 292 404 456 506 ~=676 344 264 252 364 414. |}. 480° 4663) 328° |) “24s 230 B23 aa 450 | 656 298 224 204 3504 360 416 648 (28S 216 184 278 338 398. eae ee ae 200 162 | 252 318 eect == | | | | | clean up obtained was no bigger than the effect (a small thermal clean up) obtained during the next 90 minutes without an electron current. Proportionality between electron currents and clean up.— Run G was repeated with the electron current reduced fivefold. The interval between the pressure readings was increased five fold. The readings obtained were practically | identical with those of run G, showing that the clean up, over this range of five-fold decrease in the electron current, is proportional to the electron current. . Number of Hydrogen Molecules cleaned u The Kate Number of Collisions Face Electrons and Maouie ; Let V =total volume of apparatus (=302 c.c.), V,=part of volume at room temp. ( =267 c.c. approx.), Vea ee i liquid air temp. (=35c.c. approx.), _ n,=no. of molecules per c.c. in V;, 1 ” ” 29 '= total number of molecules ; then n'=n,V,+n,Vo=my (V1: 4+ = V> ). 1 N =number of electrons emitted per unit time, > =mean free path of electron (in V,) ; then N (2 —e-) dt = number of electrons which collide with molecules between the filament and gauze in time dé. («=1 cm.) Hydrogen and Nitrogen by Electron Impacts. 783 Let 6=fraction of collisions resulting in disappearance of the molecule from the gas, and ‘dn' = decrease in the total number of molecules in dt. 1 Then dn'=bN (1 —e~)dt, | i ae (V, ae v.) PN (leave, Ny n, the number of molecules per c¢.c. in V, is proportional to the pressure p. Hence n,y=ap (a=3°55 x 10" at 10~-° mm. pressure at 0° C.). \ 1 a In V. 1 Ve =bN(1—e ajdt. 1 Hence Numerical values can be substituted for all the symbols on the right-hand side and so } can be calculated. Before doing so, it is well to consider some possible implications of the equation. If the mean free path, d, of an electron is considerably larger than the path (1em.) from the filament to the gauze, then the factor Ga...) may be written = which in turn is proportional to the pressure p. Hence the equation becomes _ d (log p) < const. Now if 6 were a constant, 2. e. if the number of molecules disappearing always bore a constant ratio to the number of collisions, we should have log p a linear function of t. The results given in Tables I. and I1. are shown in fig. 2 in which the ordinates are logy) p. In no case does logy) p appear to be a straight line, the curves all show a decreasing rate of disappearance. This can be explained on the assumption that the surface takes up the gas which has disappeared and that as the area free to take up gas diminishes, the rate of clean up must diminish. Thus superposable curves were always obtained when starting with the same initial pressure and using the electrons of the same energy, provided that Suna Log, of pressures 784 Prof. A. Ll. Hughes on Dissociation of Fig. 2. Hydrogen and Nitrogen by Electron Impacts. 785 each time there had been a thorough outgassing. If the out- gassing were omitted, other conditions being the same, the rate of clean up was diminished considerably. For the two lower pressures (the columns C and D, Table I.) the log curve corresponding to the lower pressure of the two has less curvature, as might be expected from the fact that the. amount of g gas which has disappeared and which retards the clean up is ‘always less in case D than in case C. (C and G are identical.) To calculate “‘b”’ for each run, it is clear that we must take the value of dp/dt when 50), for this is the time when “6” is least affected by the supposed inability of the surface to take up all the dissociated gas reaching it. It was found convenient to deduce dp/dé at t=0 from the initia] slope of the corresponding log curve by meaus of the relation ap ogg tao dt dt the time unit was 1 minute. N the number of electrons emitted in one minute was found to be 2°44 x 10!" (from the electron current of 650 microamps.). There remain to be calculated the free path X for the elec- tron, and n,/n, for the ratio of the densities of the gas in the parts of the apparatus at liquid air temperature and at room temperature respectively. When the mean free path of the molecule is considerably less than the distance apart of the walls of the apparatus the ordinary gas laws hold, and n,/n, = p./o,=1,/T, where T, and T, are the absolute temperatures. However, as Knudsen * ee found, when the mean free path of the molecule is considerably less than the diaméter of the tubes, another set of gas laws is applicable, from which we get n,/n;= p/p; =,/1,/T2. We shall refer to these two sets of laws as the high pressure, “H.P.”, and the low pressure, “ L.P.”, laws respectively. The critical distance in this apparatus is akout 3 mm., this being the distance between the inner tube and outer tube at the level of the surface of the liquid air, for this is the place where the temperature transition occurs. Cite hydrogen molecule has a mean free path of 3 mm. at 290 x 10-° mm. pressure ain OP MOe)) There would have been less ambiguity if the experiments could have been carried out at pressures much below the critical values, where the “low pressure” laws apply accu- rately. This, unfortunately, would have meant restricting the range of observations to within a very few mm. at the * Knudsen, Ann. der Phys. xxxi. p. 205 (1910). The pressure unit was 10-°mm., 786 Prof. A. Ll. Hughes on Dissociation of top of the McLeod gauge. It was decided, therefore, to start at higher pressures in spite of the fact that sometimes neither set of gas laws would apply accurately : 905 Los pee =3°25 if “high pressure” gas laws apply. ny 90 = ; ; a = -2 =1'80 if “low pressure”’ gas laws apply. » = mean free path of an electron in the experimental tube at 90°K. ‘This was taken as 4,/2xm.f.p. of a molecule. There will be two values according as to whether the “high pressure” or “ low pressure” laws apply. Jeans gives 11°6 x 10-° em. as the m.f.p. of the hydrogen molecule at 760 mm. pressure and 0° C. The values of “6” calculated for runs A, B, C, and D (Table I.) are given in Table III. For the initial pressures A and B, only the “ H.P.” values are admissible. TasieE III. Number of molecules disappearing. ~ Number of collisions between electrons and molecules. Values of 6” Accelerating Potential=71 volts. | [ | ! { A B ‘6 D iE Ne ee ee Initial Pressure — —, 5080 2160 942 200 *10-5 mm. x10-45 mm. x10-5 mm. x10-9 mm. mfp. ef H, [“H.P.”law.... -053cm. -124cem. | .-286 em. 1:30 em. ' molecule cae Laws: — — | "520 em: | eS 2aayeme= | ie ise Eee Vila tug. sat 129 136 Weal: (115) | eae? Taw cc. Mice: ee a | “168 178 | | The initial pressure in C is in the region where neither law is strictly applicable. The real value of “6” will, there- fore be between *122 and ‘168 for C. For the pressures used in D, “U.P.” laws are evidently to be used and the “ L.P.” value of “6” should be taken. It is seen that “6,” which measures the clean up per collision, increases progressively as the initial pressure decreases. At the lowest pressure, D, about 1 out of every 6 collisions results in the disappearance of a molecule. Here the conditions are the simplest, the electrons practically never make two collisions, and the atoms resulting from a collision with an electron have a good chance of reaching the walls and condensing there. | “HP.” law.| “100 | 133 | -122 | -088 | -064 | -056 | -029 | Mey Fic? j | Hydrogen and Nitrogen by Electron Impacts. 187 In Table IV. the values of “6” are given for approxi- mately the same initial pressure ( 950 x 10~° mm.), but for different accelerating potentials. Runs H, F,....K (Table TI.) were used. (The initial pressure in run L, which is included, was only 430 x 10~° mm.). ABE) LW Walues of ‘ 6.” Sa) NC A eT So eectgy WI key lg rey eu) Ta a Dire Meee aie eshn (ile | volts. | volts. | volts. | volts. | volts. | volts. | volts. | volts. | | “LP.” law. | 141 | 180 | 168 | -122 | 088 | 077 | 041-0045 | | | | For 8°9 volts, ‘b” seat not be measured, it was certainly less than ‘0005. The values for “0” are shown in fig. It will be seen that “6” rises rapidly between 13 and about 40 volts, and does not change much after about 60 volts. (An earlier experiment showed that “6” was much the same for 600 volts as for 70 volts; these later and more accurate experiments indicate a drop between 140 and 290 volts. It is proposed to investigate this further.) 150 200 250 300 Acceleratin g volts. Control Experiments with Helium.—Helium was admitted to a pressure of 950 x 107° mm., and a run was taken with 71 volts accelerating potential. The clean up was less than one two-hundredth part of that in the corresponding hydro- gen run (G). The helium was purified for a short time over charcoal in liquid air. The slight clean up was no doubt due to impurities, probably hy rdrog en. as the apparatus was not outgasse:| for more than half an hour, nor were excessive precautions taken to secure pure holtura: 788 Prof. A. L). Hughes on Dissociation of Discussion. It is necessary to give reasons for believing that the disappearance of the gas is due to dissociation of hydrogen into atoms which condense on the walls. Langmuir™*™ found that pure hydrogen at a low pressure in a bulb containing an incandescent filament disappeared. On carrying out the experiment with the bulb, or a side tube, immers -d in liquid air, a peculiar effect was noticed. After the pressure decrease had been in progress for some time, the liquid air was removed (the filament heating current being cut off) and the pressure rose. On replacing the liquid air, the pressure fell a little, but not to its original value. ‘There was, therefore, as Langmuir terms it, a non- recondensible gas. Langmuir accounted for this on the supposition that hydrogen molecules on impact with the very hot filament were dissociated into atoms, and that these atoms, 1f they had a clear run to the w alls, would condense on them, the effect being the more marked at low temper- atures. On removing the liquid air, some of the atoms would come off and re-combine with other atoms to form molecules which could not be condensed. Thus Langmuir gives a natural explanation for the ‘‘non-recondensible”’ gas. The more atoms already on the surface, the greater the chance of an atom impinging on the surface, striking one of them 2nd recombining to form a molecule, which leaves the surface and so reduces the apparent rate of dissociation of the gas. From experiments on the transfer of beat through hydrogen from an incandescent filament, Langmuir caleu- lated the amount of hydrogen dissociated. He states that, under the most favourable conditions, the observed decrease In pressure corresponds to only 1/7 of the amount actually dissociated, and may often be much: less. Non-recondensible gas was obtained in these experiments. The following is an illustration from one set of observations. Hydrogen was admitted to a pressure of 548 x 10~° mm. and cleaned up by electron impact to a pressure of 100 x 10~? mm. The liquid air was then removed, the pressure rose, the liquid air was replaced and the pressure fell to 263 x 107? mm. Hence the amount of non-recondensible gas was proportional to (203—100) x 10-° mm., and the amount of hydrogen which originally disappeared was proportional to (548— 100) x10-> mm. Thus the amount of non-recondensible gas formed 37 per cent. of the «mount of hydrogen cleaned up. Then, without outgassing, more hydrogen was admitted until * Langmuir, Journ. Amer. Chem. Soc. xxxvii. p. 451 (1915). et) Hydrogen and Nitrogen by Electron Impacts. 189 the pressure was again about 550 x 10-> mm. On repeating the experiment the yield of non-recondensible gas was o£ per cent. The procedure was carried through twice again, the yields rising to 67 per cent. and 81 per cent. Thus, taking the presence of non-recondensible gas as a criterion of dissociation of hydrogen, there is strong evidence in favour of the view that hydrogen can be dissociated by electron impacts. The progressive rise in the yield corre- sponds to the decreasing amount of surface available for the atomic hydrogen. Let us consider some alternative explanations. There may be chemical action between some gaseous impurity (which must obviously be gaseous at 90°K) and the hydrogen when ionized by electrons. It is extremely unlikely that the impurity would be present in sufficient quantity to account for a clean up of as much as four-fifths of the hydrogen time after time. It may be urged that hydrogen scelhann ionized combines with the nickel, or some- thing on it or on the glass. But in neither of these Be eeatnes is there any reason to suppose that the com- pound would show the exceedingly characteristic non- recondensible effect. We may now take up the behaviour of ionized hydrogen, which in some way may show the non-recondensible effect. . Take first the positively charged hydrogen molecule. Such a charged molecule would be driven to the filament, and to account for the clean up found in many runs, a layer of hydrogen 100 molecules deep would be formed on the filament. Again, a clean up of this type would be unaffected by removing the liquid air if the filament were kept hot. Such was not the case. ‘Take now the case of negatively charged hydrogen molecules. These would be driven to the gauze. If they stuck there by virtue of their charge, a very small clean up would lead to surface charge sufficient in amount to annul the field and reduce the electron current almost to zero, which is not the case. If, on the other hand, they gave up their charge, there is no reason to suppose that they would stay on the surface any more than any other uncharged moleeule. Another objection is that electrons with energies less than 10 volts w an be as likely to form a negative ion by uniting with a molecule as electrons with oreater energies. The clean up, however, below 13 volts is very small. Still another reason against this view is that various investigations have shown that the negative ion in hydrogen is an electron, proving that there is no tendency for a hydrogen molecule to unite with an electron. 790 Prof. A. Ll. Hughes on Dissociation of Franck, Knipping, and Kriiger * have recently published an account of the ionizing potentials of hydrogen, and they suggest that there are two types of ionization. The first type is ionization, without dissociation, when the electrons have energies in excess of 11 volts. The second type of ionization is accompanied by dissociation. Since 3°6 volts measures the energies required for dissociation and 13°5 volts measures the theoretical energy required for ionizing one atom, there should be an ionizing potential at 3°64+13°5=17-1 volts, corresponding to dissociation and ionization of one of the atoms. They verified this experimentally. Similarly they found a radiating potential at 13°6 volts corresponding to dissociation combined with radiation from one of the atoms (3°6 + 10°2=13°8 volts). They also verified the existence of an ionizing potential at 3°6+2x13°5=30-6 volis, corre- sponding to dissociation and ionization of both atoms. On this theory, then, one would expect dissociation to accompany these tvpes of ionization of, and radiation from, hydrogen. On account of the first type of ionization (11 volts), it is possible that all ionizing collisions are not all dissociating collisions. Wendt ft has found that if hydrogen is driven through a tube across which an electric discharge is passing, the hydrogen afterwards contains a small amount of triatomic hydrogen H;. In his experiments, the pressure was high enough to insure that the mean free path of a molecule was considerably less than the diameter of the tube, so it is natural to infer that the first step in the formation of H; is the dissociation of H, by electron impacts into atoms H, which in turn unite with molecular hydrogen at the next collision to form H;. It is difficuit to see any way of accounting for H; without the intermediate step of the for- mation of H. Dempster ft has found evidence for the view that the formation of H is a necessary step in the formation of H;. The hydrogen spectrum contains several series (e.g., the Balmer series and the Lyman series), which are, according to Bohr’s theory, characteristic of the atom. It thus appears that there are good grounds for believing that atomic hydrogen is formed in the electric discharge, and probably as a result of direct impact of electrons on molecules. * Franck, Knipping, and Kriiger, Verh. d. Deutsch. Phys. Ges. xx1. p. 728 (1919). + Wendt, Nat. Acad. Sci. Proce. v. p. 518 (1919). t Dempster, Phil. Mag. xxxi. p. 438 (1916). Hydrogen and Nitrogen by Electron Impacts. 791 It appears then that the simplest explanation of the dis- appearance of the hydrogen in this investigation is that it is due to the dissociation of the hydrogen into atomic hydrogen which condenses on the walls, especially at low temperatures. The experimental value “‘” tor the yield of atomic hydrogen per collision is probably too low for several reasons. (a) Some atoms may collide with other atoms forming ordinary molecular hydrogen which does not condense. (6) Some atoms may collide with hydrogen molecules forming H; which may not condense as readily. (c) An atom may hit a spot on the cold surface already occupied by another atom. A hydrogen molecule will be formed and will leave the surface. Hence this disseciation does not contribute to the pressure decrease. (d) If, after an appreciable clean up of hydrogen, the gas is completely pumped out, and electrons are accelerated so as to bombard the gauze and glass, it is found that there is a considerable evolution of hydrogen, much more than after a thorough outgassing. ‘This effect no doubt goes on all the time during a clean up, and tends to reduce the rate of clean up, especially towards the end of a run. (This evolution due to bombardment increases, for a given electron current, with the accelerating potential, but not so quickly. Calculation shows that the total heating effect at the gauze of such a bombardment is absolutely negligible.) These effects are more clearly marked at the higher pressures, and no doubt influence somewhat even the initial values of the clean up. (e) It may be that not every atom which strikes a cold surface condenses. Langmuir states that the measurements of the clean up ina cold bulb in which hydrogen is being dissociated thermally by an incandescent filament, gives under the most favourable conditions a value only 17 of the amount of dissociation actually occurring as deduced from the measurements of heat transfer through the gas. This factor is no doubt determined largely by the dimensions and shape of the bulb and the ecnditions of the experiment, and it would not be justifiable to seas our experimental determination of “b” by 7 to get the real amount dissociate d. Nevertheless it is significant ‘that the highest value of ‘0,’ viz.°18 (Tables ITT. and IV.), is roughly 1/6 of the maximum possible yield, 7. ¢., one dissociation per collision. The hydrogen curve in fig. 3 shows the yield for different accelerating potentials. It “attains a value not far from its maximum at abont 70-100 volts. As the electrons under 792 Prof. A. Ll. Hughes on Dissocration of our experimental conditions may collide with any velocity between zero and that corresponding to the applied accele- rating potential, the curve which would be obtained if the electrons always collided with the energy corresponding to the applied potential would attain its maximum at a lower value, possibly somewhere between 40 and 80 volts. (One disadvantage of working at higher pressures, e. g. A in Table III., is that the electrons are forced to make most of their collisions close to the filament, so that their energies on collision are less than would be inferred froin the applied field.) On the basis of Franck, Knipping, and Kriiger’s view of ionization, there are good grounds for believing that ioni- zation and dissociation of hydrogen are bound up with one another. So far as this writer knows, there are no results eiving the efficiency of ionization per collision as a function ae she energy of the electrons. i. e. the fraction of collisions between electrons and molecules resulting in ionizations. Johnson * has made some experiments on the total ionization produced in a gas by electrons. He found that the accele- rating potential had to be raised to about 50 volts before he obtained one ion per electron passing through the gas. This is not quite the same thing as saying that a collision between a hydrogen molecule and an electron possessing energy corresponding to 50 volts always results in ioni- zation, but, for want of more direct information, we may tentatively assume that an electron of such energy will give ionization at practically every impact. If, at the higher voltages, ionization of the 17 volt type predominates, i.e. dissociation of the molecule combined with ionization of one atom, then each collision would result in the dissociation of the molecule and the production of one ion. This is what our investigation suggests, if we make use of Langmuir’s result, that the rate of clean up measured, under the most Pecnonea ale conditions, only one seventh of the amount of dissociation really taking place. (It is admitted that the application of Langmuir’s numerical result to this investi- gation is open to question, but it seems certain that the clean- up measurements underestimate the amount of dissociation.) The view that the efficiency of dissociation is unity, 7. e. that every collision between a molecule and an electron will result in dissociation when the electron has enough energy, has an attractive simplicity. * Johnson, Phys. Rev. x. p. 609 (1917). Hydrogen and Nitrogen by Electron Impacts. 793 No measurable amount of dissociation could be obtained when electrons of energy 8°9 volts were used. One might expect some dissociation at, or above, 3°6 volts, since this corresponds to the measured work of dissociating the hydrogen molecule. Hither no dissocation at all occurs, or else it is produced in so small an amount as not to be detected in these experiments. The latter would demand very special circumstances, either in the momentary state of the hydrogen molecule, or in its orientation at impact, to give dissocation at hese low values. ‘The alternative view is that dissociation cannot occur until radiation of the proper type (the 13°5 volt effect, corresponding to dissociation and radiation from one atom) or ionization (the 17:0 volt effect corresponding to dissociation and ionization of one atom) takes place, 2. e. dissociation by electron impacts must ac- company some other ettect. The “clean up” of hydrogen by say an electric dis- charge in the presence of sodium or potassium, is probably due “6 combination between the metal and the atomic hydrogen produced. It is possible that lining the experi- mental tube with sodium would show a more rapid clean up than in the experiment if the chemical forces were stronger than the forces controlling temperature condensation. NITROGEN. Preparation.—Nitrogen was prepared by warming a solu- tion of ammonium chloride and sodium nitrite. Ten litres were collected over water. This was passed slowly through calcium chloride, soda lime, red-hot copper turnings, and phosphorus pentoxide, into a reservoir R of one litre capacity permanently attached to the apparatus (fig. 1). Several litres of the nitrogen were used to wash out the a apparatus before finally filling the reservoir. Small quantities of nitrogen could be admitted into the apparatus by manipu- lating the stopcocks 8, and 8y. Method of Hxperiment.— Experiments were conducted in the same way as those on hydrogen. Results. The clean up of nitrogen for different accelerating potentials is shownin Table V. The experiments on nitrogen were not as extensive as those on hydrogen. No experi- ments have, up to the present, been c sarried out on the effect of varying the initial pressure. 794 Prof. A. Ll. Hughes on Dissociation of TABLE V. Electron Current 650 microamps. Initial Pressures approx. const. Unit of pressure= 10-5 mm. ay : | | Exp. tube at | Exp. tube E in liquid air. 20°C. Time ss jo Ase geese ie ai Ju tI ee interval Vayu.) Pow a Qualthak Sen | {mins.). : 291 | 145 72) | 29-5) 23:87) 18 2a ee volts. | volis. | volts. | volts. | volts. volts. || volts. volts. 596 | 566 520 | 520 | 552 650 | 900 | 890 | 364 | 376 | 376 | 456 | 520 | 650 || 662*|- 724 | 988 | 304 | 294 | 416 | 494 , 630 || 580 | 642 | 944 | 964 | 264 | 392 | 464 . 630 || 532 | 578 | 919 | 994 | 934 | 382 | 444.) 619° ||) 480 ese 181 | 204 | 208 | 348 | 420. 612 | 444 | 5067 144 | 176 192 | 316 | 404 600 | 392 | 450 118 | 162 | 166 | 282 | 388 | 586 || 354 | 404) 101 | 141 | 144 | 274 | 370 ) 580 || 318 7 ee7ame 94 | 122 | 182 | 260.) 348 | 580: | 392 1) 3eey 88° | 119 | 128.) 255 | 344 | 572 | 260%) 328 | { St Ot Or St Ove Co Oo Oo Ce * The first and last time intervals in S were 4 mins. The other intervals were as shown in the first column. Runs § and T were taken under ditferent conditions from the rest. In run § the whole apparatus was at room temper- ature so that mercury vapour had access to the experimental tube. In run T the U-tube was surrounded by liquid air to freeze out mercury from the experimental tube. The initial pressures were chosen so as to give the electrons much the same mean free path as in theruns M,N,.... lis | ? ah . Number of Nitrogen Molecules cleaned wp The Ratio ~ AO NEES | Number of Collisions between Electrons and Molecules The value of “5” given in Table VI. was calculated for nitrogen in exactly the same way as for hydrogen. The calculations of various mean free paths were based on the value given by Jeans in his ‘ Dynamical Theory of Gases,’ viz. 5°7 x 107° cm. at 760 mm. pressure and 0° C. It was rather surprising that the clean up without liquid air around the experimental tube (runs § and T) were as great as they were, ; Hydrogen and Nitrogen by Electron Impacts. 799 Tasie VI. Values of “6” for nitrogen. PMD. ieee mmmemaii tims yey ly mae oot Se gia [ep 201) Swe 29s 2a AGS | G2.) Ze | volts. volts. | volts. | volts. | volts. | volts. | volts. volts. “bh” “AP.” Law) -141 | -095 | 070 028 (0135 0019 11-091 mm. The ch ange of pressure due to cooling the tubes by liquid air was only that corresponding to the temperature change. Hence this effect is located at the filament. The only substances in this experiment that can account for this type of clean up are platinum, BaO, and SrO, and possibly the copper leads to the platinum. The etfect calls for further investigation. Phul. Mag. 8. 6. Vol. 41. No. 245. May 1921. 3G 798 INssociation of Hydrogen and Nitrogen. Summary. The disappearance of hydrogen and nitrogen at low pres- sures when an electron stream is passed through them has been investigated. For hydrogen, no disappearance was obtained unless the electrons had energy ebove 13 volts. The rate of disappearance rose rapidly as the energy of the electron was increased to about 70 volts, after which no rapid change was noted (the rate appeared to diminish somewhat when the energy of the electrons was raised from 150 to 300 volts). For nitrogen, the rate of disappearance was at first much less than for hydrogen, but when the energy of the electrons was raised sufficiently (roughly 200 volts) the rate of disappearance of the nitrogen exceeded that for hydrogen. Reasons are given for believing that this disappearance is due to the splitting of the molecules into atoms when electrons collide with the molecules, and that these atoms condense on the adjacent surfaces particularly if they are cold. The maximum rate of disappearance of hydrogen occurred when electrons of energy corresponding to 140 volts were used. About one molecule disappeared for every six collisions. Langmuir’s work on the thermal dissociation of hydrogen showed that the clean up, even under the most favourable circumstances, accounted for only one-seventh of the amount of dissociation. Hence it is possible that when the electrons have the right amount of energy a dissociation may occur almost at every collision. A peculiar clean up, apparently due to absorption of nitrogen by a platinum filament covered with BaO and SrO when cooled to the temperature of liquid air, was noticed. The author wishes to express his thanks to Mri A. E. Harkness for bis assistance in taking some of the obser- vations. Physics Department, Queen’s University, Kingston, Ont. Canada. December, 1920. 799, LXXI. Note on the Abnormality of Strong Electrolytes. By Davip Leonarp CHAPMAN and HERBERT JOHN (FEORGE *. [* a series of papers recently published in the ‘ Journal of the Chemical Society ’ (Trans. Chem. Soe. exiii. pp. 449, 627, 707, and 790 (1918)) J. C. Ghosh has endeavoured to explain the anomalous behaviour of strong electrolytes. From his theory the electrical conductivities and the osmotic pressures of solutions of salts can be deduced, and the values thereby calculated for the ranges of concentration examined are in good agreement with the experimental results. This theory has attracted much attention, and on the whole has been favourably received, although also it has been sub- jected to a searching adverse criticism from several points of view by J. R. Partington (Trans. Far. Soc. xv. p. 98 (1919)). For a deductive theory it embodies an unusually large number of assumptions. Thus the author postulates that (1) the dissolved salt is completely ionized ; (2) the mean disposition of the charged ions is regular, and similar to the arrangement of the atoms when they assume a crystalline structure ; (3) the component ions of a salt molecule form a com- pletely saturated electrical doublet, and the work necessary for separating the component ions of- a molecule is the electrical work, A, done in moving the ions constituting a doublet from their fixed mean distance in the solution to an infinite distance apart ; ’ (4) the free ions (conducting ions) are those whose kinetic energy is greater than the work to be done in separating them to an infinite distance. In addition to these the author makes other implicit and stated assumptions which it is not necessary to specify here. Our present purpose is merely to indicate what appears to us to be a false deduction from the postulate (4) quoted above, and to show that, if the proper correction is made, the theory in its present form fails to account for the facts. It is sufficient to consider the simplest case to which the theory has been applied—namely, that of a uni-univalent salt such as potassium chloride. According to Ghosh, it follows, in this case, from postulate (4) that the kinetic energy of a free ion must exceed 5, 2.e., half the work * Communicated by the Authors. 3G 2 800 = On the Abnormality of Strong Electrolytes. required to separate the ion from its partner; and he states that the number of ions which satisfy this condition is given by the expression Tes Ne 2kt, eerie pe (A) in which N is the total number of ions, / the gas constant for a single molecule, and ¢ the absolute temperature. The latter statement is we believe fallacious; for, according to a well- known result in the kinetic theory of gases, the number of molecules which have a velocity in excess of ¢ is m being the mass of a molecule and « its ree On changing the variable from c to pene pe this number 7 2kt becomes A fee) AN pay — 72 le = UOT Ue 2 z O which, after integration by parts, assumes the form In terms of the probability integral defined by Pitre erl e=—,) e-*da Tr? ; L this number becomes N(S tigen oct erf 2%), . 3. (Vide Jeans, ‘The Dynamical Theory of Gases,’ 2nd ed. pp. 34, 35.) However ee ~—_ in expression (A) is equal to 2)”, and there- > he fore the number of free ions as estimated by Ghosh is ING STO 2 ese) ee rr Now, the number of free ions divided by the total number ROR neon. uw, being the molecular conductivity at dilution », fo Discontinuous Flow of Liquid past a Wedge. 801 and yu, the molecular conductivity at infinite dilution. There- fore, according to Ghosh, the value of this quotient is e~’, ’ es e 2 >) whereas, according to (B), it should be — xpe-*" + erf xp. TT The values of e-”? and 1— erf 2 are tabulated in the appendix to Jeans’s ‘Treatise on the Dynamical Theory of Gases.’ With the aid of this table we have calculated and compared below the two sets of values of Lo ye eo om (@). HS Poe ie : 0-2 0:96080 0:99413 03 0:91393 098075 0-4 085214 0:95623 05 077880 - 0:91889 A comparison of the two columns of figures 1s sufficient to show that the val pM 1 Nat, as the values OL calculated from the expression (C) are in good agreement with the experimental numbers, the values of the same quotient calculated with the aid of expression (B) must show a considerable discrepancy. Accordingly, Ghosh’s theory in its present form is not in agreement with the facts if the number of ions whose kinetic energy exceeds a specified value is correctly given by the commonly accepted formula. Jesus College, Oxford. LXXIT. On the Discontinuous Klow of Liquid past a Wedge. By W. B. Morton, M.A., Queen’s University, Belfast *. N the well-known case of two-dimensional motion solved A by Bobyleff, a wedge is set in an infinitely extended stream in such a way thata stream-line divides at the apex of the wedge and runs along the two sides into the twe surfaces of discontinuity which extend to infinity. This requires, for a wedge of given angle set in a given manner relative to the stream, that the breadths of the two sides should be in a _ definite ratio. The question arises as to the character of the motion when this ratio is departed from. * Communicated by the Author. S02 Prof. W. B. Morton on the Discontinuous A survey of the conditions of the problem may be ob- tained as follows. Start with a single plane lamina set at angle a to the direction of the stream at infinity. The stream divides at a point on the upstream side of the Centre. Now build out a second plane from the upstream edge of the first, at angle 8 with the stream, until it reaches the free surface of the liquid. This happens when the ratio of the second breadth to the first is less than the Bobyleff value. Up to this point, of course, the former state of motion persists, the second face of the wedge lying entirely in the region of dead water behind the first. But now a new kind of motion sets in: the stream-.ine, which passes round the angle of the wedge, is interrupted where it runs along the — second plane near its edge. Beginning at the corner, it first sweeps round a pocket of dead water enclosed between it and the second plane, it then becomes tangent to the plane and runs along it to its edge and thence to infinity. There is evidently a point of inflexion before the stream-line joins the plane. As the second plane is extended, the point on the first plane, where the stream divides, moves towards the- corner, until we reach Bobyleff’s case. Still further extension gives the reverse change in the character of the motion, the réles of the planes being interchanged until we get to the one-plane case round the second plane, with the first plane lying entirely in the dead water. It is proposed in the present note to discuss the general or transitional case. The treatment is quite straightforward on the well-known method of conformal representation, but it derives some interest from the fact that there are two different constant values of the velocity on the free portions of the stream-line, one along the infinite branches and another, smaller than this, round the pocket of dead water. The two critical breadth-ratios can be found from the two known solutions, Kirchhoff’s and Bobyleff’s. The expressions obtained are, in general, complicated, but become manageable for the special case of a right-angled wedge (e= 5 a). Taking the breadth of the “first plane” as unity, the breadth of the second when it just touches the free surface is sing sin # sin 6(1— cos « cos @) -sin3(6+4) Tsina+4 (cos a— cos 6)? © sin $(@—a) , (1) where « is the inclination of the first plane to the stream and @ is given by cos0=cos a/(2+cosa). And the value Flow of Liquid past a Wedge. 803 when the stream divides at the corner is an (“sin’ (2+) . sin? (a—@).sin 2 An sin? (b+a) XSI — a) sli 2a age) 1) 2 (2) These values are plotted in fig. 1, in the curves marked Fig. 1. Breadth-Ratio B K O bs iia 30° 60° 90 K and B respectively. It is obviously sufficient to let « vary from 0 to 7c In the general case let y be the angle between the sides of the wedge. The diagrams on fig. Z give the boundaries in the planes of z=a+vy, w=ptiny, O=log(—Q and the arrangement of the corresponding points along the real axis of the auxiliary variable v. Here Q is the velocity dz niet, — )=loe— +10, Te) log gr : SO4 Prof. W. B. Morton on the Discontinuous along the infinitely extended free surfaces, introduced ex- plicitly to keep the dimensions right, as Greenhill does in his ‘* Report.” The diagrams are not drawn to scale. The tye; 2: A B Ea? || © GS A Ber Gas § D € G F G H A B C2 ip - CO Ad f g al a a2) values —1, +1, «© aze chosen for the points H, B, C, leaving A, D, F, G to be fixed by the conditions of the problem. Indicate the corresponding values of « by small letters, then d, f, g are negative and a of doubtful sign, and the transformations are given by da ae =a), aQd om me) (Ud) .) eee at Tt will be seen that the analysis applies to a more general case than that with which we started, for there is in it nothing to imply that the second plane passes through D, the upstream edge of the first. What we really have is the eS Flow of Liqud past a Wedge. 805 case in which the left-hand free stream-line is caught for a portion of its length by a second plane set anyhow with respect to the first. The four quantities to be disposed of, a, d, f, g, are determined by the angle vy between the pianes, the angle of impact « on the first plane and the two co- ordinates which fix H, the outer edge of the second plane, with respect to DB. If arbitrary values are assigned to dq, j is determined by y and then a by @ by the relations : > t 1 | (u—f)(1—wu?)-2(u—d)-2(u—g) = du=y. . (3d) 1 and (awa d)-u—g) hana Bist) In other words, we can ensure that the planes shall make an assigned angle with each other, and that the stream shall meet them in an assigned direction, but the breadth and position of the second plane relative to the first can only be found by trial. Thus, to obtain solutions for a wedge, we could keep, say d, fixed, and work out the configuration of the z-plane for different values of g, until we found the value which made the second plane pass through the edge of the first, repeating the process with other d’s until a range of wedge-cases, with varying-breadth ratio, could be ob- tained. This lengthy procedure is considerably shortened when the wedge is right-angled, for then, taking the origin of z at D, the required condition is the vanishing of x at the point G. Reverting to the general case, let ()' be the velocity over the free stream-line bounding the pocket DFG. Then () ()/ log = HG on the 2 diagram -—] ‘i $s ( (w—f)(w?—1)-2(u—d)-2(u—y) due. (7) vy its imaginary Along DFG the real part of Q is log ae y part is zy, say, where " yan | (u—/f)(w?—1)-2(u—d)-3(g—u)—idu, ° (8) d xX is minimum for w=/ corresponding to the inflexion F and then rises to y at u=q e & 806 Prof. W. B. Morton on the Discontinuous Taking the origin at D we have along DFG *u as a" “exp O . (a—u)du= a | exp (x)(a—u)~* du d (9) If the real part of z vanishes at G (when the planes form a right-angled wedge), then cay) J , | cosy .(a—u) *du=0. 72 eee d The Q-integrals can be expressed by elliptic functions, but the formule are clumsy and inconvenient for purposes of numerical calculation. For the z-integrals one is obliged to have recourse to mechanical or arithmetical quadrature. I have carried out the calculations * for a rectangular wedge whose faces make angles of 45° with the stream. For this purpose the integrals were first transformed so as to cover the range from 0 to $m of an angular variable. Putting w= —cos 26 in (5) the equation giving f becomes -/{ (— d—cos 26)-#(—g —cos 20) -3d6 0 = : + | cos 20(—d — cos 26)-3(—g—cos 20)-2d@. (11) 0 (—d, —g, afte positive quantities greater than unity). The two integrands having been tabulated for 23° intervals the integrals were calculated approximately by Simson’s rule and f found. It is to be noticed that the values assumed for d and g must be such that the resulting value of f lies between them. ‘To get a rough idea of the restriction thus imposed on d, g, we may reason as follows. We get an approximation to the integral on the left of (11) by giving the integrand the constant value (dg)-?, corresponding to the half-way value of @: this gives 7/2 /dg. The integral on the right has a small negative value, say —Az. Thus we get as an approximate value —f= Vdg(3+A), and this has to lie between —g and —d. It follows that * T am much indebted to Miss L. Beck, B.Sc., for help in carrying out the computations and integrations. Flow of Liquid past a Wedge. 807 —d is something like four times —g and that —/ lies close to —g. This means that the inflexion on the stream-line is near the point G, which is physically obvious. Having obtained jf, and using the values already faboleted: it 1s easy “to get a table of the integrand which is equivalent to that on the left hand of equation (i onndG)e | Since "1 1 a=‘ we have | =] , so we have to find the value of 0 a = at which the transformed integral is bisected, say @, then a= —cos 26). The integral in (8) is now transformed by the substitution u=4(d+9)+4 (d—g) cos 20 and the value of (7—y) found by step-by-step integration. Tinally, the integrand in (10) transformed to @ is tabulated and’ the integral found by Simson’s rule. Having done this for an assumed value of d with different values of g, it was possible to find by inter- polation the g which makes the second plane pass through the edge of the first. Then the other integrals which specify the details of the motion can be evaluated for the special values of the constants d, g, f, a. The integrations were carried out partly arithmetically and partly by use of a Coradi integraph. The results are shown on fig. 3. The abscissa is the ratio of the breadth of the second plane to that of the first. The curves begin at the value °18, which is the ordinate of the curve K on fig. 1 for «=45°. This gives the breadth of the second plane, built out at right angles to the first, when it just touches the stream. The corresponding values of the constants are easily found to be d=—(3+ V2), g=f=—1, a=3( 2-1). The other cases actually calculated are indicated by the positions of the marks on the upper side of the horizontal axis. They correspond to the values ~d= 6 — y= 1-049 6-5 1:099 7 1-156 10 1558 20 3°023 808 Discontinuous Flow of Liquid past a Wedge. The graph DG/DH shows the shrinking of the pocket of dead water to nothing as the Bobyleff case is approached, when the breadths of the planes are equal. ‘The actual Bie. 3. a length of the pocket passes through a maximum, as indicated on the graph marked DG/DB. The approach of the point of division of the stream to the corner is shown by DC/CB, which is plotted on a tenfold scale. Q'/Q is the ratio of the velocity along the boundary of the pocket to that along the ultimate free stream-lines. i 809 7 LXXIIT. The Motion Be a Simple Pendulum after the String has become Slack. By Anraur Taser Jones, Ph.D, Associate Professor of Physics at Smith College, U.S.A. * Inrropucrion.—Under the above title Professor W. B. Morton recently published t a very interesting note. The pendulum is supposed to swing in a plane, the string to be inextensible, and the velocity of the bob to be sufficient to carry it higher than the point of suppert, but not sufficient to bring it ‘to the top of its cireular path. At some point, higher than the centre of the circle, the tension of the string vanishes and the path of the bob becomes a parabola. At the point where this parabola intersects the circle, the string tightens again and jerks the bob out of its parabolic path. In ideal cases this jerk may be thought of as perfectly elastic or as perfectly inelastic. If it is “perfectly elastic it reverses the radial component of the bob’s velocity, and if it is perfectly inelastic it destroys this radial component. In the case which Professor Morton has discussed the jerk is treated as perfectly inelastic. Throughout the present note the jerk is treated as perfectly elastic f. * With a real string the jerk is far from being perfectly elastic or perfectly inelastic, so that the case of a real pendulum is an intermediate one. The general case—It the jerk is perfectly elastic, each path consists, in general, of a circular are followed by an infinite series of parabolic arcs. [or the first parabola Professor Morton points out that the level of no velocity is given by Pome O sic matenesaiih ul iain CL) where / stands for the distance from the centre of the circle up to the level of no velocity, + for the radius of the circle, and « for the angle which the vertical diameter of the circle makes with the radius to the point where the string first slackens. On the present hypothesis no energy is lost in the jerks, so that (1) gives the level of no velocity throughout an entire path—that j is, for every one of an infinite series of parabolas, If we take the origin at the centre of the circle, and * Communicated by the Author. + Phil. Mag. (6) xxxvii. p. 280 (1919). I have checked Professor Morton’s results and obtained the same expressions that he has. ‘There is a misprint at the bottom of p. 282, where the equations should read : J (a2) = 3 C08 a1, f'(@3) =} C08 ar. { This problem may, of course, just as well be thought of as having to do with the motion of a particle which slides and bounds in a single ae inside of a frictionless spherical cavity, the impacts being perfectly elastic, 810 rof. A. Taber Jones on the Motion of a if the motion of the bob on the first parabola is toward the right, then the first parabolic arc starts at the point (—rsina, +rcosa), and ends, as Professor Morton shows, at the point (+7sin 34, +rcos3«). If we take the origin at the initial point of the sth parabolic are, the equation of the sth parabola may be written Y ae r N= YP pana Ahs cos? a, (=) 2 [ue where 2, stands for the angle of elevation at the initial point of the parabolic are, and A, for the distance from this initial point up tothe level of no velocity. For s=1 we have a= and A,/r=3cose. For s=2 we find 64 cos’ « —112 cost a+ 48 cos? 2a—3 64 cos® a— 144 cos? a+ 96 cos? a—17- and h ; =. ~* —4 cos a[9—8 cos? a]. 4 tan a= tan o Fae ers) For s>2 the expressions for e,, and probably for hs, become very complicated, and I have not attempted to use them. To find the path which corresponds to any given a, I pro- ceed as follows. After using (1), (2), and (3) to find the equations of the first and second parabolas, I determine the co-ordinates of the end of the second parabolic are by a process of successive approximation. Knowing these co-ordinates and the equation of the second parabola, I find the angle of incidence of the second parabola upon the circle, take the angle of reflexion as equal to the angle of incidence, and so find #;. I can then use (1) and (2) to find the equation of the third parabola. By this method the path which corre- . sponds toany given « may be found to any desired degree of accuracy and for as many parabolas of the series as may be desired. The general manner in which the path changes with changing @ may be traced to the beginning of the fourth parabola from the curves in fig. 1. Special cases.—In the cases which are especially interesting the paths repeat, and so form figures that are déscribed periodically and may be thought of as somewhat ana- logous to the well-known Lissajous curves. These cases fall into three classes :— (1) Cases in which the end of some one of the parabolic ares is tangent to the circle, (2) Cases in which the end of some one of the parabolic ares is perpendicular to the circle, (3) Cases in which some one of the parabolic ares de- generates into a vertical line. From the first class are to be excluded those cases in Simple Pendulum after the String has become Slack. 811 which the arc in question is identical with the first parabolic are. Otherwise this class would include all the paths which repeat. Cases of the third class occur only when the bob travels upward on the vertical part of its path, and reverses its motion before reaching the upper half of the circle. This can oceur only when the level of no velocity lies below the top of the circle--that is, for «>48° 11’. The case a= 90° may be thought of asa limiting case of this third class. In all the cases which are periodic the paths are symmetrical with respect to the vertical diameter of the circle. In order to find the values of « for the various special cases, a study is first made of a series of paths like those shown in fig. 1. From this study some idea is gained as to Fig. 1. e=55° the neighbourhood in which certain cases are to be sought. A calculation of several paths in one of these neighbourhoods gives data from which the value of « for the case in question may be found to any desired degree of accuracy. I have aimed to have my values for @ correct to the nearest minute. Fig. 2. y ile Una ea ——___- ft a=60° a=43°55' a=57°42' a7 4¢! I think I have found all of the periodic paths which do not involve parabolas beyond the third. In the first class S) ee ) ~ there are the four cases shown in fig. 2. In one of these 812 Prof. A, Taber Jones on the Motion of a cases the end of the second parabolic are is tangent to the circle, and in three of them the end of the third parabolic are is tangent to the circle. In the second class there are the eleven cases shown in fig. 8. In one of these cases the end of rd Fig, 3 [First Part]. @=70°12" the first parabolic arc is perpendicular to the circle, in three the end of the second parabolic arc is perpendicular to the circle, and in seven the end of the third parabolic arc is perpendicular to the circle. In two of these cases the com- plete figure has not been drawn. In these two cases 1t Simple Pendulum after the String has become Slack. 813 appeared that it would be easier to imagine the rest of the figure, by remembering that it is symmetrical, than it would be to follow the complete figure. Jn the third class there are three cases—one in which the second parabola degene- rates into a straight line, and two in which the third parabola degenerates into a straight line. Two of these three cases are shown in fig. 4. As to the,third case, it can Fig. 4. a = 55°3' a =71°9' be seen from the paths shown in fig. 3 that, when the bob is on the third parabolic are, it is moving toward one side of the figure for «=57° 51’, and toward the other side of the figure for a=57° 52’. One of the desired cases must therefore exist for a value of a which lies between these limits. It is very noticeable that the values of « for which these periodic cases occur are not at all uniformly distributed throughout the possible ninety degrees. Their distribution is indicated in fig. 5. On the three horizontal lines are Fig. 4. 6° 20° 40° 60° 80° shown the valnes of a for the cases in which there are in- volved (1) only the first parabolic arc, (2) only the first two parabolic arcs, (3) only the first three parabolic ares. It will be seen that, with only one exception, all of the cases here considered are included between the limits of «=50° BG vee 2”. Northampton, Mass., December 28, 1920. Phil. Mag. 8. 6. Vol. 41. No. 245. May 1921. ot LXXIV. The Spectrum of Helium in the Extreme Ultra- Violet. By Huco Fricke and THEODORE Lyman * HE study of the Spectrum of Helium in the Schumann region,, which was published some five years ago, yielded results which were difficult to interpret t+ ; and the conclusions drawn from a more recent attack on the same subject t, though interesting, were by no means final. Both investigations suffered from the same defect, for to obtain any lines at all when hclium was used in the vacuum tube it was necessary to employ a strong disruptive discharge, an electrical condition sure to introduce impurities by its action on the walls of the tube and upon the electrodes. The upshot of the whole matter was that, though the region betwen 1700 and 600 A.U. contained a considerable number of lines, only those at 1640 and 1215 could be attributed to helium §. It has been pointed out that these wave-lengths fit the relation v=4N € = =) and therefore probably form members of the enhanced spectrum series. Hven now, however, the origin of these lines cannot be regarded as perfectly certain. It has become increasingly evident audine the progress of the present investigation that the arrangement proposed by Hicks || by which a considerable number of lines, many of them certainly due to impurities, were made to fit the formula for the enhanced spectrum, is without justifi- cation {]. The immediate cause of the renewal of the attack on the problem was the discovery by several investigators of a resonance potential in helium corresponding to a wave- length of about 600 A.U. ** The improvements introduced were two in number : first, by the employment of a vacuum spectroscope with a grating of but 20 cm. radius the gas absorption was greatly reduced ; * Communicated by the Authors. + T. Lyman, Astrophysical Journal, xliii. No. 2, p. 89 (1916). t T. Lyman, Science, xlv. p. 187, Feb. 1917. § T. Lyman, Nature, civ. p. 314 (1919). | Hicks, Nature, civ. p. 393 (1919). 4] T. Lyman, Nature, civ. p. 565 (1920). aX OF, Horton & A. ic Davies, Proc. Roy. Soc. xev. p. 408 (1919). F, Horton & D. Bailey, Phil. Mag. xl. p. 440 (1920). J. Franck & i, Knipping, Phys. Zeitsch. xx. p. 481 (1919); Zettsch. f. Physik, i. p. 320 (1920). K. T. Compton, Phil. Mag. xl. p. 553 (1920). Spectrum of Helium in the Extreme Ultra-Violet. 8195 and second, by the use of a continuous current in the discharge-tube the chance of impurities was minimized. The result of these improvements was the discovery of a fairly strong line at 585 A.U.° The existence of this radiation was confirmed by an observation with the vacuum grating spectroscope of 97 cm. radius, long used in this laboratory. It is interesting to note that the resonance potential corresponding to 585, 21:2 volts, agrees rather with the results of Franck and Knipping than with those of other investigators. The significance of this fact will be con- sidered presently. It may be concluded that, apart from the two lines already mentioned as probably belonging to the enhanced spectrum, but whose origin is even now somewhat doubtful, only one line in the extreme ultra-violet can be ascribed to helium with any certainty at present. The description of the experimental arrangements and a detailed discussion of results follows : The vacuum spectroscope containing the grating of 20 cm. radius was so arranged that its joints could be closed with Khotinski cement. This method of sealing together with the small volume facilitated the production of an excellent vacuum, and assisted in preserving the purity of the gas under examination. The discharge-tube was of quartz, the cathode being of aluminium and cylindrical in form, about 2-4 cm. long with a hole 5 mm. in diameter ; the anode was of tungsten. The cathode was always placed at the end of the tube néar the slit. The helium was purified by char- coal and liquid air, the spectroscope being protected from mercury vapour by U tubes refrigerated in the usual manner. The gas when originally prepared was free from impurities. However, as it was impossible to heat the whole apparatus, the helium not infrequently showed the presence of traces of hydrogen and the oxides of carbon when examined after it had been admitted to the spectroscope. The gas was usually at a pressure of eight tenths of a millimetre, the current being between twenty and forty milliamperes. : The dispersion of the grating was 846 A.U. to the mm., the region contained in the length of the photographic plate extending from the slit image to the neigbourhood of 1200 A.U. Under these conditions the strongest line on all our plates occurs at 585 A.U.; its intensity bears a constant ratio to that of the direct image of the slit. The other lines which are usually present at 686, 860, 972, 992, 1026, and 1176 3 H 2 816 Messrs. H. Fricke and T. Lyman on the Spectrum show varying relative intensities ; none of them occur on all our plates. Their behaviour indicates that they are due to impurities. This conclusion is confirmed by the fact that we have found lines at 686 A.U., 860, and 1176 when we employed a high-potential vacuum spark between carbon terminals — with our small spectroscope. Millikan* has obtained radiation from the same source at nearly identical wave-lengths, namely 687°3, 858°6, and 1175°6, using a grating of greater dispersion than ours. The line 992 has already been attributed to an unknown impurity f ; 972 and 1026 are due to hydrogen f. Hicks ¢ has suggested that as the lines 972, 992, 1026, and 1086 fit the formule for the enhanced spectrum of helium, they must belong to that gas. The fact that the line 1086 is not found on our plates is sufficient in itself to disprove this idea. When we undertook to confirm our observations by means of the large spectroscope, it was only after several trials and with a very long exposure that we were able to obtain a faint but unmistakable record of \585. It was accompanied by fairly strong lines at 1216, 1200, 1086, and 1085, all of which have been observed before with the same instrument and are known to belong to hydrogen or to some other im- purity. In the visible spectrum the discharge showed traces of hydrogen and carbon in about the same intensities as when the small spectroscope was used. In addition, nitrogen bands appeared very faintly at the end of the exposure. This impurity, not present with the smaller grating, may account for the appearance of the lines at 1200, 1086, and 1085. The region covered by the photographic plate extended from 500 to 1300 A.U. Owing to its faintness and to the fact that it was separated from lines of reference by a considerable distance, an exact measurement of the wave-length of this resonance line is impossible at present; however, the value of 585+2 may be regarded with confidence. The feeble character of X 585 may be attributed to the long eas path of the large spectroscope whose grating has a radius of curvature of 97 ci. as against the radius of 20 cm. of our small instrument, if we are willing to admit a certain amount of selective absorption by helium in the neighbourhood of the line in question. This hypothesis of selective absorp- tion is made necessary by the fact that a disruptive discharge * R, A. Millikan, Astrophysical Journal, lii, p. 47 (1920 ). + T. Lyman, Astrophysical Journal, xiii. No. 2, p. 89 (1916). t Hicks, Nature, civ. p. 393 (1919). of Helium in the Extreme Ultra- Violet. 817 in helium examined with the large spectroscope yields strong impurity lines of shorter wave-length than A 585. It may appear curious that this resonance line was not discovered in previous researches when a disruptive dis- charge was employed. This may be explained in two ways : first, on the ground that the impurities known to be liberated by the disruptive discharge carry the current to the exclu- sion of the helium; second, by reference to the work of Compton, Lilly, and Olmstead* on the minimum arcing voltages, in which they have shown that when current density is high it is possible to obtain an are at about 8 volts with the emission of the ordinary helium spectrum, while the resonance potential line requires upward of twenty volts. It would seem, therefore, that with high current density it might be quite reasonable to expect a bright helium spectrum in the visible spectrum without the resonance line in the extreme ultra-violet. In conclusion, it is interesting to note that, if we follow the speculations of Bohr, Sommerfeld f, and others {, two resonance potential lines in helium might be expected corre- sponding to the two principal series of this element. The wave numbers of the limits of these two principal series are respectively 32031 and 38453, these figures corresponding to potentials of 4:0 and 4°8 volts. Taking the ionization potential of helium as 25:2 volts and subtracting from it the values 4:0 and 4°8, we obtain the values 21°2 and 20°4 as the two values of the resonance potentials in helium. The line 585 corresponds to the value 21:2: that we have only this line in the spectrum and not the line corresponding to 20°4 volts appears to agree with the work of Franck and Knipping. Jefferson Physical Laboratory, Harvard University, December 1920. * K.T. Compton, E. T. Lilly, & P. S. Olmstead, Physical Review, xvi. p. 282 (1920). Compare T. C. Hebb, Phys. Rev. xvi. p. 375 (1920). + Sommerfeld, Atombau u. Spektrallinien, 2nd Ed. p. 287. { J. Franck & P. Knipping, Joe. cit. fests. ] LXXV. Note on the possibility of separating Mercury into its Isotopic Forms by Centrifuging. By J. H. J.. Pooug, M.A.* T appears certain from the recent work of Dr. F. W. Aston on the mass spectra of the elements that mercury is really a mixture of several isotopes of varying atomic weight. Dr. Aston ina letter to ‘Nature’ (Dec. 9, 1920) states that his most recent results have shown that it consists of at least six isotopes with atomic weights of 197, 198, 199, 200, 202, and 204. The exact resolution of the four isotopes between 197 and 200 has not yet been definitely deter- mined, but the existence of some such isotopes and also the two of atomic weight 202 and 204 may be taken as definitely established. The maximum difference between the isotopes of mercury would amount to 7 units, but as we are ignorant as to the proportions of the various isotopes present it is impossible to deal with the question of their separation when centrifuged in a rigid manner. I propose to assume that the mercury can be treated as a nearly equal mixture of two isotopes only which differ in atomic weights by 4 units. This seems to be a fairly justifiable assumption to make, as the mean atomic weight of the two heaviest isotopes whose exact atomic weight is most accurately known, exceeds that of ordinary mercury by over 2 units. As regards the equilibrium state of a mixture of two liquid isotopes in either a gravitational or centrifugal field of force, the case is not very clear. Drs. Lindemann and Aston have dealt with the subject as regards gaseous isotopes in a paper in the Philosophical Magazine for May 1919. Their application of the results obtained to the case of two liquids is, however, not so plain. ‘he following discussion will perhaps throw some light on the problem. Let us assume that the two isotopes differ only in mass, 2.e., the molecular volumes and all other properties are the same for the two. Further, let us assume that mercury is incompressible, so that the total number of molecules per c.c. is constant. Consider first the case of a column of mercury in an ordinary gravitational field. It is obvious from symmetry that if equal volumes of both isotopes are present on the whole, then at the central section of the column the number of molecules per c.c. of each isotope will be the same. let us accordingly take this point as our origin and consider the equilibrium of a layer of mercury at a depth & below this. * Communicated by the Author. On separating Mercury into its Isotopic Forms. 819 Let n,j=number of molecules per c.c. of isotope A at this depth, n2=number of molecules per c.c. of isotope B at this depth, m,=mass of molecules A, mg =mass of molecules B. Then, since we have assumed that mercury is incompres- sible, ; Ny +N,=a constant, dn, dng doe HO Also the density in the layer=nym,+n m,. Consider an A molecule. It is acted on by a vertical force downwards due to its weight of m,g, and a vertical force upwards due to flotation of = (nym,+ngmz)g since the volume of a 2 ale M+ No Hence there will be a net downward force on an A mole- cule of molecule = 1 \ E coe (nym, + nana) |g __ 22(— M19) me Ny + Ng 5 T£S is the area of the column, the total force on all the A molecules in a layer of thickness da due to gravitational effects NyNo ny Ny + Tho (m,—m.) .S. dz. Now we have assumed that the mercury is in equilibrium hence there must be a force acting which will counterbalance this effect. Since we have also assumed that the two isotopes A and B are exactly alike except as regards mass, and that the total number of molecules per c.c. is constant, it is plain that the attractive force on a molecule due to the sur- rounding molecules will on the average be zero, and no re-distribution of the two classes of molecules will affect this result. : It would thus appear that the only force which could counterbalance the tendency of the heavier molecules to move downwards would be due to the increase of the osmotic pressure of the latter owing to an increase in the number of 820 Mr. J. H. J. Poole on the possibility of separating them present per c.c. This really amounts to the fact that the number of A molecules which diffuse out of the layer downwards owing to the gravitational effect must be equal to the number which diffuse into the layer from below owing to the increase of concentration downwards, so that the total number of A molecules in the layer remains constant, 2. e., the mercury is in a state of statistical equilibrium. Let W= average molecular energy. Then the osmotic pressure=23Wn where n=number of: molecules per cc. Hence we have at once : Ity7b5 2W odin =— y+ No (m,—my)g. da dn, din, a (ay — Mo )g aa ny lee W z din, dns But —— == — = dz TR (ai ian, . & (mi, —Im oo be EAN ae 5) SSS Se dz, Uap ang | eae \ | n 2 (m,— My) Gx gg 2 eee fp) W Now, since we have chosen our origin so that ny=n, when “z=0.C must be zero. Hence ny 3 (my— mM) ga log = aR 5 Ics: . : 3 RO But W= IN 5 where R is the gas constant, 6 the absolute temperature, and N the number of molecules per gram-molecule. - oe — NGm— me) ga Hie R@ : It is probable that the mercury molecule is monatomic, as all metals appear to be so. Hence Nm;=one gram-atom of isotope A and N(m,—m,) = M,—M, grm., where M,; and M, are the atomic weights of the two isotopes. Mercury into its [Isotopic Forms by Centrifuging. 821 Thus finally we have loo (M,— M,)yr meena It we are dealing with a centrifugal field of force, the equation obviously is Ci TOE ON iis Tigh. — =>. =. o'r . dr Ny Ny 2 W 2 fon USB TES oun DN which gives Jaya als oe a ae e Ds 2RE : In this case we cannot eliminate C so easily as we do not n : know == for any given value of r. Vlo As, however, we are not very certain as to the exact conditions, I propose to deal with the question as though the centrifuge tube was in an uniform gravitational field of magnitude equal to the centrifugal field at the central section of the tube. ‘This is of course only an approximation, but it will be probably good enough to enable us to obtain some idea of the magnitude of the result to be expected. The particulars of the centrifuge used in our experiments on the effect of centrifuging liquid Jead (see Phil. Mag. March 1920) are approximately as follows :— Length of centrifuge tube ............ =6 em. Distance of inner end of tube from Ceniire OL Lotanlon i a =— On. Number of revolutions per sec. la) Hence acceleration at centre of tube =w?r ==) x 2am) x i, and at outer end of tube, log = yee, ax UDO x 2a )e xox 3 7 83715 3005108" Since #=air temperature = 300° Abs. (approx.), Eye Sala cle Weg aya. M,—M,=4 hal! X27) (; and # is to be measured from centre of tube, and is there- fore=3 cm. This gives finally that a 1-003. We have now to consider what effect this small change in the concentration of the two isotopes would have on the mean density. ey 822. On separating Mercury into its Isotopic Forms. Let b= , d,=density of isotope A, Nae d.= ry) oy) B. Gee i 15 Then hese =1:02 very nearly. And the density of the mixture 1 44 1g en + dy) = = (£102 +1). Putting k=1:003 and k=1 respectively, we find that the difference of the densities at the outer end of the tube and at the centre equals ‘000015d,. The difference between the densities at the outer and inner ends will of course be double this, but still it only amounts to one part in 30,000. It would probably be extremely difficult to detect this differ- ence with any certainty, and thus experiments with this centrifuge would not lead to any really satisfactory results. In this connexion it is of interest to note that J. N. Bronsted and G. Hevesy (Nature, 106. p. 144, 30 Sept. 1920) claim to have effected a certain separation in mercury of about this magnitude by fractional distillation at low pressures. They state that their density determinations can be trusted as accurate to 1 in a million, but it would be difficult to deal with so large a quantity of. mercury, as they did, in centri- fuging experiments. However, it would certainly seem quite feasible to construct a centrifuge in the form of a hollow disk which could be run up to at least 60,000 revs. per minute; and such a centrifuge, if its other dimensions were approximately the same, would give a difference of density of about 1$ parts in 1000, which should be quite easily detectable. The amount of separation might further be increased by successive experiments, but the results obtained would apparently hardly justify the expense entailed in constructing the special centrifuge. Iveagh Geological Laboratory. January, 1921. Norse.—It might be possible to separate liquid neon by this method, if a centrifuge could be run ata sufficiently low temperature. The case of neon is rather more hopeful as there is a difference in density of about 10 per cent. between the two isotopes, and working at such a low temperature would, of course, increase the separation effect very largely. LXXVI. The Gravitational Field of a Particle on Einstein’ s Theory. By F. W. Hitt, M.A., late Fellow of St. John’s College, Cambridge, and G. B. Jerrery, M.A., D.Sce., Fellow of Unversity College, London ™*. “la solution of Hinstein’s contracted tensor equation G,,=0 for a single attracting point mass may be expressed by means of the line element ds? = — eddy? —e" (dO? 4.7? sin? Odd?) +erd?, . (1) where 7, 0, @ are polar co-ordinates and A, w. v are functions of + only. On substitution into the equations G,,=0, it is found that the resulting equations are insufficient to determine 2, uw, v. One relation between them must be laid down, and this corresponds to the way in which the radius vector 7 is imeasured. Perhaps the best known form of the solution is that for which ~=0, in which case we have f ds? = —y dr —7r? d@?—1? sin? Odd? +ydt?, . (2) where y=1—2m/r and m is the mass of the particle. De Sitter { gives approximate solutions for which A=p and A+ 2u+v=0. Difficulty is sometimes felt in applications of the solutions for which X and wp are different, and this is often met by writing r+m for 7 in (2) by which, neglecting squares of m/r, we have ds? = —y-1(dr?+1r7d@? +7? sin? Odd?) +ydt?, . (3) y having the same meaning as before. The purpose of this paper is to show that there is an exact solution for which A=p. In the general theory with co-ordinates 2, %, #3, X4, we have stoma ne de. where the occurrence of the same suffix twice in any term indicates that that term is to be summed for values 1, 2, 3, 4 of that suffix. The sixteen quantities g,, form a sym- metrical covariant tensor whose determinant is denoted by g. The contravariant tensor g?” is defined to be the * Communicated by the Authors. ore + Cf. Eddington, ‘Report on the Relativity Theory of Gravitation, p. 46. eRe t ‘Monthly Notices, Royal Astronomical Society,’ Ixxvi. p. 699 (1916) ~— 824 Mr. Hill and Dr. Jeffery on the Gravitational minor of g,, divided by g, and we introduce the Christoffel three index symbols: ; Og O0nas Opa’ B) Seg Pe pe ee Pos T} i a9" ( 02, r 02, OL ): Kinstein’s equations of gravitation in free space are then Ge = - 2 for, at} a8 {Od, By uf OF at | 2. 16 7 ag taam BY —y— lon atx log ¥—g= 0. In our case, we have from (1) 2 pl LER Oo alt i = ee > Gea = 2c", 933 = — 1" sin? 6 ee a eee g = —rt sin? Oat”, OT 8 Tot Ue i OS Sc5, and GPP = Nb Gore It is then found that of the 40 Christoffel symbols only 9 survive, and their values are easily calculated. Sub- stituting these into (4), it is found that there are only three equations between A, u,v. As given by de Sitter *, these are : ; 2 i: Ve ! ! pit ay +— pid! + gh — gh — ah + ay” = 0, betAT 1 + Ory! + dre" + br(v' —2X') +4rp'(p'+4v'—3r')] = 0, bu!" 4a! + do! + 4v'’—tX') = 0. Putting X=p, we have ay) aaa — 0’ +40 Te == 0, (5) Wa 3 ! u I ILS He, lie 6 NX: ome ad ar eh Sie = (0, (6) ] —bY oy =p =O, (7) * ‘Monthly Notices, Royal Astronomical Society,’ Ixxvi. p. 712 (1916). Field of a Particle on Einstein’s Theory. 825 Adding these three equations, we have the following equation in X only : i 4 ! / 2r eg 3X? = which on integration gives e=(1tz),- hi 220, ainsi) where m is a constant of integration and a second constant of integration has been checen so that X->0 as r-> wo. Substituting in (6), we obtain an equation in v which readily integrates to give 1-F\: Ca ae ° . 5 5 : ° (9) eee It is then necessary to show that (8) and (9) satisfy either (5) or (6), and this presents no difficulty. The line element may therefore be written without approximation det = —(1422) (er ade tre sin2oagy 1-5 +( 4 dt?. (10) La The constant m is most readily identified with the mass by considering the approximation of (10) when m/r is small and comparing it with (2). In fact, (10) may be obtained from (2) by means of the transformation ran(l+s-), RAAT akan gnfieae ho tt aan (GlalD) and then dropping the suffix in 7. The advantage of the form (10) lies in the facility with which it can be transformed from one set of co-ordinates 826 On Systems with “Propagated Coupling.” to another. For example, in Cartesian co-ordinates it becomes : ae at=—(1+ +) (da? + dy? +dz2) + ( 4 dt?. 10 ‘et It cannot be too clearly Loca ae that the difference between the two forms of the line element given in (2) and (10) depends only upon the way in which we agree to measure the radial distance 7». The form (10) can therefore give no physical result which is independent of radial measurements which is not already implied in (2). For example, the equation of the orbit © of a particle is obtained from dx, Ax, lity : “ge FAs a} ae a 0. . 2 eee For a particle moving in the plane 0=$7, the form (10) gives du m 6mu een omen 77 8...) Oe (13) il while (2) gives the more usual form a u dq? where, as usual, w=1/r. These two differential equations have different solutions ; but if we investigate a fact which does not depend upon radial measurement, the advance of the apse line per revolution, (13) and (14) agree in giving 3m?/h’. tu = 7+ Bmw’, ot ees LXXVII. On Systems with “ Propagated Coupling.” To the Editors of the Philosophical Magazine. GENTLEMEN,— Proressor A. W. Porter, F.R.S., and Mr. BR. HivGagee B.Se, have published in No. 243 of the Philosophical Magazine a paper “ On Systems with Propagated Coupling” (March 1921, p. 432). In the first part of this communi- cation some interesting experiments are described, but in the second part (p. 434), in the paragraph headed ‘“ (ii.) Simple On Systems with “Propagated Coupling.” 827 dynamical case illustrating maintenance of vibrations,” the authors give what comes toa theoretical treatment of what happens when an ordinary telephone receiver is connected toa battery. Professor Porter and Mr. Gibbs arrive at the unexpected conclusion that this telephone receiver will produce a continuous sound. This obviously cannot be the case, and the erroneous result arrived at is due to an unfortunate oversight of a minus sign. In fact the second equation of p. 435 must be —a0+ (moa 75 +K)y=0 instead of ; a? pi ites 6 +aC+ (mi Se ss +K)y= if the first equation 1s maintained. This error, which is repeated a few times (pp. 437, 433, 440), just makes the maintenance of free undamped vibrations, which the examples are intended to illustrate, impossible in all cases treated, none of the differential equations given having purely imaginary roots. In order to obtain spontaneous oscillations a variable resistance must be present in the electrodynamical system, as was the case in the authors’ experiments, where a tele- phone receiver was coupled acoustically to a microphone transmitter, and not to another telephone receiver. tam, Gentlemen, Yours truly, BaALTH. VAN DER Pou, Jun. Physical Laboratory, Teyler’s Institute, Haarlem (Holland). To the Editors of the Philosophical Magazine. GENTLEMEN,— We must thank Dr. van der Pol for pointing out the unfor- tunate slip in our paper in connexion with the sign of «. We do not agree with him, however, in regard to the necessity for assuming a variability in the resistance—though this variability was certainly present either in the micro- phone or in the valve set. It may be pointed out that the experiment works equally well with Brown magnetophones. The effect of the propagation of the mutual action is to create a phase-difference: between y and © to an amount 828 - Geological Society. depending upon the separation of the reacting parts; and this, in certain positions, is equivalent to a reversal in the sign of «, which thus becomes of the same sign as given in the paper. Since the position for which this will occur is different from what we supposed, it must be admitted that the detailed algebra requires to be restated. ALFRED W., Porter. | REGINALD EH. Gress. March 15th, 1921. LXXVIII. Proceedings of Learned Socteties. GEOLOGICAL SOCIETY. [Continued from p. 684. } May doth, 1920.—Mr. G. W. Lamplugh, F.R.S., Vice-President, in the Chair. f ; ‘HE following communication was read :— ‘A Natural ‘‘ Kolith” Factory beneath the Thanet Sand.’ By Samuel Hazzledine Warren, F.G.S. The paper describes a section in the Bullhead Bed at Grays, where the conditions have been favourable for the chipping of the flints by subsoil pressure. There is evidence of extensive solution of the Chalk beneath the Tertiary deposits, and the differential movements thus brought about have occasioned much slickensiding, and remarkable effects in the chipping of the flints. In the author’s opinion the section affords the most complete and conclusive evidence hitherto obtained in support of the theory of the origin of the supposed Holithic implements by purely natural ‘agencies. ‘There are not only the simpler Kentish types, such as notches, bowscrapers, and the like, but also the larger and more -advanced forms of rostro-carinates which are characteristic of the sub-Crag detritus-bed. Careful digging enables the pressure-points of one stone against another and the resultant chipping effects to be studied in detail; and in many instances the flakes removed can be recovered and replaced. | A few examples are more than merely Kolithic in character. If such exceptional examples were removed from their associates, and also from the evidences of the geological forces to which they have been exposed, no investigator could be blamed for accepting them without question as of Mousterian workmanship. Individual specimens may often deceive: in order to distinguish a geological deposit of chipped flints from the débris of a prehistoric chipping- floor, it is necessary to base one’s judgment upon fairly representa- tive groups, and also to take into consideration the circumstances in which they have been discovered. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. a. = ™ ([SEXTH SERIES.) 7 . £, KF Be y 1GOF = ; ¢ bal Ze oth JUNE 1921. “~< ~ LXXIX. The Dielectric Constants of Electrolytic Solutions. Ba ix. arty. BSc. A.* I. HH values given by various observers for the change in the dielectric constant produced by dissolving electrolytes in water show considerable divergence. To take the case of copper sulphate—Drude’™, Coolidge”, and Palmer*’ were unable to distinguish any difference between the dielectric constant of pure water and of solutions of this salt, whereas Smale? asserts that a 0:05 normal solution has a dielectric constant 15°5 per cent. greater than that of water. Now, according to the Nernst-Thomson * rule, the dielectric constant of a solvent is intimately connected with the degree of ionization of an electrolytic solute; and it is therefore of importance that we should have a knowledge of the effect of the solute on the dielectric constant of the solution. It seems to be pretty generally assumed that this effect is to raise the dielectric constant—a view which is supported by Smale’s experiments with aqueous solutions and by Walden’s © experiments with non-aqueous solutions. The observations of Drude !*?!, Coolidge’, Hichenwald*, and Palmer * are * Communicated by the Author. Phil, Mag. S. 6.:Vol. 41. No. 246. June.1921. 3 830 Mr. R. T. Lattey on the Dielectric generally overlooked*. This may be because these ob- servers have usually been somewhat guarded in their statements. Drude!’, for example, says: ‘Man wiirde daher auf eine geringe Abnahme der Dielectricitats- constante mit wachsendem Salzgehalt zu Schliessen haben. Wenn auch diese Schluss wegen der Grosse der Beo- bachtungsfehler nicht mit voller Sicherheit vorliufig zu ziehen sein mag, so geht doch aus der Beobaclitungen zweitellos hervor, dass die Dielectricititsconstante des Wassers durch Auflésung eines LHlectrolyten selbst bis zu der Leitfahigkeit K=11.1077 jedensfalls nicht ver- erdssert wird.” Practically all methods by which one can determine the capacity of a condenser whose plates are separated by a conducting medium involve the use of alternating currents ; these may have a comparatively low frequency (2C0 to 300 radians per second, as in the experiments of Franke *, Heerwagen*)*, and Rosa‘) or a considerably higher frequency (10° to 10'°, as in the experiments of Niven”, Harrington ®, Nernst*!*, . Marx *-9,- Cole *, -’iamgden ee Colley}, Lampa®, and many others). With moderate frequenc-es (not exceeding 10° or 10%) it is possible to use forms of Wheatstone’s bridge (see 1, 26, 27, 30, 38, 41-44, 48, 54, and 59). Silow’s method of charging two electrometers simultaneously by ths same source of E.M.F. and comparing their deflexions when each is filled with a different dielectric, has been applied over a wide range of frequencies (see .d—-8, 28, 31, 32, 46, 47, 49, 51, 52, 55, 57). It is not proposed to discuss either of these methods here, but it may be pointed out that the results obtained from them are only trustworthy when proper corrections are made for the current passing through the condenser by conduction. Imperfect insulators under the influence of an alternating field may behave in one of two ways: (a) they may convey a current in accordance with Ohm’s law and dissipate the power of the current at a rate proportional to 27, or (b) they may cause an absorption of electrical power which is out of all proportion to the conductance they display to a constant field or when only slow oscillations are used. The second form of absorption has been studied * Lecher’s observation ®’ that a condenser had the same effect on Ruhmkorf oscillations when it was filled with water as when it was filled with 10 per cent. sulphuric acid has been quoted in favour of the view that both these liquids have the same dielectric constant {Palmer**). It really shows that both behaved as conductors. Constants of Electrolytic Solutions. 831 by Cohn and Arons®® and by Drude’, and is known as anomalous absorption or dielectric polarization. It has been shown by many observers (see 2, 3, 14-23, 25, 13, 44, 61), and confirmed by the present investigation, that aqueous solutions show only the first type of absorption (that due to their conductivity) so long as the frequency used is not too high. The experiments of Drude”, Colley 1”, and Rukop*® on water, and of Wildermuth *® on solutions, tndicate that at very high frequencies anomalous absorption inay become noticeable. In most cases in which high frequency has been hitherto employed, the current was in the form of highly-damped trains of waves. It is proposed here to confine attention to continuous waves of pure sine form; the treatment of ‘such cases is comparatively simple, and the general con- clusions are similar to those obtaining for -waves of more complex types. In a circuit consisting of an inductance L, a resistance R, and a capacity C, whose dielectric has a resistance r, the Figs 1 (OCOD AAA eye ti Cc relation between the amplitude of the applied or induced HLM.F. (E) and the amplitude of the current in the re- sistance R is given by the equation iD ns Ais / ” 2 ne 1 i (7) mo (R+ ip cae) a \PL~ (G4 p 0 (1) where p=the frequency. If such a circuit is adjusted so that 2 is a maximum, then the conclusions drawn from the experiment may depend on the method of adjustment employed. A. Tuning by adjustment of L. This is essentially Drude’s method!*?%. Z will clearly be a minimum when (pl) -t=pC+p7!C-1r-7. Now, if C, is the capacity of a condenser employing a good insulator and which gives a maximum current with the same value of Las does C, then we may eall C,; the apparent value of C; and it follows that Cy=C+p-?r-?C~}, or C, is greater than Oe l.2 832 Mr. R. T. Lattey on the Dielectric the true value of C, and that such experiments will give too. high a value for the dielectric constant of the substance under investigation unless proper corrections are applied. Comparison of the charges imparted to two condensers by an alternating E.M.F. will require correction depending on the inductance aud resistance of the leads between the con- densers. ° The most complete series of observations of this kind is that carried out by Smale®*’, and his results for the ratio of the dielectric constants of certain solutions to that. of water are given in Table I. TaBLe I. Ratio of D.C. for Solutions to that of Water (Smale). Normality ...... 0-001 0-002 0:005 0:008 0:010 0-020 0-030 0-050 Solute HCl ...... 0:99 1083 1-064 1090 1-126 dd 6) irene LOS 7 018 1:034 1:070- 1S) | alot CusOze-. #17012 1-017 1-050, 1,086, 1°128 Seas The apparent rise in the D.C. due to dissolving an elec-- trolyte in water is most marked in the case of the best conductor (HCl), and increases as the strength of the solution increases. It will be seen from what follows. that the true D.C.’s for these solutions are probably less. than that of water. B. Tuning by altering the size or the distance between the plates of the condenser. As the capacity is increased, the resistance will diminish in such a way that the product Cr remains constant. The condition that Z in equation (1) may be a minimum is. D4N then p°CL=1+R/r, or, using the same notation as before, C=C,(1+ Rr), and the uncorrected values of the D.C. of a solution obtained in this way will be lower than the true values. The method was used by Marx, but was. only applied to water of low conductivity—z. e., the ratio R/r was negligibly small. C. The electrolytic condenser is connected in parallel with a variable air condenser, and tuning oltuined by varying the latter. The electrolytic condenser is then removed and the air condenser readjusted. This was the method used in the present work. The essential features of the circuit are shown in fig. 2. Oseil-- lations were induced by a neighbouring circuit emitting a Constants of Electrolytic Solutions. 833 ‘continuous waves, and the capacity C was varied until the current in L was a maximum. Fig. 2. Fig. 3. A more complete scheme of the circuit is shown in fig. 3, in which D represents a current detector and r the leakage resistance across w. The impedance Z of such a circuit is given by 2 Daa if Li = (R+ 1+ p?r(C+ =, +( ey : ‘ pur pC+petp ir ?(C+a)7* cm a) And on removal of w and 7 this becomes i IEN2 2 92 Bare oat Z? = B+ (pL at a) Ae ey If i and I are the maximum currents obtainable in these two cases, and C, and Cy the values of C which give these currents, then Co = ne 9 eal als )= on eS Them eae eee) ee ke (1 5c} CR. 20, Whence e : iE 2 r g 2 ? fo (K on Gy) == D4 ee er 12h 74 a2 ls... ‘17 184 = -202 2 ae 182 1565 1515 1:87 1995 +1585 118 155 Ls 162 142 1395 1-70 “WhO 1G), 2096). 195 5 eee 163 1345 132 1:58 121 1015S 0815— 108 Bee: 145 130 1275 1:49 101-0895 = 069—~Ss-0889 oe 1860 1-280 188 078 0735 049 ‘072 ax. oterencos an Gn Gin Ge a ear as 36385 “413 -343 eile eoriether | :2545 :236 ‘252 235 Avid ny bee eet 1865 180 7-179 crm eke 149) Sidon 1148) 7-149 Chins | 124 +1285 125 195 oterepareter o 0955 0985 -:090 098 The correction Co(I?/?—1)/(K?+1—L/i?) in the case where the distance between the plates was three threads of the screw varied from 0°055 to 0°134, and it will be seen that concordant values of w were obtained. IV. CoRRECTION FOR INDUCTANCE OF LEADS. $ A condenser of capacity C in series with an inductance / is equivalent to a condenser of capacity C/(l—p/C). With high frequencies the term p*/C may be appreciable even when / is small. The values given in the above tables under Fig, 5. L L, Cc the heading «x will not therefore be true values of the capacity of the condensers ; to obtain these, a correction must be applied for the induction of the leads joining wv and C, §38 . Mr. R, T. Lattey on the Dielectric Let /, = inductance in series with C and je 9)? 99 oy) 99 wv. Equation (1a) must then be modified, and becomes a(R) +r og BPE BY) OME plat +B? foes AES 7 lel ae + le re O xr (1l—pla)—l ~ Lap, § (1 —p le)? +p where pa = and (4a) If tuning is obtained by varying ©, then a is invariant, and the condition for maximum current is ee Dee Bo BED ang So a ee Kee ane where ©): = Cof(1— ply Co). The values of / and J, can be found by using an electrolyte with a high value of r (e.g. pure water). Hquations (4a) and (5) then become Co C v 1—p7l,C) 1p, * 1=plz or 1p DG 1+ p’?C,(i—h) 40°C (+1) 4 p20, where D=the dielectric constant of water and Ca=the value of « when air is used as dielectric. The term p*l,?CC, is negligible, and therefore (Cy—C)/Cu will be a linear function of C. For conducting dielectrics, _ pila? +A(1+plA) } po (1+ pA)? | 27.\\2 and pr= plat pee) : Lo) eo ( | Or J ee A= p- i ete cs Can Apiyad Constants of Electrolytic Solutions. 839 In such a case pl can be evaluated by utilizing the fact that the product of the resistance and capacity of a condenser containing a given dielectric is independent of the form of the condenser, and hence 7 2 pii(a+" >) +2 = constant. ole Ae eee Oe a The values already quoted were obtained at a time when the importance of this correction for the leads was not realized. They are consistent with equations (5) and (6), but as sufficient care had not been taken to keep / constant, it was not thought worth while to attempt to apply the corrections for J. The above equations may also be used to deduce the resistivity of an electrolytic dielectric : if A is the molecular conductivity of a solution containing n gram mols. per ¢.c., then its specific resistance is 1/An, and if the solution is put into a condenser, then the resistance of this will be (An36007C,)~* ohm, where Cy, is the capacity of the condenser in millimicrofarads when filled with air. Hence CE et) 8) 3000TNCg( pia t+ (1+ plA)?/a) When this equation is applied to the observations of Table V., A for 0:000755 normal KCl solution is calculated to be 105 (Koblrausch gives 129 at 18° C.). VI. EXPERIMENTS witH IMPROVED Form or APPARATUS. The final form of the apparatus is shown diagrammatically Fig. 6. \ Transmitter | in fig. 6. This was designed so as to reduce Land R to a minimum ; two rectangles of brass strip connected in parallel 840 WEE IRE ae Lattey on the Dielectric were used as receiving inductances (Lj, L.). These were connected to two adjustable condensers, the other plates of which were connected to two circles of brass strip (13, Ly) also in parallel with one another. Between these two lay two turns of thick copper wire connected to a thermal converter and a millivoltmeter. It was found that with this arrangement the resistance for high-frequency currents in the principal receiving circuit was about 1/5 ohm. It is probable that this is not, properly speaking, the real ‘resistance,’ but a function of this, of the mutual inductance between the two circuits and various leakage effects. All of these will vary when the frequency is changed. The various values of K found at various times were used to calculate R, and the results are shown in Table VI. TasiE Vie d (metres) ...... 33°2 35°5 39°2 45°5 45°5 45°5 45°5 R(ohms) ...... 306 "269 "346 211 210. 216 ‘206 A (metres) ...... 45°5 45°5 45:7 45°7 46:2 47-3 R<(Olingis)- "2-5-4 “209 ‘190 "220 “200 192 “230 These agree fairly well with the empirical formula RA=9'68. Experiments with other circuits over a wider range of wave-lengths showed that this formula is jus- tifiable. The condenser used in these experiments consisted of two coaxial cylinders ; the inner one could be withdrawn from inside the outer by means of a screw. Mr. H. W. B. Gill kindiy calibrated this condenser for me with air between the plates. It was connected to C by two stiff parallel wires, in order that the inductance of the leads might be constant. Experiments with water and with elycerol in the condenser enabled a series of values of pil and pl, to be found ; these were in satisfactory agree- ment with 1 = 343 cm. and /, = 67 cm., assuming the dielectric constant of water to be constant ‘over the range of frequencies employed. This assumption is justified by series of experiments to be described later (see Section VR): Table VII. gives the results obtained for water at various wave-lengths (A) and various temperatures. These have Constants of Electrolytic Solutions. 841 all been reduced to 18° C., using Coolidge’s value for the temperature coefficient *. TasBLE VII.—Dielectric Constant of Water. x 7 OO. 1D). De SO UC) ih da, 80:3 So se ee 14 84:25 82°85 Baer omer nn Mean 82-05 UChr Peer is. Welk 80:2 799 By OMe eee ar tc 18:2 82:2 82:25 AU ane a os 19-1 81-05 81:45 TEES eae 16'8 78:2 77:8 AAO ey ee ily 82°5 82:15 A oe utes 18:2 81-2 81:25 AON ue 11°8 84-9 82:7 ANAS Thea Ooi 14 BAO 82°6 ASI Cee i 176 81-0 79:85 APO alee eae 12:2 80:95 78:95 AG OR lin eee 76:7 AGs Gh Mere 191 73:0 78:4 Gieoia e 18:6 81-4 - 81:6 GlEG se 17°4 81:25 81:05 Om on ue 19°8 82-05 82:7 SO se 17-2 80:7 80°4 + UR aN pene ae 166 80°7 80-2 CB ae eae Le 82'8 82°65 Meera gerne Wie it 81°05 * The temperature cvefficients found by various observers vary between —0:28 per °C. and —0°495. Coolidge’s value is in fair agreement with that found by Marx and Heerwagen. : “ AD Observer. Kegs ORE: RROsae ee Oe ee 778 0:348 Gol oe ee wee 74-7 0°3383 nicerwarens? 7 eh a. 80°52 0362 iramiceicceren. eee 81:6 0-504 AA bea aea ge eta cae T4227 0-473 PUGS 2h Boe ee 81:4 0-368 OBR AS Ue ale ae a 80°6 0°39 ee eae 80:42 0-425 Coollidwet Ms ous: 81:25 0352 alterna leet hen eawe rh tr dcscss 0°358 at 60 ~ 0:507 at 10®~ 0-570 at 38 .106&~ INGER Gee ie saves nes 80:0 0:28 SY ES es YON 82:3 0-362 INI eet eh 5 econ ba: 79:4 06382 842 Mr. R. T. Lattey on the Dielectric Some experiments were also made in which the leads were cut down to a minimum; this reduced J to 285 ems. ‘The results are indicated by an asterisk. Tasie VILII.—Dielectric Constant of Glycerol. (All these observations were made at 17°-18° C., except that for \=45°5, which was at 14° C.). r. D. G32 eat ae 52°85 23) aan eer 49°85 (UES eee 53°05 “OSes 50:9 Lee ae ee 51-95 So Sauce Caan ese SPE Ie seal oe 52°15 Z(G Wareee een eg 49-5 (SER ee ee eee (53°05) AO Sassen 49°05 SIR Spee eee oie 01:0 peas ev ee 50°05 Mean ...... 51°15 In the case of water the most careful determination seems to be that of Turner *, viz. 81:12. The somewhat varied values by other observers found for glycerol indicate that D for this liquid is dependent on the frequency ; at the same time it must be borne in mind, when comparing the results of different observers, that it is by no means easy to obtain glycerol free from water or to prevent its contamination by water. The various values found in the literature for D and for the square of the refractive index (n”) are collected below :— Observer. Metres. iD). n2, Thevamepeeen. a0.) 10 56°2 Drude 2% 1% 20, 21 2 oc O15 ee 25-4 G73 £9 st a 16°5 V.., Linnie epee emer: -0°085 oe 14-1 Hickert “42 0057 [ 14-4 Mercezyne eae 0045 sie 16°8 lampa? te 5 eee 0-008 ee 3:4 0-006 ook 0-004 2°62 bi Constants of Electrolytic Solutions. 843 VII. DreLectric CoNSTANTS OF SOLUTIONS. Four substances were chosen for investigation :— Sugar: a typical non-electrolyte. Potassium chloride : a uni-uni-valent electrolyte. Copper sulphate: a bi-bi-valent electrolyte. Tetraethylammonium napthalene-@-sulphonate: a uni- uni-valent electrolyte with comparatively large ions. (a) Sugar Solutions.—Three concentrations were used ; the traces of saline impurity in commercial sugar gave the solutions an appreciable conductivity (the 20-per-cent. solution had a conductivity comparable with that of 0:001 normal KCl solution). TABLE [X.—Dielectric Constants of Sugar Solutions. Wave-length in air=45°5 metres. K=38°4. Percentage of sugar ...... 6°84 11°92 20°6 Temperature (°©.) ...... 13 14 16 Wfor: solution! ... i kcccsceee: 81:2 69°8 Gi9'e.) Witor water ?...c.-) c.02 Ce 82°8 82°45 81°75 IBC O sneeonnmonr Mer Oren carn ccst: | 0-985 0°85 The last horizontal row gives the ratio of D for the solutions to that of water at the same temperature. Inter- polation of Harrington’s results? gives 0:975 and 0-96 for the corresponding ratios; Drude’, using waves of 2 metres length in air, found for the square of the refractive index of a 40-per-cent. solution the value 67:5. He noted “strong absorption ” equivalent to that of a solution having con- ductivity 3x 10~", though the actual conductivity, as tested by the ordinary methods, was only 6 x 10—°. (b) Potassium chloride.—In the case of the more con- centrated solutions the maximum current was so small that accuracy in adjusting the condenser was very poor. The necessary corrections are large and depend on a number of observations, and hence the values obtained for D are untrustworthy and are not quoted. It will be seen that very fair agreement was obtained between values of the mole- cular conductivity (A) and those obtained by Kohlrausch*. * As the apparatus was designed with a view to making the effects due to conductivity as small as possible, the accuracy with which A may be determined is also small. 844 Mr. R. T. Lattey on the Dielectric TABLE X.——Dielectric Constants of KCi Solutions. (metres) ese: 34:2 dg eare nent a scice 22-9 Memyp.-(°/ CG.) ee lO Normality ...... 000755 D for solution .... 80°25 D for water ...... 81°95 IAGO Peer ore 0-98 Nia Seen a ae ers 147 A (Kohlrausch). 1216 (c) Copper-sulphate Solutions. X was in all cases 45°5 metres and K =57°5. Temperature (° C.).... Normality 1) forsolution:.:.-25-5. DOr Water ns-2o-c ne Ratio (d) Tetraethylammonium Solutions. eeeres sere eee > fe ree reer eer ceesace 45°d 45°5 45°5 45°5 45:5 384 45°9 57°5 Dies 459 15°4 10°6 16°8 16:0 11-4 000753. “OO151T - :0075s"> O1on ‘O151 75°0 Tad 66°25 81°95 83°65 81°45 0-915 0-93 0-815 121°5 101 [os POUT 91°53 - 121°6 106°1 120-4 116935: 1052 TABLE OXI, ifs 12-2 14:8 14:8 ‘00114 “00228 00456 ‘0228 15:2 18°2 [39 81°08 82°75 82°15 0:93 0:945 0:90 89°8 13°F 67:0 41°7 SES 84:5 765 55°35 - X and K as for copper-sulphate solutions. Memperature: (°C) .2.<:. Normality......... D for solution ... _D for water ...... eee teecee AB Ica Nelle 166 14 ee 002 ‘005 ne 76°3 69:0 Ph here 81°55 82°45 a5 eee 0-935 0-85 38°8 32:5 naphthalene - 8 - sulphonate 14-2 ‘010 32'8 The agreement between the values obtained for the mole- cular conductivity and those obtained by Kohlrausch at smaller frequencies indicates that frequency has very little influence on this quantity. In the case of the tetraethyl- ammonium salt no data are available for comparison, but Constants 0; Electrolytic Solutions. 845 data for similar acids and bases indicates that the con- ductivity at infinite dilution should lie between 30 and 40. It has already been pointed out that the agreement above referred to is an indication of the absence of anomalous absorption or dielectric polarization. Some attempts were made to calculate the correction factors in difficult cases by using Kohlrausch’s data, but uncertainty as to which of the observations was to be considered faulty rendered the attempt fruitless. . The ratios of the dielectric constants of the solutions investigated to that of water have been plotted against concentration (in grm. equiv. per li.) in fig. 7. The rela- tionship appears to be linear and practically independent of the nature of the salt. Ratio of D to that of Weter -002 2004) -006 -008 -O10 Normality of Solution. Wik: The Dielectric Constant of Water is independent of the Frequency between Wave-lengths of 1% and 52 metres. A receiving circuit was made up, consisting of (i.) an inductance lL, which could be varied to suit the frequency employed but which was kept constant during each series of observations, (ii.) a variable condenser ©, (iii.) a small inductance coupled with the detector D, and (iv.) a fixed air condenser C,. The latter could be replaced by one Piul, Mag. S. 6. Vol. 41. No. 246. June 1921. 3K 846 Mr. R. T. Lattey on the Dielectric having the same external dimensions, but with a greater distance between its plates. When this was filled with Fig. 8. water its capacity (w) was not very different from that of G,. By noting the value of C, which brought the circuit into tune with wave of known frequency, first with C, and then with xz in the circuit, the relative values of C, and 2 could be caleulated. The results are shown in Table XIII. TasBLE XIII.—Capacity of Water Condenser (.), taking Air Condenser as 0°465. Inductance. L, Ly, i Lz. lige ix L,. Mean. Napili e = a “ e .. ‘5637 +587 550 19900. 602° 5540 517-5640 Sees eee 105o5 ee =o: .. °B730 .... *5687 5968-580 Dy eee). 5818 -s52G Ky ec: a7 5965 5697 TG i oa 5RE 33-4 oo, ec) 5733 | oe te io eae 'BB4B oo... 26042 EST oe” 5502 2 oe. in su is eee: fies ey ee a a 2 Bs | BB GB ce. GGL. y io vf on 2 G56 Mean. 570 -569 -579 -5645 ... 1°56b> -SA7mONESES The only corrections applied to the above results is a small one due to the conductivity of the water; the variations from the general mean are only such as may be expected from the probable experimental errors of the method, and show no trace of any connexion with frequency. Marx* has shown that D for water is constant between X=11'1 and Xu. =17°8 metres. LX. SumMMARY. The methods in use for finding the dielectric constant of water and of solutions are criticised, and a method is described by which the necessary corrections for con- ductivity may be applied. Constants of Electrolytic Solutions. 847 The electrolytes investigated appear to lower the dielectric ‘constant of water, and are in this respect analogous to the majority of non-electrolytes. The Electrical Laboratory, SOR De OG es ORS Oxford. Abevg, v. Baeyer, Berg, Cole, Cohn & Arons, 3” s9 Cohn, Cohn & Zeemann, Coehn & Raydt, Colley, A. R., Coolidge, Drude, 79 Kekert, Hichenwald, Everscheim, ”? Franke, Hertwig, Harrington, Heerwagen, ve) Joachim, Klein, Lampa, v. Lang, Lecher, Marx, 9) Mercezyng, Nernst, ” ) REFERENCES. Wied. Ann. lx. p. 54 (1897). Ann. Phys. xvii. p. 80 (1905). Ann. Phys. xv. p. 807 (1904). Wied. Ann. lvu. p. 290 (1896). XXVl. p. 454 (1886). Xxxul. pp. 138, 381 (18838). XXXVlil. p. 42 (1889). xlv, p. 370 (1892). i lvit. p. 15 (1896). Ann. Phys. SO 105 HU (INO) xiii. p. 809 (1914). Phys. Ztschr. x. pp. 829, 471 (1909). Wied. Ann. lxix. p. 125 (1899). liv. p. 852 (1895). ly. p. 633 (1895). lviu. p. 1 (1896). lix. p. 17 (1896). Mf Pie xe paOOO) Stay: ela 66 (1897). Ztsehr. Phys. Chem. xxiii. p. 267 (1897). Wied. Aun. lxiv. p. 131 (1898). yi xv p. 4997 (11898); Ann. Phys. viii. p. 336 (1902). Verh. Deut. Phys. Ges. xv. pp.307, 422 (1913) Wied. Ann. Ixil. p. 571 (1897). Ann. Phys. viii. p. 539 (1902). » X&ill. p. 492 (1904). ”) 9 3? Py) 9? 19 7) ” ” ) Wied. Ann. 1, p. 163 (1893). Ann. Phys. xlii. p. 1099 (1913). Phys. Rev. viii. p. 581 (1916). Wied. Ann. xlviii. p. 35 (1893). yy Mlix. 11272) (1898). Ann. Phys. Ix. p. 570 (1919). Archiv. Math. Astron. Phys. 1918. Medd. k. Vetenskapsakad. Nobel Inst. iii. p. 1 (1918). Wien. Ber. cv. (2a) pp. 587, 1049 (1896). “te 3 - p. 253 (1896). Wied. Ann. xlil. p. 152 (1891). Phil. Mae. xxxi. p. 181 (1891). Wied. Ann. Ixvi. pp. 411, aa (1898). Ann, Phys. xii. p. 491 (19038 Phys Ztschr. iv. p. 531 (1903). Ann. Phys. xxxii. p. 1 (1910). Ztschr. Phys. Chem. xiy. p. 622 (1894). Wied. Ann. lvii. p. 209 (1896). a a p. 600 (897 y 3K ? 848 Mr. H. Carrington on the 44, Nernst & Lerch, Ann. Phys. xv. p. 836 (1904). 45, Niven, Proc. Roy. Soc. (A) Ixxxv. p. 189 (1911). 46. Palmer, Phys. Rey. xiv. p. 38 (1902). 47. a +, Xvi. p. 267 (1908). 48. Ratz, Ztschr. Phys. Chem. xix. p. 94 (1896). 49. Rosa, Phil. Mag. xxxi. p. 188 (1891). 50. Rukop, Ann. Phys. xlii. p. 489 (1913). dl. Smale, Wied. Ann. lvii. p. 215 (1896). 52. =A 7 eipye25 (1897). 53. Speyers, Sill. Amer, Journ. xvi. p. 61 (1908). 54. Tang}, Ann. Phys. x. p. 748 (1903). 55. Tereschin, Wied. Ann. xxxvi. p- 792 (1889). 56. Thomson, Phil. Mag. xxxvi. p. 320 (1893). Nernst, Ztschr. Phys. Chem. ay p- 530 (1894). o7. Thomas, Phys. Rev. xxxi. p. 278 (1910). 58. Thwing, 9 ike Poeo (1895). Wh ae Phys. Chem. xiv. p. 286 (1894) 59. Turner, »» -XXXv. p. 385 (1900). 60. Walden. Bull. Acad. St.’ Petersb. vi. pp. 305, 1055. (1912). 61. Wildermuth, Ann. Phys. viii. p. 212 (1902). 62. Yule, Wied. Ann. 1. p. 742 (1893). Phil. Mag. xxxvi. p. 531 (1895). LXXX. The Modul of Rigidity for Spruce. By H. Carrineton, B.Sc., M.Sc. Tech., A.M.I.Mech.E.* [ Plate IX.] HE investigation described below was conducted during the war in the College of Technology, Manchester, for the Air Board and the ~ Roy al Aireraft Establishment. Aeroplane designers found themselves at some disadvantage, | owing to the lack of information about the elastic properties of timber, and so far as the writer is aware, no published records of values of the moduli of rigidity, determined on the assumption that the wood had three planes of elastic symmetry, were then available. Torsion experiments on prisms of wood were conducted by Gerstner t (1833) and Pacinotti & Perit (1845), who used the old Coulomb theory that plane sections remained plane during strain to calculate the values. A short table of values of Young’s Modulus along the grain, Poisson’s Ratio, and the Modulus of Rigidity for beech and pine, based on the experiment: results of Chevandier & Wertheim § (1848) and of Mallock || * Communicated by the Author. + ‘The Theory of Elasticity,’ Todhunter & Pearson, vol. i. art. 810. t Ibid. vol. i. arts. 1250-1252. § Ibid, vol. i. arts. 1812-1814. || Proc. Roy. Soc. Lond. vol. xxix. pp. 167-161. : Moduli of Rigidity for Spruce. 849 (1879), and assuming the partial ellipsoidal elasticity of St. Venant to hold, is given in a paper on ‘lhe Torsion resulting from Flexure in Prisms with Cross-sections of Uni-axial Symmetry only,” by A. W. Young, M.A., Hthel M. Elderton, and Karl Pearson, F.R.S. (Cambridge Uni- versity Press, 1918). The shortness and incompleteness of the table—for neither the humidity nor the density of the wood 's given, both of which have an important bearing on the values of the el: act that the authors had to reter back prior to 1880, indicates further the lack of information about the elastic properties of timber. The values of the moduli were determined from torsion experiments on prisms of spruce of rectangular cross-section. Most of the prisms were about 12 inches long, and the length of the longest side of any cross-section did not exceed 1+ inches. A photograph of the apparatus used is shown iy JElle Deis itera The two shafts could revolve freely in ball-bearinzs, and each shaft was fitted with a pulley, one of which was fixed when an experiment was being performed, ‘The torque was transmitted to the other pulley and thence to the specimen by weights placed in a scale-pan, which was suspended from the periphery of the pulley by a fine wire. The effective radius of the couple was 2 inches. The shafts were provided with jaws, into which the test pieces were fixed by wooden wedges, and alignment of the axis of the test pieces with the axes of the shafts was effected by a scribing-block. One of the shafts had a slight end play, so that longitudinal fention was avoided. It was found possible to obtain cou tinuous stress-strain curves with increments of load of . 20 lb. in the scale-pan, equivalent to increments of torque of 4}, lb. inch. An optical method was used to measure the angles of twist. Two frames, each supporting a small mirror, were fixed to the test piece by pointed set screws, so that the axis of each of the pair of screws in a frame was in a line inter- secting the axis of the test piece at right angles (see Pl. TX. fig. 1). When the frames were fixed in position, the distance between the axes of each pair of screws was that over which the angles of twist were measured. This distance was usually about 2 inches. The magnitudes of the angles were obtained by noting the apparent motion of the cross-wires of a pair of telescopes along scales placed opposite to the mirrors and reflected down the telescopes. Hach scale was about 50 inches from the corresponding mirror, and was 850 Mr. H. Carrington on the placed so that the line of sight between the mirror aid the corresponding scale mov ed in a plane perpendicular to the axis of the test piece. When conducting an experiment the load in the seale-pan was increased by equal amounts, and the scales were read after each increase. ‘The scale readings were then plotted against the torque, and resulted in a curve whose initial portion was straight. The slope of this straight portion was. involved in the calculation of the moduli. Although the main object of the experiments was to obtain values of the moduli, it was considered worth while to determine values of the elastic limits and moduli of rupture when possible. These were calculated from the values of twist and torque where the curves ceased to be straight and from the maximum torque. For most of the experiments plane scales were used, and the error introduced was negligible. In other cases, where the angle of twist at the elastic limit was considerable, curved scales were employed. The prisms were obtained from balks representing four different trees. In every case the specimens were cut from portions over 12 inches from the pith, and since the lengths of annual layers in the circumferential direction did not exceed 13 ins. corresponding with any cross-section, or about 2 ins. beer pontine with the tested length of any prism, the layers were sensibly plane surfaces. The specimens were cut so that the annual layers were parallel, either to opposite sides or ends, and the values of the moduli were calculated on the assumption that the specimens had three planes of elastic symmetry. The direction of the grain will be noted by ZOZ, the direction perpendicular to this and normal to the annual layers by XOX, and the direction perpendicular to the other two, which is thus tangential to the annual layers, by YOY. When a specimen with its length in the direction ZOZ is twisted about its longitudinal axis, two values of the moduli are involved, i.e. wyz and wer (nyz denotes the modulus of rigidity corresponding with shear strain along planes OZX and OXY, i.e. along the direction YOZ). The moduli were calculated from the slopes of the stress- strain curves and the dimensions of the test pieces. Two methods were used, and in order to explain them, consider a prism of rectangular cross-section twisted about the longi- tudinal axis, and let the two values of the moduli involved be denoted by #, and wy. Also let M=twisting-moment corresponding with the angle of twist per unit ‘length Ty Modul, of Rigidity for Spruce. 851 26 and 2c=lengths of the sides of the cross-section, and let Oy [ee >1. Then M=y,7bc°8, where ¢ M1 tomy ee te° Bnei / B=) 3 (=) 5 = = 1 -tanh a fen / 2! When LA ise B= e Ee EN ae 0°1875 with an error of less than 0°1 per cent, and therefore 0°1875 (=) bha= bes (1- 0-630£ ae ay Thus the calculation 3 #, by this equation required a knowledge of py. This method could have been used for all the experiments, but in some cases it would have‘necessitated excessively thin test pieces. This will be evident from a consideration of (1) formula (1); for if / is greater than, or about unity, He and since it is only approximately known, then ; must be small in order that the error in the value of i 0-630 4/#) shall be negligible. Excessively thin pieces were undesirable because, for instance, of the impossibility of measuring the thickness with sufficient accuracy, owing to the yielding nature of the material and the difficulty in getting the surfaces planed parallel. Also in cases where the thickness of the pieces was in the direction of the width of the annual rings, thin pieces were in most cases specially to be avoided, because the thickness would have been comparable with the width of the rings, and hence the proportion of autumn to spring wood may have varied from piece to piece. In the case of the second method, two experiments had to be performed on each test piece. After the first experiment the piece was cut so that the ratio of the sides was altered. * Todhunter & Pearson's ‘ Elasticity,’ vol. ii. arts. 27 & 47. + Ibid. art. 47, and ‘ Experimental Elasticity, Searle, art. 40. 852 Mr. H. Carrington on the With the above notation let the suffixes 1 and 2 refer respectively to the first and second experiments, and let i) Py & ahaa Then My = 17101¢,78; and M, = fbi To0nly oes : B> abs 710,¢° Mo Therefore eae °M, MM ba, [Ma and ein Pat 4 | PS Ag ae ea by Me biC2 My Values of @ corresponding with different values of Eve were calculated by St. Venant*, and in fig. 2 a Fig. 2. RELATION BETWEEN B & a Ks, (See Todhunter & Pearson's ‘ History of Plasticity vo ’ vol. i. art. 47.) curve is drawn from the values. Since the ratios (2) and (3) were known from the experiments and ee of the * Todhunter & Pearson’s ‘ Elasticity,’ vol. ii. art. 47. Moduli of Rigidity for Spruce. 853 } hls l po test pieces, the values of 8), Ps, a VE and Aa] 2 1 1 could be obtained by trial from He curve, and hence the values of yz, and py. The writer is not aware of any previous record of the second method, but by making the ratios of the lengths of the sides of the cross-sections such that the ratios of were equal to 1-4 or 1°5, and avoiding the steep part 1 of the curve corresponding with values of 8, greater than about 4°6, it was found capable of the accuracy aimed at throughout the experiments, 7. e. an error of less than 1 per cent. Six types of experiments were performed :— (1) Torsion about axis XOX, breadth of cross-section in direction YOY, thickness in direction ZOZ. (2) Torsion about axis XOX, breadth of cross-section in direction ZOZ, thickness in direction YOY. (3) Torsion about axis YOY, breadth of cross-section in direction ZOZ, thickness in direction XOX. (4) Torsion about axis YOY, breadth of cross-section in direction XOX, thickness in direction ZOZ. (5) Torsion about axis ZOZ, breadth of cross-section in direction XOX, ALicnaaes on dineeaion 0 ONG (6) Torsion about axis ZOZ, breadth of cross-section in direction YOY, thickness in direction XOX. From experiments type (1), values of wry were obtained. From experiments type (2), values of pew and pry were obtained. From experiments type (3), values of wyz and wry were obtained. From experiments type (4), values of way were obtained. From experiments type (5), values of wys and muza were obtained. From experiments type (6), values of wyz and uzx were obtained. For most of the experiments of types er and (4) equation (1) was used. Since the values of My a and eae [jc v : y FP’! were somewhere about 0-2, it was not necessary to pew make the pieces excessively thin in order to calculate pay with the required accuracy, and in many cases the cross- sections were made square. [for all the experiments of the other types, with a few exceptions, equations (2) and (8), in conjunction with the graph in fig. 2, were employed, 854 Mr. H. Carrington on the In addition to the moduli, the following properties were- also measured for each test piece. The density when tested,. density when dry, humidity, shrinkages in the directions XOX and YOY, and the number of annual rings per inch. In order to obtain the density when dry, the humidity and shrinkages,' each test piece was dried in an electrically- heated stove at 104° C. for from 24 to 48 hours immediately after testing. From the measurements and weights of the pieces before and after drying, the properties were calculated, obvious precautions being necessary because of convection currents and the rapidity with which the pieces absorbed. moisture. The number of annual rings per inch was deduced. from measurements of the thicknesses of the rings by a finely divided scale. . The values of the elastic limits and moduli of rupture were calculated by expressions of the type Le=pztst.7 iy, . 2 2 er b * where Eee >1 and Z, is the stress in the middle of the- ow side 2b corresponding with the angle of twist per unit length 7. Values of y corresponding with different values. b > of ee | were calculated by St. Venant, and are eee : reproduced in Todhunter & Pearson’s ‘ Hlasticity.’ In caleu- lating values of the elastic limit, the twist per unit length to. be used in equation (4) was obtained direct from the torque- twist curves. In calculating values of the moduli of rupture, it was assumed that the material remained elastic right up to the maximum torque, and the twist was taken as pro- portional to the twist at the elastic limit. The results of the experiments are given in Tables L., I.,. and Jil. The balks are distinguished by the letters A, B, C, D, and about 10 values of each constant were obtained for each balk. The means of the values are tabulated and also the means. of the other properties. In order to give some idea of the variations in the properties across each balk, the probable variations of the values from the means were also tabulated. These were calculated on the assumption that the values obeyed the laws of probability and Peter’s formula * was. used. * “Theory of Errors and Least Squares,’ Weld, New York (1916). 855: lity for Spruce. igu 7 R Moduli o | donee. quUrT OTST 6-4 CFS 971 | S86 6-6 GG-G 6-8 66-1 PEELE ee et ee { ainjdny jo “por | c.g 0686 99 | 009% G-9 0263 G9 (Gigs ee (‘ur “bs/‘q1) (24) oangdny jo “poy | PG COG2 | = 010 OVNI | ra OSé I 1G (OSs id oe ee “Cur bs/-qt) (24) quarry oseigy | P-L 91-6 8:8 | 82.6 LL P31 8-6 S26. ae (-ut-bs/qt 501) (247) Aqrpiuay Jo “port | ——_——_—- | ee | | ee SS SS LG 61 V9 | 96:8 6-F #8-8 Le 63-8 oo sadeseaicns eas sree (agatd 4saq Lp jo | ig a eS eee Se ee a ae ae ees oe ee ene ee ee Ee f z 9) od Q5R UII | 64 26-1 Och oT 0s FO-€ P01 To KOK as oe ee eee OL-3 LZ Guta aL Z-9 0-81 2.9 8-11 SUOOSCOdoOOabOGG eee cecetcee SepOIIUG i neha) god dINISLOYL | ¥-61 g.eT °.4G F-01 G.8G 1-92 Tees O-LT eae pee ae "ar ded ssuit jenuue Jo ‘ON | 09-1 G-FG We | PG OE G-1E C6-G VGC et re resesssreroereeses (ae nn/“qq) Aap Aqisueq aG.T 1.9 OF | 1G a Orr €-€8 OF-G Bo fe Cay no/-q]) paqsey ueym Aqisuo *yue0 10d ‘quad 10d | 5 | yueo aod i ‘quedo wed : : “sy UIUI SJUOUL | syuotu SJuUdUL UvVIUL WOU "3 uUvBOU WOT] | UBoul wWO0.1J Ix uBvoll Woda UOIYTIABA gluse IL) cours woeose Peioonuca gee eal, UOIYRIIVA siaeoe OT yyeq ayquqoig jo UvIfL alquqorg jo UBITAL erqrqorg jou OT a[quqorg jo UvITA | a ai i. ra) ai a "| Wav], Mr. H. Carringion on the 856 PP G6-T T-L1 OF-T Cé-1 ‘quedo a0d UvaUT MOLY UWON}VLUBA a1qeqou ‘ad 69-G “SJUOUL -1t0dxo [Tf Jo uvo py ‘dad ‘qt1e0 od Uva WOLY WOLBLUVA elqeqoig ze) “squat -11edxe QO] JO UveTT ze) | LG ee 0-4 6-F L-61 0-6 6-§ ‘quao rod UVIUl WO] UOLZVLIVA a[quqoig ‘a | "8] WOU JO Wed ‘a } | | 6:8 —————————— LL 1-0 G9-G CPG ‘quedo sod “aveul WOdj -170dxo CT | UOLJVIABA a[quqoul A "S]UOU -1t00X9 TT] | jo uvay Vv {Uy Ose see veeccnrecses (ur ‘bs/"qq) (v2) eanydngy jo ‘pow (ut *bs/*q]) (wz) yloury o1jse yg seers Cur “bs/"qy ,OT) (vem!) Aqrprsiyy Jo ‘pow XOA Spopcn Hsseneeeeres (ggord 480} Lap Jo XOX SUOISUAIIP UO “quedo aod) oseyarayg Ree: Selene il vio ReMi cliche ike | einjdny Jo “poy | “[[ 81av I, 857 for Spruce. waicy g 2 Moduli of R OF-T “"yue0 s0d uvou WOal UOTYVIABA S14eqord a "S7 U9 UM -140dx9 GT] JO uveyy 0 | 09-9 3-01 [== OSG 0-G1 | 686 6-6 PLS 0-41 68-6 6-G 81-1 LG L-61 g-1 6&1 0-0¢ PPG 09-6 8-46 OL-G eeu ‘qyueo aed | a “uvoul THO] a ia UOIPCLIVA jo uvoyy aaedora a 19) -"yueo aed uBeMl WLOTT UONRLIvA a[quqoug ‘a 6-66 60-4 CVS g9-G Pl L-81 v-0€ ee 8-66 *s}UOUL -110d x0 [] jo uveyy “a 666 v6 6-9 & FL 06-6 OF-G “yueo wed UvOT TOL UOMCIIVA 9[qeqord "Vv “TIL STV yy, 18.9 LIE 99-7 LL 16-6 9L-1 0-11 0-81 9-€6 6-46 "$] U9 OT -110dxa [] jo uvopT a\' 5 J eee | eanydnay Jo ‘poy, secre eoesere aoe (‘ut'bs/*qq) (fiz) arngdny jo "poy seo seeosse ce mes Cur*bs/q]) (Aa) quay orqsepay ses Cur ‘bs/‘qr QT) (Arn) Ay1pisny jo ‘poz KOA Me Eee eee * (aoe1d-4804 Aap jo sucIsueMIp UO “yus0 ted) eseyutayg | Seesesex = LID jad 9.14810 Ty ‘ul god sSula [enuue Jo ‘ON seeeeee Cayno/-qq) Sap $qsuacy | seecceoeeer rece = (az °n90/"q]) pejse ueym AGIsuaqy cee 858 Mr. H. Carrington on the It should be noted that where two experiments were performed on one test piece yielding a value for two of the moduli of rigidity, only one value of the elastic limit and modulus of rupture could be obtained. Thus the means of the elastic limits and moduli of rupture usually corresponded with a smaller number of results than the number of experi- ments given at the heads of the columns. For each experiment the torque was increased at half- minute or three-quarter-minute intervals, and the amounts of the increases were such as to correspond with from eight to twenty readings up to the elastic limit, so that asa rule the elastic limit was reached in about 10 minutes or less. From a consideration of the tables, it will appear that the mean values of the percentages of moisture differ little from 12 per cent. ‘This sensibly constant humidity was obtained by keeping the prepared test pieces in the laboratory away from the sun, for about one month before experimenting upon them. Photomicrographs of normal, tangential, and axial sections magnified about 40 diameters are shown respectively in Pl]. IX. figs. 3, 4, and 5 for balk A, these being also typical of similar sections for the other balks. All the baulks were straight-grained and free from knots and shakes, and represented the good average spruce which was available for aeroplane manutacture early in the war. Relations between the Moduli and the Physical Properties. At the commencement of the research only balk A was experimented upon. It was then found that values of a modulus obtained from pieces which had been nearly adjacent to each other in the balk often varied considerably. This suggested that the variations might be connected with the shrinkages, which were accordingly measured. No relations, however, between the values of a_ particular modulus and the shrinkages or any of the measured physical properties could be found. From an examination of sections of the pieces under the microscope, differences in structure often appeared, and it seemed likely that investigation along these lines might reveal reasons for the variations. Such refinement being impracticable, it was decided to consider the variations as characteristic of the structural nature of the material, and to experiment upon three other balks, with a view to the possibility of finding relations between the means of the moduli and physical properties. Modul of Riadity for Spruce. 859 An examination of the means from Tables I., I]., and II]. resulted in the curves shown in fig. 6. Fic. 6. 5000-———— 3 | #4000] A e | | ; : Wena S y S | | gay ® , 00 Oe yf ay a x Lea 6 ; 2) ee 2b OQ x | Ai | o WA S 2000 Pay el 0 : | J Y) nN 4 x ~ Wixy. Sx. Sy. (16.17 unres) ey) 2 — Ay ate 1O 20 30 Den Sity Dry (les. per cu. Fé.) Here Swz and Sy denote respectively the shrinkages in the directions XOX and YOY per unit length of dry timber, 860 Profs. C. V. Raman and Pihalbatie tt Banerji on and the products pyzSy, pzwSe#, and paySaSy are plotted against the dry densities. It will be noted that the points. involving pay lie sensibly on a straight line, whilst those involving pyz and wee lie approximately on two parallel straight lines. Relations similar to the above have been obtained for each of three values of Young’s Modulus and six values of Poisson’s Ratio for spruce, which the writer hopes shortly to publish, and these were found to agree with the sym- metrical Davies of a body having three planes of elastic symmetry *. It may be noted that the number of annual rings per ae is not involved in any of the relations. The writer wishes to thank the late Principal, J. C. M Garnett, M.A., for providing facilities for conducting the “aaenrel. Badieieo Professor (GL Stoney, F.R.S., for many helpful suggestions. He is further much indebted to Mr. A. L. McAulay, B.Sc.Tas., for valuable assistance with the experiments. LXXXI. On the Colours of Mixed Plates—Part I]. By C. V. Raman, M.4., Palit Professor of Physics, and BHABONATH BANERJI, M.Sc., Assistant Professor of Physics, University of Calcutta + 1. Inadequacy of the Elementary Diffraction Theory. Olas optical phenomena exhibited by mixed plates have been described in detail, with illustrations, in the first part of the paper. We now proceed to consider their explanation. It is obvious that the phenomena must be classed as laminar diffraction effects. But, as has already been remarked in the introduction to the first part of the paper, many of the features observed differ from what one might expect on the usual elementary theory of diffraction phenomena. Conspicuous amongst these is the special character of the diffraction-halo seen surrounding a distant light-source viewed through a mixed plate of uniform thick- ness. Many of the observed features of this halo are not explicable on the elementary diffraction theory: namely, the succession of dark and bright rings of widths rapidly increasing from the centre outwards, the perfect blackness of * Love, ‘The Mathematicai Theory of Elasticity,’ Art. 73. if Communicated by the Authors. the Colours of Mixed Plates. 861 the dark rings in the outer part of the halo, and the obviously composite structure of the inner part of the halo. In seeking for an explanation of these effects, a clue is furnished by the observation already recorded, that when a mixed plate is observed by the light diffracted by it, the whole surface of the film does not appear luminous, but only the laminar boundaries or lines of separation of the two media forming the jilm. The optical effects of mixed plates are thus, in fact, the optical effects due to the scattering or radiation of light from laminar diffracting boundaries. It is necessary to study the manner in which an individual laminar boun- dary scatters or diffracts light incident on it; to compare this with the indications of theory, and from the observed effects to infer the aggregate result of the scattering by a large number of such boundaries irregularly situated on the film. These points we now proceed to discuss. 2. Examination of Mixed Plates by the Method of the - Foucault Test. The most convenient way of examining the scattering of light by a laminar boundary in directions nearly coincident with that of the incident waves is by the method of observation known as the Foucault knife-edge test or the Tépler Schlieren method. The theory of this method was developed by the late Lord Rayleigh on the basis of the usual elementary treatment of diffraction phenomena *, and it was shown by him that a discontinuous laminar boundary should appear as a luminous line when examined by the method of the Foucault test. A beautiful illustration of Rayleigh’s theory is fur- nished on examining a clear piece of mica by the Foucault test, when it will be found that the striz or boundaries in the mica between regions having slightly different thick- nesses shine out as vividly-coloured lines cf light in a dark field+. To study the phenomena of mixed plates by a similar method, the following arrangement is suitable. Light from a small circular aperture illuminated by an incandescent filament lamp falls upon a good achromatic lens, and is brought to a focus at a distance from it. Two * “On Methods for Detecting Small Optical Retardations and the Theory of the Foucault Test.” Phil. Mag. Feb. 1917. + “On the Colours of the Strize in Mica,” C. V. Raman and P. N. Ghosh. ‘Nature,’ October 1918. See also P. N. Ghosh, Proc. Roy. Soc. A. Series, vol. xevi. p. 257 (1919). Phil. Mag. 8. 6. Vol. 41. No. 246. June 1921 3 L 862 Profs. ©. V. Raman and Bhabonath Banerji on pieces of good plate-glass, pressed together with a drop or two of water mixed with air between them, are placed in front of the achromatic lens. The film thus enclosed between the glasses forms the “‘ mixed plate,’ which is observed through a telescope placed with its objective just behind the focus of the achromatic lens. The appearance of the film, as seen with this arrangement, depends on the form of the aperture or stop regulating the admission of light from the focal plane of the Jens into the object-glass of the observing-telescope:—(A) With an aperture placed cen- trally in the focal plane so as to admit the light coming to the geometrical focus but cutting off the diffracted light, the water-air boundaries in the mixed plate appear as coloured lines in a bright white field. (B) When a central stop is placed symmetrically so as to cut off the light coming geometrically to a focus, and an annular aperture surround- ing the stop admits only diffracted light into the observing telescope, the whole field appears dark, except the water-air boundaries, which are seen apparently doubled, shining out as two brightly-coloured lines of light running parallel to gach other and separated by a fine perfectly dark line coinciding with the exact outline of the boundary. The colours seen depend only on the thickness of the film, and are independent of the size or shape of the boundaries. They are complementary to those seen in case A. A magni- fied photograph of the film under these conditions, showing the apparent doubling of the boundaries, was reproduced in fig. 4 of the Plate accompanying the first part of the paper. (C) When, instead of a symmetrical annuJar aperture, we have only a small aperture placed eccentrically in the focal plane admitting diffracted light into the observing telescope, then the full outlines of the water-air boundaries are not seen, but only two small portions of each closed boundary, such that the normals to the boundary at the two points visible are parallel to the radius vector joining the focus with the aperture placed in the focal plane. With such an excentrically-placed aperture the phenomenon of the doubling of the boundaries noticed under (B) does net occur, and we merely get a single luminous coloured line in a dark field running along the portions of the boundary visible. The phenomena described above are closely analogous to those exhibited by the striz in mica*. The observations show clearly that the laminar boundaries in a mixed plate * P. N. Ghosh, Joc. cit. the Colours of Mixed Plates. 863 “act as centres or sources of diffracted radiation. Hach element of a laminar boundary may be regarded as sending out two streams of radiation—one on the more retarded, end one on the less retarded side of the wave-front. In directions nearly coincident with that of regular trans- mission of the incident waves, these two Jpne are of practically equal intensity and of opposite phases. In such directions the phenomena observed are in agreement with the indications of Lord Rayleigh’s theory, according to which the colour of the laminar boundary, as seen in the Foucault test, should be complementary to the colour of the central fringe in the laminar diffraction pattern produced by it. For \ very small angles of diffraction, therefore, the elementary diffraction theory gives results sndelh are substantially valid. As we shall see presently, this ceases to be true when we consider larger angles of diffraction. 8. Lhe Unsymmetrical Scattering of Light by Laminar Boundaries: Normal Incidence. Very simple observation suffices to show that the scatter- ‘ing of light through larger angles by laminar boundaries exhibits features not indicated by the elementary theory. For this purpose, a thin film ‘of liquid mixed with air enclosed between two glass plates is placed normally in the track of a strong pencil of light from a lantern, and viewed obliquely by the eye with or without the aid of a magnifier. It will be noticed at once that the edges of separation of diquid and air diffract light in a strikingly unsymmetrical manner. p— b?¢)(P— Q) e 2«K(a+6)(P+ Q)(aP +6Q)’ _ «(a+b) Ae Ss ice ~ Kak +Q) = (P+Q + ( _ Aer —{atHA—aP_ DQ) | - (9) © An? (a+ b)(aP + bQ) mittee 2) CQ) a 2«*(P + Q) Tt is at once evident that B+ 8 and B—B, and hence B, are positive. ‘That C is positive follows at once from its definition in (8). Distances will be obtained from the instrument on the assumption that @ in (&) may in general be given its mean value zero. Thus the quantity known to the surveyor as the constant of the instrument is the distance from the vertical axis of the instrument to F minus B. C divided by the multiplier for the staff readings, usually 100, is the correct separation for the siadia lines. @ and Y are lengths which specify minimum errors, and it will be apparent in a moment that both have the same sign as x’, At the two end points for which M=P and M=Q, @ has the value +1, and takes the other extreme value —1 for the intermediate point where M = (aP +b6Q)/(a+b). Obviously accuracy is lost if either a or 6 be made zero, since the error does not then rise to a stationary point and suhsequently fall. The character of the error, which is determined by the ratio of 8 to y, will usually be specified, and two special cases call for particular consideration. In the one cause the magnitude of the possible errors is made independent of the distance measured by assigning the value zero toy; this is the case usually considered, and is secured by giving a and d the values —q and W~—p respectively. A better choice both practically and theoretically is to relate the magnitude of the possible error directly to the distance under measurement. In this case @ must vanish and a and 6 are equal. Before proceeding further it will be verified that in these two cases @ and y agree in sign with «’. For when y=0, if p is the magnification for the far distance and a and } have the positive values of the square root,a>0. On the other hand, since the power is less for the far point than for Phil. Mag. 8. 6. Vol. 41. No. 246. June 1921. 3N 894 Mr. T. Smith on the Accuracy of the the near, P Qewys A+h 2«?(P + Q) The two factors P—R and Q~—R are positive, and thus both the terms dependent on the range have the same sign. If the focussing lens is positive, the stadia lines must be closer together than the separation determined by the collimator method; if negative, the separation must be greater. The most satisfactory method of spacing the lines is to determine their correct distance by calculation. For this purpose, somewhat greater accuracy than a slide- rule gives is required. : 3 The significance of the expressions found for the constants and errors is more readily appreciated by expressing them in terms of other variables. It has hitherto been necessarv to regard M asa function of m defined by equation (6) to enable d to be expressed in the form (8). This is no longer necessary, and equation (7) shows that it may be regarded as a power, and thus be expressed in terms of the powers of the two lenses and of the position of the focussing lens in the telescope. The equations required are A = 2e+K'—lxk’, M = «(1+ £). } (10) where J is the constant sum of the separations of the focussing lens from the stadia lines and from the objective, and & is the excess of the former distance over the latter. The separations are of course measured from the principal planes, so that the length of the telescope will be rather greater than / plus the space required for the eyepiece. ‘The second formula confirms the result, of which extensive use has already been made, that M and «’ have the same sign. The presence of the objective and graticule at the CC Internally Focussing Telescope in Tachometry. 8995 two ends of the telescope requires € to fall within limits ‘somewhat closer than ee ee ee DS ut it must not be forgotten that if the lens is moved ~ by means of a rack and pinion or similar device the range of movement is restricted to one-half of that Tae by these limitations. Substitution from (10) in (9) yields results which are highly suggestive of the ee which should be followed to obtain high accuracy in the estimated distances; but these will not be detailed here, as a method is deseribed later by which a teiescope can be designed to give any required degree of accuracy. The simplest method of applying (9) to any given system is to express M in terms of d, «, x',and / or A. “The formula required is found at once by ‘squaring (6) and eliminating m by (5), with the result M = —k'/ed+ JS f(AtK!/ed)?—4An*t, . 2 (12) where the square root agrees in sign with «’. It should be noted that the values of @ and } need not be found. For if Lis the value of M for which 0= —1, B+B = «/kL i and eee ecko) Le (Cy Ne Cae Let Instead of fixing L independently, it will be determined in the two cases here considered by assuming in turn B=0 cand y=0. In the former case B is known from P and Q, which are found by giving d in (12) the extreme values 7 and n for the far and near points. Thus L and hence C—y are known. C+y¥ is found from the values of P and Q. When y=0, L is given by the solution of (13): I= A—2K(C—y) + 2K,/{1—A(C—y)+«?(C—y)?t, (14) where the square root has the positive sign, and the value of © determined fron: f and n is used. Perhaps the clearest insight into the meaning of C is gained by noting that 0 becomes unity at each end point, and therefore the power for both these distances satisfies d= —B+8+(C+y)(dK+e'/e) . . . (15) ‘by (4) and (8), and hence, if the powers corresponding 3N 2 896 Mr. T. Smith on the Accuracy of the to the distances f and m are for the moment denoted by F and N, : ey a? ay ~ fF—nN B—8 = (Cty) = and fn(F—N) Pane These two expressions may if preferred be used in place of the equations in (9) not involving a and 0. The converse problem of designing a telescope to have given constants may now be considered. It will be supposed that f, n, B, C, B, y-are given and that «, «’, and / are to be found. From (15) | 3(A+P)(C+y) = 14+{B—-8—-(C+y)«/e}/ Ff 3(A+Q)(C+y) = 14+{B—B—-(C+y)k’/e}/n. Also from (9) 2(P + Q) = «'/«(B— 8), and therefore B—B—(Ct+y)«'/« = yg, . a(A-P)\Ct+y) =y(l—-g/P), 2(A-Q)(Ct+y) =y(l—g/n), and | where eee SS y= A(C+y)—1 | ne and 1 1 Leal | The second equation for C in (9) may be rewritten in the form (A—P)(A—Q) = 4«’y9/(B—B8), giving by (16) e(C+y)’ =y{1+ BB) 9/n) It is now possible to determine y by substituting from (16) and (17) in the relation implied by (13), with the result 4y{yg(C —y) + 2(CB + By) }(B+8){1+(B—B)g/fn} = iy(B+8+g)+28}(C+y). (18) When y=0 the solution is B -- is \> B=s y\B+A) 2\g_B+a)* jr 1 a V(B—B)(B—B+fng) ; Internaliy Focussing Telescope in Tacheometry. 897 sane for = 0, yy C+) G-;)+ 8y( i+ 7 (3+ aS lee S OW ae Lt = 8B (+3) (57) It will be of interest to take the latter case in some detail. When a negative focussing lens is employed, since by (17) y 1s necessarily positive, it is essential that (C+y) (7-2) +87(5 45) (B+ <0 30 Internally-focussing Telescope with positive focussing lens recording distances between 20 and 1000 accurately to 1 part in 1000. This may be regarded as an equation giving a lower limit to n or an upper limit to B, neither of which may be closely 898 Accuracy of the Internally Focussing Telescope. approached. No such restriction occurs when the focussing: lens is positive. Fig. 1 illustrates the results obtained when the focussing lens is positive for various values of B with C=1, y=001, B=0; f=1000, n=20; and tree corresponding results for a negative focussing lens with: y=—'001. All the calculations required are easily carried [eriee5 25 Internally-focussing Telescope with negative focussing lens recording distances between 20 and 1000 accurately to 1 part in 1000. out on a slide-rule. The systems illustrated in the diagrams. conflict with no mechanical restraints. The mean position of the focussing lens approaches the stadia lines as B increases in both series of telescopes. 4 Heat Loss by Convection from Wires in Stream of Air. 899 The condition that the constant of the instrument should be zero may be conventionally represented with fair approxi- mation by Eo Ps hy and this is satisfied if Kol, fe = OOO (Aol we pally pS 1 So8l The leading differences between the two series are that when the focussing lens is positive the powers are com- oD paratively small, and theretore larger apertures may be employed ; and, on the other hand, aenia negative focussing lens is used the length of the telescope is reduced, the system resembling a tele-photo lens. It is doubtless this feature which has led to the general adoption of the negative lens. In, this investigation no account has been taken of the errors arising through imperfect focussing of the system or inexact reading. The latter introduces an error of the same general type as y,and a possible error of this type is therefore always present. If care is taken in use, the errors due to inaccurate focussing should not be large, and when the focussing lens is positive it is possible, provided the system is properly corrected for aberration, to adopt a construction which ensures that the possible errors from this cause are quite negligible. The National Physical Laboratory. January, 1920. LXXKYV. Zhe Heat Loss by Convection from Wires in a Stream of Air, and its Relation to the Mechanical Resistance. Hoy i El Davis, Sos 1. Introduction. [ lias been shown fT that some published results for the cooling of cylinders in a stream of air agree excellently wita Boussinesq’s formula, which may be deriv ed from con- siderations of similitude t, and which may be written in * Communicated by the Author. + Davis, Phil. Mag. xl. p. 692 (1920). { Rayleigh, Nature, xev. p. 66 (1915). 900 Mr. A. H. Davis on the Heat Loss by the form Hjk= (luc/k), .... anes where H=heat lost per second per unit length of the wire, per degree temperature excess above streain temperature. k=heat conductivity of the fluid. ce=specific heat of the fluid per unit volume. v=velocity of the fluid stream. /=diameter of the wire. It was pointed out that since for a given gas cv/k is constant { being the kinematical viscosity), the above reduces to H/k=F (lp). . .- = re Here vl/v is the familiar variable in hydrodynamics de- termining fluid resistance and turbulence, so the equation in this form involves a relation between the thermal and dynamic effects of a fluid stream. The dynamic effect is represented by the formula R/@v’?) =e); . nn here R=fluid resistance per diameter length of the wire, p=:density of the fluid. The present paper extends the investigation of the agree- ment of the similitude equation with published data for heat loss from cylinders in a stream of air, and determines the relation between the thermal and dynamic effects of the stream in this case. Osborne Reynolds*, considering the heating surface of boilers, pointed out that, from the identity of the two molecular phenomena by means of which convection of heat and of momentum (surface friction) were carried on, their dependence on the conditions of motion would be the same. Formule he deduced for cool liquid flowing through a hot pipe indicate that bee loss Pr) x Temper ature difference between fluid and . unit area pipe. « Fluid resistance per unit area. « Reciprocal of the velocity of the fluid in the pipe. * Reynolds, Proc. Manchester Lit. & Phil. Soc. 1874. See also Stanton, Report Adv. Committee Aeronautics, p. 45 (1912-13). Convection from Wires in a Stream of Arr. 901 Stanton * showed experimentally that for the same pipe and range of temperature the heat transmission varied as v’-l, where n is the index of v in the ordinary resistance formula. | 2. The Convection Constants of Arr. Below are given the formule taken in this paper to represent the variation with temperature of the convection constants of air. Langmuir f, dealing with convection of heat from the point of view of conduction through a stationary film of gas in contact with the hot body, has eollected formule for the convection constants for various gases. From his paper we have, T being the absolute temperature, ~ ANG 5 EOS een aera 1+124/1 1+ 0:0002T 14 124/10 viscosity (7)= k=4°6 x 103° cals. per cm. per sec. pera Ce wen (op and sinee for air p=0'001293 (273/T) gm. per c.c. it follows that 425 x 105%) 122 é v(=7/0) = oo Wel SeCarraiiias ae cue (0;) 3. The Thermal Relation. For convection loss from thin wires in air, the most complete and satisfactory experimental results are those given by Kingt, and he obtained to represent them a formula which may be written HEE Beant Ore . D) where v has been put for k/c. This agrees with the theory advanced by King and with (2), except that B and C vary slightly with the temperature excess at which the wire works, and that C increases with the diameter of the wire. He gave empirical coefficients which allowed for this, and * Stanton, Phil. Trans. exe. p. 67 (1897). + Langmuir, Phys. Rev. xxxiv. p. 401 (1912). t King, Phil. Trans. A, cexiv. p. 373 (1914). 902 Mr: A. H. Davis on the Heat Loss by supposed the slight variation in B and C to be due to the change in the values of the physical constants of air. To get a relation between H/k and vl/y it is necessary to ussign values to these physical constants. While throughout this paper the temperature of the air stream is taken as. 15° C., the values of & and v appropriate to any experiment with a hot wire in the stream are neither those for air at the temperature of the wire nor those for air at the temperature of the cold stream. Consequently, as a first approximation, the values taken are those for air at the mean of these temperatures. The relation between H/k and yl/v is then found to be practically independent of the temperature excess. Hig. 1 shows this and will be referred to later. In section 2 are given the formule representing the variation: of k and v with temperature. With respect to the dependence of C on the diameter / of the wire, it is not possible by rearranging the similitude: equation to obtain a term in which / occurs without v, nor is this possible by introducing the temperature coefficients of & and v. It is possible, however, by taking into account the free convection from the wire, as well as the forced con- vection. It seems improbable that this is the complete solution, for with the higher wind speeds the effect is still appreciable when variation with diameter of the free con- vection loss must be negligible. The true explanation may lie in the increasing lack of rigidity in the finer wires, which to eliminate end correction were generally of the same length (23 cm.) as the stouter ones. The former, vibrating in the wind and presumably following the changes. in air-pressure more closely, might yield less heat. King himself mentioned the existence of vibration, and that higher air speeds were impossible owing to risk of breakage of the wire at high temperatures. To cover the whole range of King’s experiments, the heat loss has been determined, using his formula, for two wires. (0:003 em. and 0:015 cm. diameter) working at three different temperature excesses (85°, 500°, and 1000° C.) in winds of four speeds (20, 100, 500, and 1000 cm. per sec.). As stated earlier, the values taken for k and v have been calculated from the formulze already given, the appropriate temperature for any case being the arithmetical mean of the temperatures of the hot wire and of the ambient cold stream (15° C.). The results have been plotted in fig. 1 and give the lower end of the “ Thermal Relation” line. The upper part of the curve has been obtained from data Convection from Wires in a Stream of Air. 903: given by Hughes* for larger cylinders (0°-43—5:°06 cm. diameter), steam-heated, in air (assumed 15° C.) moving with speeds of 200 to 1400 cm. per sec. These data have been plotted on a graph + having heat loss as ordinate and vl as abscissa ; values read off from this curve give points in the upper part of fig. 1. Bigad: GO King, (C-003 cm.) uw (6-035 cm.) Kenpelly, Rugees, Boo + Log (R/pv'l*) & Log (H/k) = , re RELATION: DYNAMICAL T fa) 1 2 3 4 Log (vi/v) Some values intermediate between the two sets above mentioned were obtained from figures compiled by Lang- muir { from a study of results given by Kennelly §. They relate to the high velocity (800-1800 cm. per sec.) values for Kennelly’s largest wire (0°0204 cm.) working at tem- perature excesses of 51° and 128° C. The graph thus obtained— Thermal Relation’ of fig. 1— is excellent, practically independent of the temperature or diameter of the cylinders involved. The data range over air-streams of from 4 to 50 miles per hour, wires of 0-008 cm. diameter up to 5 cm. piping, and temperatures from that of boiling water to a bright red heat. The curve may be further examined by comparison with the results of experiments by Kennelly and Sanborn ||, who investigated the heat loss from a wire in air at various pressures, the wire being mounted on a whirling arm ina * Hughes, Phil. Mag. xxxi. p. 118 (1916). t Davis, loc. cit. t Langmuir, Trans. A.I. E. E. xxxi. (1) p. 1229 (1912). § Kennelly, Trans. A. I. E. E. xxviii. (1) p. 868 (1909). | Am. Phil. Soe. Proe. hii. p. 55 (1914). 904 Mr. A. H. Davis on the Heat Loss by special chamber. Table I. is calculated from the data given in table ii. of their paper. As before, allowance has been made for variation of k& and v with temperature. It has been assumed that, as indicated by the kinetic theory of TABLE I. Convection at various air pressures. Wire diam. 00114 ecm. v=970 em. per see. Values of vy given for a pressure of 1 Atmosphere (=1-012 x 108 bars). Room temperature about 20° C. Pressure 390° C. CxCeeS 538° C. excess. (bars). log v=1°56 log vy = 1°67. log £=5'95 log k=4-00. 10&x log v/v. log H/A. log vl/v. log H/&. £0 = “= 1:97 1:33 3°99 2:07 1:36 as fe3/ Us — 1:68 1-18 2-00 1:78 | 1:20 wee eat: 102) eal ct 48 | )\ 108 1:38 1-05 vee TE Se ee 1-01 0-90 gases, both the conductivity k of a gas and its viscosity 7 are independent of pressure. The kinematical viscosity v has consequently been taken as inversely proportional to the pressure of the gas. The data of Table J., if plotted on the graph of fig. 1, agree satisfactorily with the curve there given for air at atmospheric pressure. A slight upward displacement above the line occurs for all the points, but this is undoubtedly due to experimental error, since two of the points relate to ordinary atmospheric pressure, and should agree with King’s results for the same condition. It is interesting to compare the above results for air with data for the heat loss in a stream of other fluid. Worthington and Malone* have shown that the convection of heat from a 0°0256 em. platinum wire in water follows the same laws as convection in air. Water carries off over 100 times as much * J. Frank. Inst. clxxxiv. p. 115 (1917). Convection from Wires in a Stream of Air. 905. heat as air, and they give the following example for the wire :— Velocity of water relative to wire... 14°85 cm. per sec. Temperature excess of the wire...... a0° C. Heat loss from the wire ....,......... 1 Wee Cally oer Gite per sec. According to tables by Kaye and Labv C= |, k=0-00136, hence il fora vle/k= 280, Now for air k/cv=1°3 approximately, so in fig. 1 the relation between H/k and vl/v for air is a relation between H/k and. 13elc/k. Plotting the above value for water on the curve regarded in this manner, gives the point X in fair agreement. 4, The Dynamical Data, The mechanical pressure on smooth wires placed in a stream of air has been given in the ‘Technical Report of the Advisory Committee for Aeronautics for 1913-14. The data relate to 60° F. and a barometric pressure of 760mm. The wires range from 0-002 in. to 14 in. diameter, and were placed in winds of 10-50 ft. per second (20-50 ft. per sec. for wires above 4 in. diam.). The results were exhibited in a non-dimensional curve, and this has been re-drawn in the ‘‘ Dynamical Relation ”’ line of fig. 1, where R/pv7l? is plotted against v/v. 5. Relation between Thermal and Dynamical Hffects. Fig. 1 gives for definite values of vl/v the appropriate values of Hk and of R/pvl?, and using these fig. 2 has been drawn showing the relation between H/k and R/pv?l? for corresponding values of vl/v. In the above form the curves are non-dimensional and general, but it is instructive to draw a curve which shows for corresponding values of vl the relation of the heat loss H to the mechanical resistance R. Strictly, this cannot be done unless vy is the same in the two cases, and presumably this condition would be satisfied if KR were measured not for a cold wire, but for one at the same temperature as that at which H was measured. Failing such measurements, how- ever, the data of Table II. ap, proximately satisfy the condition, for here, for an air stream of 15° C., the mechanical pressure “906 Mr. A. H. Davis on the Heat Loss by R of cold wires is compared with the heat loss H from wires -at a temperature (100° C ) not far removed. ‘The table has -been drawn up from data already considered, and the results Fig. 2. 4 @ re) los R/p ‘are plotted in fig. 3. To facilitate comparison with previous work H/k is given instead of H, taking for & the value 5°77 x 10-* appropriate to the temperature of the ambient air, Similarly, R/p is given. This does not alter the shape Convection from Wires in a Stream of Air. 907 of the curve. It may be mentioned that a curve between H/k and vl/py plotted from this table is practically identical with that, of fig. 1, where allewance was made for the change of the physical constants with temperature. Peis JUG, Relation between R/p and H/k. p= 0-145 log v=1°16. k=577x107%. log k= 5°76. vl R log — log 7 log ae 1015 0-81 1°88 King (0 003 cm.). 1316 0:93 2°03 1-015 0°84 1°88 (0:015 cm.). 1714 1:12 2°51 2:015 1-26 3°01 ol4 178 5°09 Hughes. 3°84 2°13 6°54 414 2°30 CUTE 4-44 2°48 N19 4-68 2°64 8:29 2-05 1:25 3°08 Kennelly. 2°26 131 3°50 2-4 1:41 3°76 2-10 1:26 3:18 240 | 1-42 3°76 As stated earlier, the theory of Osborne Reynolds gives for fluid flow through a given pipe H« R/v. This simple relation does not hold for wires in the region considered, for taking the straight line part of the curve of fig. 3 it is seen that Piece Ap OGOXe ys... oe (ONE * The value taken for v in the ‘ Report of the Advisory Committee for Aeronautics’ differs slightly but inappreciably from this, and so no allowance has been made for the difference. + With respect to the reality or otherwise of the unexplained dip in the “‘ Dynamical Relation” of fig. 1, little effect is made in fig. 3 by smoothing out the dip entirely. Smoothing appears, however, to make more uniform the tendency of the relation to take the Reynolds form as vl/y increases. 908 Dr. C. Davison on the Annual and and for this region it may be shown from fig. 1 that roughly Hav and Re« v?, Consequently, He Rj’... . ne There is, however, most probably a tendency for the relation to take the simpler form given by Reynolds as the size of the cylinders is still further increased. This follows since R tends to be proportional to the square of the velocity, and since Hughes found that the heat loss H tends to he proportional to v. January 1921. LXXXVI. The Annual and Diurnal Periodicity of Earth- quakes in Japan. By Cuares Davison, Sc.D.* HH annual and diurnal periodicity of earthquakes has been considered by several writers—by Prof. C. G. Knott in 1886, by Prof. F. Omori in 1894 and 1902, and by myself in 1893 and 1896 f. Since the year 1896, more complete records have become available. The present paper deals with the earthquakes of a limited area—the empire of Japant. The method of investigation adopted is the rough form of harmonic analysis * Communicated by the Author. + C. G. Knott, ‘“ Earthquake frequency,” Japan Seis. Soc. Trans. vol. ix. pp. 1-20 (1886,; C. Davison, “The annual and semi-annual seismic periods,” Phil. Trans. 1898, A, pp. 1107-1169; C. Davison, “On the diurnal periodicity of earthquakes,’ Phil. Mag. vol. xlii. pp. 463-476 (1896); F. Omori, ‘ Annual and diurnal variations of seismic frequency in Japan,” Publ. Earthq. Inv. Com. No. 8, pp. 1-94 (1902). } The records analysed below are contained in the following cata- logues :— Milne, J. (1). Japan Seis. Soc. ‘Trans. vol. x. p. 61 (1887) ; vol. xiii. pt. i. p. 93 (1889); vol. xv. p. 101 (1890); vol. xvi. pp. 56 and 83 1892). : Milne, J. (2). Japan Seis. Soc. Trans. vol. ii. pp. 4-14 and opposite p- 88 (1880) ; vol. vi. pp. 382-85 (1888) ; vol. vill. pp. 100-108 (1885; ; vol. x. pp. 97-99 (1887); vol. xv. pp. 127-184 (1890): Brit. Assoc. Report. 1886, pp. 414-418; 1887, pp, 212-213 ; 1888, pp. 435-437; 1889, pp. 295-296 ; 1890, pp. 160-162 ; 1891, pp. 123-124; 1892, pp. 93- 95; 1893, pp. 214-215 ; 1895, pp. 114-115; 1897, pp. 133-187 ; 1898, pp- 189-191: 1899, pp. 189-191 ; Brit. Assoc. Seis. Com. Circulars, vol. i. pp- 29-80, 90-92, 142-144, 223-225. Milne, J. (3). “A Catalogue of 8331 Earthquakes recorded in Japan between 1885 and 1892.” Seis. Journ. Japan, vol. iv. pp. 1-867 (1895). 7a Diurnal Periodicity of Karthquakes in Japan. 909 Known as the method of overlapping means, which gives results of sufficient accuracy, considering the nature of the materials ™. Annual Periodicity. In Table L., the resalts, with certain stated exceptions, refer to the whole country; the maximum epoch occurs about the end of the month given; for the semi-annual period, the eariier of the two epochs is given, the second of course being six months later ; in estimating the amplhi- tude, the average monthly number of earthquakes in each ease is taken as unity. The amplitude, as Prof, Schuster has shown, should exceed the value of ,/(7/n), where n is the number of earthquakes f, unless the epochs agree in corresponding records. In Table I1., similar results are given for the earthquakes recorded at special stations {. Tables J. and IT. lead to the following results :— G.) The maximum epoch of the annual period for the whole country occurs from December to March for strong earthquakes, and in September—October for slight earth- quakes. In the destructive and slight earthquakes recorded Milne, J. (4). ‘‘A Catalogue of destructive Earthquakes, A.D. 7- 1899.” Brit. Assoc. Report, 1911, pp. 649-740. Omori, ’. (2). ‘‘On the After-shocks of Earthquakes.” Journ. Coll, Sci., [mp. Univ., Japan, vol. vil. pp. 111-200 (1894). Ono, H.(2), “ Notes, on the Barthquake Investigation Committee Catalogue of Japanese Earthquakes.” Journ. Coll. Sci., Imp. Univ. Japan, “vol. xi. pp. 389-437 (1899). Omori, F. (3). “Annual and Diurnal Variations of Seismic Fre- quency in Japan.” Publ. Earthq. Inv. Com. No. 8, pp. 1-94, (1902). Omori, F. (4). Note on the Annual Variation of Seismic Frequency in Tokyo and Kyoto.” Bull. Harthq. Inv. Com. vol. 1. pp. 17-20 (1908). Omori, F. (5). “Note on the Annual Variation of the Height of Level of Lake Biwa.” Jbid., pp. 51-57. - Omori, F. (6). ‘List of the stronger Japan Earthquakes, 1902- 1907.” Jbid., pp. 58-88. * Phil. Trans. 1893 A, pp. 1108-1112. See also Boll. Soc. Sism. Ital. vol. iv. pp. 89-100 (1898). In the present paper, I have not taken account of the Rr eee lengths of the months, such refinement being unnecessary. + A. Schuster, Roy. Soc. Proc. vol. xi. pp. 455-465 eae! ). i che records are instrumental at Tokyo, Hikone, * Sapporo, Nagano and Oita ; non-instrumental at Numazu, Hamamatsu, and Kyoto ; partly instrumental at the other stations, the seismographic record beginning with the year 1888 at Wakayama, 1891 at Ishinomali, 1894 at Niigata, 1895 at Hakodate, and during the omitted year in the remaining ve cords, Phil. Mag. 8. 6. Vol. 41. No. 246. June 1921. 3.0 on on the Annual and aVIs D 1 Je Die 910 “THyIqQnop #o. | ‘ady cg. | eg | aa. | Sen PI. | Av | Ol. | “eu ; AVY “ayy Il. —ady | LT. ~ AON i “ULE QT. | S8W | PL |_-oog Og. ‘Une elie ‘une 3 “UU OT. | AVI | LT ac qg. | Avy | 1g. | 09d iene ‘po Oe STN) sol radon ar Ue oz. | vump | at. | Alto || Sacelyye OT. | VI : "1d V 0%: dy Bl: aon A’ 80: eae Il. | “det ’ rae 3 029] Re ee UNG nO 190 eo. | Av | OT. | 9d dury (yoodsy ‘[dury bypod sy ~--——_ + = | yenuUr-1tttag > [e nuu V | *() soyenbygaeq JO “ON 46 6é 3 66 -(g) Up (9) ttoutg (g) taowig 66 (7) OUT TT “A{LLOUINW Ge O681-G881 L061~906T FOGT-Z06I F68T-88T 6681-1001 6681-1081 "‘paooory JO uOLyRAn (TL ‘sopiur ‘D3 6.Q = 1 'DST » Per aCe a CC Cnr a pesescesece ROX 66 é 66 66 eer eres esto tee toeresreene . purl 4c 3 66 UA04S9 AA. Pr aC et ee eceee rere sees Vas ia} 6é a3 Gk SOCIO RGR CAGE Hur, "PUT LOPUN ‘SJOLMSIP OLUISIOs ULeysey LStp OLuts! ae avetata Taxa? sirstniaie: . go0URSIP joke qv UL LALO 66 $6 “BLOM LO OG ce Ze cA e (a3 (73 OT 6 66 66 “ DUBT WOT SOTLUE @ JNOGR ULSILO ‘soyenbyywo ouLeuIqng Pe ee were eee ee ee reese sence teens . re ‘bs OOT uByy SSO] 66 6c 11 ‘bs QOOT par OOT usomyoq ‘ i y It DS OOOT Wey toqvoad wore poqanyst(y i ‘soyenbyjavo Arvurpag 66 66 prtom eporym ‘soyenbyyave oarjon.4ysoqy —_— “polto DTUISTOS [vnuuy— | HAV YT, x 19. key | ¢h | ‘00g gr. 6G “ | « |seneeeeseseeereeneeees nsyemumeyy | ors FG. “ART PG. ‘a(t FI. 091 (¢) T10UI GEST-C88I Jeseeadiarssaae rWloeeusente nzeuiny, | 90. | aem—vdy| tr | “Me | G0. ie te | ee yasys “| S Zl: “UReP ee | “qdog | @le 856 (Z) L0WUO 8681-008 soyenbryave satjonaysop | S G0. Ran OT. Rarer en: Ore] Fe S9sT-162 ee is deeds ojohyy = GT: MIE 80: TENE: Ss x0: 99F : 668I-F681 ‘G6ST-T68T Sei ear me eftaounsy 7 | = OT- CHENE 6G: “ady 90): LOL e GESTSIORI OSM sOGCI teat es BNO ISA | | < CO. ‘ady ge. ‘gny 90). FEOT & BGERT-II oleae te Cae “ TyVULOUTYST | ze 88: WUE 9L- TE €I- 981 = 6681-681 €681-0681 | es Gate ee a yenemey | = IZ: ‘inp -Keqy QT. ‘dag G0. zee 6s EGRI-LE8T ‘E821 lovagogaaccoqae5D sao0D000 O06 * oanwayy | cS LO ady XI. SOC | Site TOT _ GERI=s68h IGSlaGSSl 5 oe PARLIN OAT nf lee ‘ ey. “ady LT. heyy 60. COP ‘ 668 I-088T [Peoeapreoscucnecncrins see Bure deye Ah | = = GF. keqy Ie. “une cI. aa “ Gesioiesi Bue eae beteeeres rene pap BLINT ie 8 0z: Le LI. ‘aI OI- cge “ BE8T-LL8T siesta Sarsias sche nies casein “ oyupoyryy eS lect “IRIN, ZZ. ULP—v9qT | ut: FOG “< BESI-LESI ejso\eis/e\oiele\ein\eis\lelelelste Belele/stereleieye '0000 UO ; = Ie. “dy 19. 2, OL. 108 ac GEST-0GRT seeeeeeeeeeeceeeneeeneseees ourde yy ms Pare “ACT PZ. ‘aq CT. Fel (¢) row BGRT-L88I | enoncoonceacnbdcaodanobed osoddeg =e i 20: AVIA Nes kenge | 60: 1¢e (¢) 10 IQ 1oBI-ae@r. 0 fe ort ~ QUONTT Re = 80 AT] 80. | idy—ey} FO. G08 (2) OUTIL | OGT-E88T “TS8I-G28T | MRS a "* oA¥OT, = | qduy} ‘yoody |ydury| ‘qoodsy "(w) -px0990yy eer es wy, oes —| *(u/2)N seyenboszuy “<{LIoyyny ronlowenticy | [eNUUY -LIllag JenUuuy JO “ON | ‘pollog ollusteg [enuuy-—T]] AIA, 2 912 Dr. C. Davison on the Annual and at Kyoto, the reversal of the epoch of the annual period is. less clearly marked ; the epoch occurs in September with the strong earthquakes and in May with the weak ones. (ii.) The earlier epoch of the semi-annual period usually occurs in spring (March to May). In the destructive and slight earthquakes recorded at Kyoto, the — of the semi-annual period are reversed. ii.) As Prof. Omori has pointed out, the maximum seismic frequency occurs during the winter ‘months in some parts of Japan, and during the summer months in others. The latter places are for the most part confined to the eastern portion of the northern half of the country. The smallness of the amplitude of the annual period for the whole country is in part due to this opposition in epoch. (iv.) Submarine earthquakes with origins not more than five or ten miles from the coast are subject to an annual periodicity with the epoch nearly the same as in land-earth- quakes, namely, in winter (Dec.—Mar.) ; those with origins at, a distance of twenty or more miles have the epoch in summer (June). Diurnal Periodicity of Ordinary Earthquakes. In considering the diurnal periodicity of earthquakes, seismographic records only are considered, ali others being useless for the purpose, owing to the varying conditions of observation which prevail throughout the day. The records for a whole of Japan an! its eastern and western districts may to a certain extent be incomplete. They include all ae in Prof. Milne’s great catalogue in which the time of occurrence is given in hours, minutes, and seconds. There may, however, be other earthquakes recorded instru- mentally in which the time is given in hours and minutes only. For the diurnal periodicity of ordinary earthquakes, we have the following results :—- Gi.) Though Bie amplitude of the diurnal ec is in each ease not much above te value of ,/(a/n), the close agree- ment in epoch throughont is sameicas to establish the reality of the period, with its maximum epoch shortly before or about noon. (ii.) For the semi-diurnal period, the average amplitude is one-half that for the diurnal period. The maximum epoch occurs at about 8 or 9 A.M. and P.M. 913 S GI 8 = = SI: 4 aa 90: 6 S 8 a 6 = OL: 6 Bs Ss CO. “Gott > 90: 8 = GO. “POLIT ree —- -S S ‘[dury | wa ey ay “qoodar cS ~- -—-—5)-- —— = [vuANIp-tuieg > 6&- NL GI. FI: OT Il. OT: ey! c0- 90: | 4oou GT co. 80: peste FO: OT. | woou ET 80: 06: SEI IT: Tei OCU G0): ‘jdury | ‘qooda 2 (u/)N ——, a as [BumMiCy “(w) soyenbuyytegy TOR ONT c 6681-L881 (g) to 6681-F68T be ee 66 66 (Z) OUTLI | GOBI-ERRT ‘TS81-GLST | ce 66 66 66 (¢) out ee | — O681-G8sT "pl000%7 “AQILOYIN\W JO uoeancy oossrae eseeee i i soe eYIO (eee ee ee ee eeese Cr | SUOYITYT snes i routines a4 atojutm § avak sjoym ‘o SPOLLO DIUISTOS [eusniqg]7— AT wlav y, breasts UeayO eecvee 66 Diurnal Periodicity of Harthquakes in Japan. 915 The following results may be deduced from this Table : Gi.) In the early after-shocks, the maximum epoch of he diurnal period occurs near midnight, or twelve hours later than for ordinary earthquakes. “The epoch of the semi- diurnal period is at first variable. Gi.) In the later after-shocks—after a month or more at Nemuro and after about two years at Gifu—the epoch of the diurnal period returns to the neighbourhoed of noon, and that of the semi-diurnal period at Gifu to about 8 or 9. Gil.) In the early after-shocks, the amplitudes of both diurnal and semi-diurnal periods are about double those in the later after-shocks. Origin of Seismic Periodicity. G.) Annual Period.—Vhe maximum epoch of the annual barometric period occurs almost invariably in December. Prof. Omori has shown that the resultant pressure on the ocean-bed is subject to an annual variation, owing to the presence of a Hig espressp: system in the neighbourhood of the Aleutian Islands*. The maximum epoch of the annual period occurs in October r at Misaki, and in September at Ayukawa, Otaru, Iwasaki, Wajima, and Hamada. The annual periodicity of earthquakes may be due largely to that of barometric pressure, but this is probably not its a cause. The summer maximum in earthquake- -frequency < the north-eastern stations may be due in part, as Brot. Omori suggests, to the variation in pressure on the sea-bed. In this part of the country, many earthquakes are of sub- marine origin. Gi.) Semi-Annual Period.—The first maximum epoch of the semi-annual period occurs from March to May, usually in March or April. That of the mean pressure on the ocean-bed occurs in April, May, or June. As the maximum epoch of the semi-annual period, whether for land or sea earthquakes or both, usually occurs from March to May, it is probable that the semi-annnal periodicity of earthquakes is due to that of the barometric pressure, the variation in total pressure on the sea-bed being due to the same cause. (iii.) Diurnal Period. —The maximum epoch of the diurnal period of barometric pressure at ‘lokyo occurs atabout 15A.M. that of wind-velocity at Tokyo at about 0.45 p.M., and that of the barometric gradient between Nogano and ‘Tokyo at about 0.45 p.m. The maximum epoch of the diurnal period of * Publ. Earthq. Inv. Com., No. 18, pp. 28-26 (1904). Bull. Earthq. Iny. Com., vol. 11. pp. 85-50 (1908). 916 Mr. A. G. Shenstone on the Lject of an ordinary earthquakes occurs about or shortly before noon, and that of after-shocks about or shortly after midnight. ‘The diurnal periodicity of ordinary earthquakes is probably due to more than one cause; that of after-shocks mainly to the diurnal variation of barometric pressure. (iv.) Semi-diurnal Period.—Vhe first maximum epoch of the semi-diurnal period of barometric pressure (the amplitude of which does not differ greatly from that of the diurnal period) occurs about 9 A.M., and that of wind-velocity about 2am. As the corresponding epoch for ordinary earthquakes occurs about 8 or $ A.m., it is probable that the semi-diurnal periodicity of ea ‘thquakes is due to that of barometric pressure. In thecase of after-shocks, the epoch varies much but, after the lapse of some time, the influence of the pressure variations tends to prevail over other causes. LXXXVII. The Effect of an Electric Current on the Photo- Electric Hffect. By ALLEN G. SHENSTONE, 1/.A.* URING the course of an experiment to determine whether there was any relation between the Hall effect in bismuth and the photo-electrie effect, it was found that the total photo-electric current was so much influenced by the current passing through the bismuth that the original experiment had to be abandoned. To determine the variation of this apparently new effect, an apparatus, shown in the attached sketch (fig. 1), was con- structed, and gave cousistent results. The bismuth plate was kept at a negative potential of 14 volts, which gave practically saturation of the photo- electric current. The electrometer was used with india-ink resistance of 5(10)* ohms to give a sensitivity of, roughly, (10)? ampere per mm. As only relative variations were being determined, no accurate determination of sensitivity was made. The work was all carried out at the pressure given by a glass Langmuir diffusion pump, in operation throughout the experiment... During the first part of the work it was difficult to know what sort of ‘interdependence to expect. This first work was further confused by the fact that the same plate of bismuth was used for a considerable number of runs, and it was afterwards found that one consistent run was all that could * Communicated by Prof. E. P. Adams, Ph.D. Hlectrie Current on the Photo-Electric Effect. COT be obtained from a plate. However, it was finally roughly determined that for any given plate current the photo- electric current increased up to a maximum, beyond which it did not go unless a larger plate current was used. With very high currents this maximum was followed by a fairly rapid decrease in the photo-electric current. The rises of iBigc ole Q Q, quartz plate; E, copper electrode, with ebonite cover, connected to electrometer; B, bismuth plate: A, battery leads; P, connexion to pump. the photo-electric current were always very slow, in some cases occupying up to two hours to attain the maximum. When the plate current was stopped, the photo-electric current at once started to decrease, at first quite rapidly, but apparently it never drops to exactly the original zero, usually going beyond it. Maxime of Photo-electric Current 918 Mr. A. G. Shenstone on the Effect of an It was proved that the effect was not dependent on the light by making a few runs with the plate screened, except when the readings were being taken. ae Fig. 2. rTTL LLL PT a Coe iO ies Current through Plate in Amperes Bismuth plate, cross-section 0:112 em.? When high currents were being used, readings were always taken with the current shut off, in order to eliminate the effect of the magnetic field. After the preliminary work it was determined to try to obtain complete curves of the dependence of the photo- electric current maxima on the current through the plate. Electric Current on the Photo-Hlectric Effect. SHUG: This was a matter of some difficulty, because of the large uumber of factors that had to be kept constant over a long time. The plate was illuminated by the light from a quartz- mercury vapour lamp, the intensity of which is very sensitive to changes in the current caused by other work in the building. A few long sets of readings were obtained, how- ever, and were all similar. A typical curve is shown in fig. 2. This occupied sixteen hours. To get these results the plate was first thoroughly fatigued by exposure to the light for about twenty-four hours. It was then found to give a very constant photo-electric current. Then a small current was passed through the plate, and readings taken at intervals, until it was determined that no change i in the photo-electric current was occurring. In this way by successive increases 1n plate current, a current was reached where a small increase in the photo- -electric current occurred. Beyond this point the increases became greater and greater up to the large current, where the photo-electric current after reaching a maximum began to decrease. Beyond this point the photo-electric current could be further increased by a large rise a the plate current, but consistency vanished at this point (2:2 amperes in fig. 2), With the current off, the plate rapidly fatigued again, usually going to a point below the original value. If, now, an attempt was made again to go through the process just: described, the results were very erratic. The increase Mected at a slightly lower value of plate current and rose very irregularly. For some increases of plate current there was an abnormally large rise followed by constancy for several further increases, followed further by another abnormal increase. The final value reached in the first run - with a given plate reached in some cases 100 per cent. increase. Sputtered Plates of Bismuth. Because of the fact that bismuth behaves very differently in the form of a sputtered film (especially the failure to change resistance in a magnetic field), several sputtered plates of different thicknesses were tried. They all gave results similar to the thick plates, except that they could be run right up to the point at which the films burned out, without any inconsistency appearing. A curve for such a plate is attached (fig. 3 A zine plate was also tried and was found to give similar results, but of smaller magnitude. oD There seem to be only two ways to account for the effect 920 Lect of Electric Current on Photo-Electric Effect. described :—(1) Temperature effect ; (2) an orientation of the eleinentary crystals of the metal. 1. The possibility of temperature making such a large change is extremely unlikely. In fact, with pure metals, itis l4r Maxima of photo -electrie current Current through plate in milliamperes Sputtered bismuth plate (on mica). generaliy accepted that temperature produces no change in the photo-electric effect. Moreover, in this experiment; currents which produced very considerable increases in the photo-electric current caused only a fraction of a degree rise in the temperature. Production of Luminosity in Atmospheric Neon. 921 2. The hypothesis of the orientation of the elementary erystals seems to be the most probable. The appearance of the effect in sputtered plates cannot be argued against this hypothesis, since the sputtered particles are undoubtedly of more than molecular dimensions. ‘The fact that no increase of photo-electric current appears until a certain minimum current is reached, indicates that a certain minimum field is necessary to orient an elementary erystal, the further increases being due to the larger numbers. so oriented. Further, the fact that a second series of readings with a plate gives the irregular results described previously, would seem to indicate that the structure of the metal had been in some way weakened. The greater rises under these con- ditions would likewise support the view. The validity of the hypothesis could, aos be best tested by working with plates cut perpendicular to the erystal axis of a large bismuth crystal. =seuueaN instead of being merely indeterminate as the ratio of two infinities.; w herefore, combining these equations and putting 1/r=u, he gets du (aa) + yu? = constant. Whence on differentiation (remembering the variability — of y) aru dP? an equation which may be trusted to behave properly. It is indeed the usual progressing gravitational equation: without the constant term responsible for an orbit. tu=smu’, . 9.) * Communicated by the Author. Notices respecting New Books. 945 But is his obtaining of (8), by aid of legitimate for the case when ds=0? I askin no controversial spirit. I too thought at one time that it was the observer’s field that was effective, but I am now very doubtful ; and it would be interesting to have the point settled before a clear experimental verdict is forth- coming. XCI. Notices respecting New Books. Transactions of the Bose Research Institute, Calcutta, Vol I. parts 1 & 2, 1918; Vol. 11. 1919. Luafe-movements in Planis, by Biers. ©. Bose, Kt., M.A., D:Se., C.S.1., C.1.E., Professor Emeritus, Presidency College ; Director, Bose Research Institute. Published by The Bose Research Institute, Calcutta. ]N these two small volumes a set of papers produced by Sir J. C. Bose and his assistants are collected. The Bose Institute was opened in 1917 and the work done up to 1919 is here recorded. Owing to the fact that the volumes are published by the Bose Research Institute at Calcutta, they are somewhat inaccessible to English readers. But as they contain detailed descriptions of a long series of experiments on which the various pronouncements of Sir J. C. Bose are founded, a critical evaluation of his work can only be obtained through a study of these volumes. Whatever else may be said, it is entirely evident that new applications of physical methods are being introduced ; and it is up to the workers who consider the living organism their own domain, to look into the possibilities with respect to this new school; for the old methods have not proved magical in hurrying on the development of knowledge in this domain, and methods which have produced great increases of knowledge in many different branches of physics might well have some help to offer. In An Outline of Physics (Methuen, 6/6), L. Sournerns, M.A., _ B.Sc., Lecturer in Physics, University of Sheffield, attempts a radical rearrangement of the subject-matter of a student’s first course at a modern university. He gives first a general sketch or outline of the subject, which is intended to act as a frame on which detailed instruction may be placed. Part II. comprises a course suitable fur general purposes, and is planned so as to allow the greatest elasticity and scope for modification. 946 Notices respecting New Books. Masor P. A. MacManon in his Introduction to Combinatory Analysis (Camb. Univ. Press, 7/6) gives an outline of the easier parts of the theory of Combinatory Analysis expounded in his. larger volumes of 1915-1916. This small treatise treats the theory of symmetric functions and builds the general results in the theory of distributions on it. It is a valuable and interesting hook. Modern Analysis. Vhird Edition. By Prof. BE. T. Waurrraxer, F.R.S., and Prot. G. N. Watson, F.R.S. Cambridge University Press. 40s. THERE is no great change in the new edition of Modern Analysis. The authors have added an entirely new chapter on Lamé’s functions which should prove of great value, and the chapter on fourier Series has been rearranged in the interests of the large number of applied mathematicians who use this book. It is notoriously difficult to please two masters and to avoid slipping between two stools: but as the book is undoubtedly of great value to applied mathematicians, a small sop of this kind is probably not out of place. There are no new features to remark on beyond these, except the price. The Heperimental Basis of Chemistry. By Ipa Freunp. Edited by A. Hurcuinson ond M. Brarricsk Tuomas. Cambridge University Press. Pp. xvi + 408. Price 30s. net. Miss Ipa Freunp, who died in 1914, taught chemistry at Newnham College from 1887 to 1912 with well-known success. Her book Vhe Study of Chemical Composition caused her name to be familiar to students who had never been to Cambridge, and was received with well-deserved praise. The present work embodies ten chapters left by her in manuscript, which have been admirably edited by two of her friends. We are told that the book was planned to consist of twenty chapters, but the portion published is. complete in itself. It shows remarkable originality of treatment,. and utters a vigorous protest against the very conventional way in which experiment is dealt with in too many of the laboratory manuals for students. It is an elementary book, but teachers and advanced students will get many valuable suggestions from the perusal of what isa sound, clear, critical, and logical account of the fundamental principles of chemistry. It appears from the sub- title that the book was not primarily intended to be worked right through in the laboratory, but any student with time and patience to do so would acquire a very good knowledge of the principles of the science. The book forms an excellent memorial to a teacher who, as the editors say, “was richly endowed with the critical faculty, keenly sensitive to fallacious reasoning, and quick to detect an unwarrantable assumption.” f 947 (i) XCII. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from p, 828. | June 9th, 1920.—Mr. R. D. Oldham, F.R.S., President, in the Chair. Mr. Careint Giusron Kwort, D.Se., LU.D., F.R.S., delivered a Lecture on Earthquake Waste and the Elasticity of the Harth. The Lecturer remarked that the record produced on delicate seismegraphs of the earth-movements due to distant earthquakes proves that an earthquake is the source of two types of wave-motion which pass through the body of the Earth, and a third type which passes round the surface of the Karth. Before earthquake records were obtained, mathematicians had shown that these three types of wave-motion existed in and over a sphere consisting of elastic solid material. Many volcanic phe- nomena, however, suggest the quite different conception of a molten interior underlying the solid crust. At first statement these views seem to be antagonistic; but there is no difficulty in reconciling them. Whatever be the nature of the material lying immediately below the accessible crust, it must become at a certain depth a highly-heated fairly- homogeneous substance behaving like an elastic solid, with two kinds of elasticity giving rise to what are called the compressional and distortional waves. The velocities of these waves are markedly d different, being at every depth nearly in the ratio of 1°8 to 1. Both increase steadily within the first thousand miles of descent towards the Harth’s centre, the compressional wave-velocity ranging from +5 miles per second at the surface to 8 miles per second at depths of 1000 miles and more; the corresponding velocities of the distortional wave are 2°5 and 4:3 at the surface and at the 1000-mile depth respectively. At greater depths these high velocities seem to fall off slightly ; but the records fail to give us clear information as to velocities at deptbs greater than about 2500 miles. Down to this depth the Earth behaves towards these waves as a highly-elastic solid. The elastic constants, which at first increase with depth more rapidly than the density, become proportional to the density, for the velocity of propag: ation becomes practically steady. About half-way down, however, the material seems to lose its rigidity (in the elastic sense of the term), and viscosity possibly falkes its place, so that the distortional wave is killed out. In other words, there is a nucleus of about 1600 miles radius which cannot transmit distortional waves. This nucleus is enclosed by a shell of highly-elastic material transmitting both compressional and distortional waves exactly like an elastic solid. 948 _ Geological Society :— June 23rd.—Mr. R. D. Oldham, F.R.S., President, in the Chair. The following communication was read :— ‘The Seandinavian Mountain Problem.’ By Olaf Holtedalhl. In the introduction a brief account is given of the history of research regarding the Scandinavian mountain problem, which deals with the superposition of highly-metamorphosed, often gneissose rocks upon slightly-altered fossiliferous Cambro-Silurian sediments. From a consideration of the phenomena in the mountain-belt of deformation, it is inferred that the age of the displaced materials depends upon the angle of inclination of the thrust-planes and their depth. Though the thrusts have extended downwards for a considerable distance, they have not, generally, in the author’s opinion, reached below the level of the pre-Cambrian plane of denudation, and no true Archean rocks could, as a rule, have been tapped. In support of these conclusions some of the tectonic features of two districts are indicated: (1) Finmarken in Northern Norway, and (2) the southern part of the Sparagmite area near Randsfjord, in South-Central Scandinavia. Brief descriptions are given of the rock-groups in Finmarken and their structural relations. Special attention is directed to the structure of the Alten district, where the main tectonic feature is a highly-undulating thrust which does not intersect the pre-Cambrian floor. Regarding the Randsfjord district, the original order of succession of the strata is indicated, from the Holmia Shale to the close of the overlying Cambro- Silurian sediments. Pressure from the north in late Silurian time developed imbricate structure in these sediments, but such dis- - placements are not supposed to have affected the pre-Cambrian floor. As investigation proceeds, it seems to become increasingly ~ evident (1) that the highly-metamorphic sedimentary rocks of the middle and northern part of the eastern mountain-belt are mainly of earlier Ordovician age, while those west of the Sparagmite region in the south-western mountain district are chiefly of Silurian age; and (2) that the igneous materials associated with these highly- metamorphosed sediments are younger intrusive rocks. December 1st.—Mr. R. D. Oldham, F.R.S., President, in the Chair. The following communication was read :— ‘An Aolian Pleistocene Deposit at Clevedon (Somerset).’ By Edward Greenly, D.Sc., F.G.S. Banked up against the craggy hillsides about Clevedon are considerable depostts which contain a terrestrial molluscan and vertebrate fauna of Pleistocene age. Most of the vertebrates were obtained from a small cave, during exploration some years ago by Mr. G. E. Male & Prof. 8. H. Reynolds. The deposits consist of sandy breccia, stony sand, loamy sand, and loam. Breccias oceur only close to the crags, but the sands may extend all across the valleys. The stones, which are sharply angular, are of exclusively Tertiary Deposits in North-Western Peru. GAY local derivation, the sands of alien derivation. The formation pre- sents a number of unusual characters, among which are a total absence of lamination and a singular vertical cleavage. From its fauna, structures, and composition, but especially from its physio- graphical relations, it is clear that the formation cannot be of aqueous origin. The breccias are manifestly local talus. The distribution of the sands and loams, and their peculiar structural characters, can, it is argued, be accounted for only by eolian action. The winds, banking them up under the lee of southern slopes, must — have been from the north; and, as the land seems then to have stood sufficiently above its present level to lay dry the bed of that part of the Bristol Channel, it is suggested that the alien sand was blown from the older Pleistocene deposits of that hollow. A comparison is made with the less, and it is shown that, out of 20 characteristics, paleontological, structural, and otherwise, the Clevedon drifts have 17 in common with that formation. Those which they have not in common are assignable to the prox- imity of Clevedon to the sea. The deposits are ascribed, accordingly, to ‘ less-conditions ’, acting in a region where local euserbinsiraness appear to have been ‘especially favourable. December 15th.—Mr. R. D. Oldham, F.R.S., President, in the Chair. The following communications were read :— 1. ‘Structure and Stratigraphy of the Tertiary Deposits in North-Western Peru.’ By Thomas Owen Bosworth, D.Se., M.A., GS. The westernmost ranges of the Andes, in the north of Peru, are of pre-Tertiary age. ‘The Hocene Pacific Ocean lay at the foot of them. The Tertiary rocks oceupy a narrow strip of country between the mountains and the sea. They are exposed in those areas which have been denuded of their horizontal cover of Quaternary deposits. The Tertiary consists of 15,000 to 25,000 feet of clay- shales and sandstones, with innumerable thin seams of beach-pebbles and shells. Thus, during the Tertiary Period, a large subsidence was In progress. The stratigraphical succession is as follows :— eee Miocene. Zorritos Formation ..................... 5000+ MobibosmHormtahOner tant eke ce. 5000 + Negritos Formation. Clavilithes Series \ Turritella Series KOCENE. 1 cee EAS 7000+ The Tertiary accumulation is greatly broken up by intense block-faulting ; between the fault-blocks are differential dis- placements of many thousand feet. It is inferred that, in the interval between the Tertiary and the Quaternary Periods, an important movement occurred along a great fault-belt parallel with the Andes. The mountains were further upraised, and the sea-floor subsided to a great depth. The uplifting of the mountains caused a strip of territory along the west side of them, 20 miles wide, to emerge from the sea. This is the littoral: it was part of the crush-belt of the great fault. 950 Geological Society. ‘Paleontology of the Tertiary Deposits in North-Western Se By Henry Woods, M.A., F.R.S., T. Wayland Vaughan, PhDs coke Cushman, hey, Saal Prof. Flerbort Leader Hasan D.Sce., EGS. 3. ‘Geology of the Quaternary Period on a Part of the Pacific Coast of Peru.’ By Thomas Owen Bosworth, D.Se., M.A., F.G.S. Throughout the Quaternary Period, the littoral has undergone a series of vertical oscillations. It has been lifted up and down repeatedly like a lid, having its edge a few miles out in the Pacific Ocean and its hinge-line in the Andes. During these processes the littoral has several times been alternately overspread with a marine deposit and then raised above the sea. The ocean-soundings show a steep 2000-foot submarine cliff at the edge of the continental shelf. It follows a fairly direct line, which passes within 5 miles of the land. This cliff is taken to be a submarine fault-scarp, marking the important fracture (Pacifie Fault) which was the western boundary of the Quaternary uplifts. The oldest and highest of the raised sea-floors (‘tablazos’) now has an elevation of 1100 feet. It extends 20 miles inland, and,. within the territory here discussed, it covers an area of 700 square miles. The inland boundary of each ‘ tablazo’ is a raised sea-eliff. The original western limit of each one of them probably was the edge of the continental shelf. Whether there was any oscillation of the deep sea-floor on the west side of the Pacific Fault is not known. The depth, 27 miles from the present coast, is 12,000 feet. The Quaternary deposits formed upon it are presumably deep-sea 00zes. The events on the east side of the Pacific Fault may be grouped into four similar episodes. Each consists of a subsidence accom- panied by marine transgression, followed by an uplift causing emergence of new land from the sea. They are as follows :— (1) The Mancora Episode. (2) The Talara Episode. (8) The Lobitos Episode. (4) The Salina Episode. Each episode obliterates all trace of any preceding one which was not greater. Four episodes have left their mark ; but probably there were many others, of which no evidence remains. The most substantial of the deposits formed during these marine transgressions is 250 feet thick. The material ranges from shell- limestone to beach-pebbles. The shells have been examined by Col. A. J. Peile, who pronounces them (probably all) to be ving species. On the land, extensive breccia-fans and valley-terraces were pro- duced, under desert conditions, during these oscillations. They are correlated with the marine terraces. In conclusion, it is considered that not one ten-thousandth part of the Quaternary history, here outlined, can have taken place within the last 500 years. INDEX tro VOL. XLI. ———_- EMGHINTEM on the distribution of the active deposit of, in elec- tric fields, 357. Atther, light, matter, on, 940. Air, on the thermal effect of a slow current of, pene past heated platinum wires, 230. Airey (Dr. J. R.) on Bessel functions of small fractional order and their application to problems of elastic stability, 200. Albumen films, on colours of, 388. Allen (Dr. H. 3.) on the angular momentum and related properties of the ring electron, 113. Alpha particles, on the collision of, with hydrogen atoms, 307, 486. Ammonium nitrate, on the pr -operties of plastic erystals Offa ls Analysis, on a method of, suitable for differential equations, 584. Anemometers, on the construction of hot-wire, 240; on null-defiexion constant current, 716. Asymptotic forms ‘of certain hyper- geonietric functions, on the, 161. Atmosphere, on problems relating to rotating fluid in the, 665. Atoms, on the constitution of, 281 ; on some dimensions of, 872. Aurén (Dr. T. Ii.) on the scattering and absorption of hard X-tays, 733. Baly (Prof. IX. C. C.) on the physi- cal significance of theleast common multiple, male Banerji (Prof. B.) on the colours of mixed plates, 538, 860. Bateman (Dr. H.) on an electro- magnetic theory of radiation, 107. Batho (Prof. C.} on the torsion of closed and open tubes, 568. Beams, on the transverse vibrations of, 81. Bessel functions of small fractional order, on, 200. Beta-ray emission from films of elements exposed to Rontgen rays, on the, 120, Bismuth, on the Hall effect in 916. Bond. (Ww. N.) on the properties of plastic crystals of ammonium nitrate, 1. ] Books, new :—Berget’s Ou en est la. Météorolopie ?, 682 ; Transactions. of the Bose Research Institute, 945; Southerns’s An Outline of - Physics, 945; MacMahon’s intro-. duction to Combinatory Analysis,. 946; Whittaker & Watson’s Modern Analysis, 946; Kreund’s. The Ixxperimental Basis of Chem-- istry, 946. Bosanquet (C. H.) on the intensity of reflexion of X-rays by rock-- salt, 309. Bosworth (Dr. T.0.) on the Tertiary deposits of N.W. Peru, 949; on the Quaternary period in Peru, 950. Brachistochrone, on the problem of the, 225. Brage (Prof. W. L.) on the intensity of reflexion of X-rays by rock- salt, 309. Briggs (G. H.) on the distribution of the active deposits of radium, thorium, and actinium in electric fields, 357. Broughall (L. St. C.) on the dimen- sions of the atom, 872 Campbell (Dr. N.) on ane physical significance of the least common multiple, 707. Capacity, on the self-, of magneto windings, 33. Carbon monoxide, disappearance of, in the electric discharge, 685. Carrington (H.) on the determination of ie s modulus and Poisson’s ratio by. the method of fiexures, 206; on the moduli of rigidity for spruce, 848. Carson (J. R.) on wave propagation over parallel wires, 607. Centrifuging, on the separation of mercury into its isotopic forms by, 818. Chapman (D. L.) on the abnor- mality of strong electrolytes, ae ytes; 799. Chemical combination, on the strue- ture of the molecule and, 510. Cireular currents, on the magnetic field of, 377. Coalescence of liquid spheres, on the, Naestend oie 8 952 Colour sequence in Talbot’s bands, on the, 877. vision, a statistical survey of, 186. Colours of mixed plates, on the, 358, 860. Compressibility, on surface energy, latent heat, and, 21. Compton (Prof. A. H.) on the pos- sible magnetic polarity of free electrons, 279 ; on the degradation of gamma-ray energy, 749; on the wave-length of hard gamma rays, 770. ‘Convection, 899. currents, on the, from heated wires, 716. Copper sulphate, on resistance of solutions of, in glycerine, 544. Coupling, on systems with propa- eated, 4382, 826. Cowley cw. L.) on a method of analysis suitable for the differential equations of mathematical physics, 584. Cox (Prof. A. H.) on the Lower Paleozoic rocks of the Arthog- Dolgelly district, 688. Crystals, on the properties of plastic, of ammonium nitrate, 1. Cyclones, on, 665. Darnley (EK. R.) on the transverse vibrations of beams and the whirl- ing of shafts supported at inter- mediate points, 81. Darwin (C. G.) on the Eallbeione of alpha varticles with hydrogen nuclei, 486. Davies (Miss A. C.) on the critical electron velocities for the produc- tion of luminosity in atmospheric neon, 921. Davis (A. H.) on the heat loss by convection from wires in a stream of air, 899. Davison (Dr. C.) on the annual and diurnal periodicity of earthquakes in Japan, 908. Descent, on curves of quickest, 225. Dielectric constants of electrolytic solutions, on the, 829. Differential equations, on a method of analysis for, 584. Distillation, on the separation of miscible liquids by, 638. Dufton (A. F.) on the separation of miscible liquid by distillation, 633. on heat loss from, ENED EX. Dunlop (Miss M. A.) on a statistical survey of colour vision, 186. Dynamics of a particle, on the New- ton-Hinstein planetary orbit and the, 143. Earth’s magnectic field, on the de- termination of the horizontal com- ponent of the, 454. Earthquakes, on the annual and diurnal periodicity of, in Japan, 908. Edgeworth (Prof. F. Y.) on the genesis of the law of error, 148. Einstein's spectral line effect, on, 747, 944; equations of gravitation, on, 823, Elastic stability, on Bessel functions applied to, 200. Electric charges, on the scalar and vector potentials due to the motion of, 485. currents, on the effect of, en the photo-electric effect, 916; on the magnetic field of circular, 377 —— discharge, on the, in hydrogen, 304; on the disappearance of gas in the, 685. fields, on the distribution of the active deposits of radium, thorium, and actinium in, 357. Electrolytes, on the abnormality of strong, 799. Electrolytic solutions, on the dielec- tric constants of, 829. Electromagnetic theory of ation, on an, 107. Electron impacts, on the dissociation of hydrogen and nitrogen by, 778. velocities, on the, for the pro- duction of luminosity in neon, 921. Electrons, on the properties of the ring, 113; on the speed of ejection of, “from ‘an atom, 120; on the magnetic polarity of free, 279; on the liberation of, in the retina and the theory of vision, 289; on the radiation of energy by accele- rated, 408, Electronic: energy and relativity, on, 96. Energy, on the radiation of, by an accelerated electron, 405. Error, on the genesis of the law of, 148. Flexures, on the determination of Young’s modulus and Poisson’s ratio by the method of, 206. radi- INDEX. Flint (H. T.) on intergration theo- rems of four-dimensional vector analysis, 389. Fluid, on problems relating to ro- tating, in the atmosphere, 665. —— discharves as affected by resis- tance to flow, on, 286. Force-transformation, proper time, and Fresnel’s coefficient, on, 652. Four-dimensional vector analysis, on integration theorems of, 389. Fricke (EL. ) on the spectrum of helium in the extreme ultra-vio- let, 814. Galileo's ‘ Dialogues,’ note on a pro- position in, 225, Gamma-ray enervy, dation of, 749. Gamma rays, on the wave-length of hard, 770. Garner (W. E.) on the activity of water in sucrose solutions, 484. Gases, on anemometers for the inves- tigation of slow rates of flow of, 240, 716; on the temperature radiation of, 267; on the disappear- ance of, in electric discharge, 685. General Electric Company's Research Staff on the disappearance of gas in the electric discharge, 685. Geological Society, proceedings of the, 158, 583, 828, 947. George (H. J.) on the abnormality of strong electrolytes, 799. Gibbs (R. E.) on systems with propagated coupling, 432, 827. Gilmour (A.) on the resistance of solutions of copper sulphate in glycerine, 544. Glycerine, on the resistance of solu- tions of copper sulphate in, 544. Graphical method for determining on the degra- frequencies of lateral vibration or . whirling speeds, on a, 419. Gravitational field of a particle on Einstein’s theory, on the, 825. Gravity, on times of descent under, 225. Green (Dr. G.) on problems relating to rotating fluid in the atmosphere, 665. Greenhill (Sir G.) on the Newton- | Hinstein planetary orbit, 1438. Greenly (Dr. E.)-on an AXolian Pleis- tocene deposit at Clevedon, 948. H, on the determination of, 454, Hammick (D. L.) on surface energy, latent heat,and compressibility, 21. 953: Helium, on the spectrum of, in the extreme ultra-violet, 814. Hill (F. W.) on the oravitational field of a particle on EHinstein’s. theory, 828. Hjalmar (E.) on precision measure- ments in the X-ray spectra, 675. Holtedalfl (O.) on the Scandinavian mountain problem, 948. Horton (Prof. F.) on critical electron: velocities for the production of luminosity in atmospheric neon, 921. Hot-wire anemometers, on the con— struction of, 240, 716. Houstoun (Dr. R. A.) on a statistical survey of colour vision, 186. Hughes (Prof. A. Ll.) on the dis-. sociation of hydrogen and nitrogen by electric impacts, 778. Hydrogen, on the electric discharge in, 804; on the collision of splhe particles with atoms of, 307 ; the disappearance of, in the alee ic discharge, 685 ; on the dissociation of, by electric im pacts, 778. nuclei, on the collisions of alpha. particles with, 486. positive rays, on the spectrum of, 558, 566. Hyper bolic space-time, on an inertial frame given by, 141. Hyperdimensions, on motion and, 647. Hypergeometric function )A,(z), on an asymptotic formula for the, 161; on a generalized, with x parameters, 174. Inductance, on the effective, of mag- neto wingings, 33. Inertial frame given by hyperbolic space-time, on the, 141. Integration theorems of four-dimen- sional vector analysis, on, 389. Isothermals of vapours, on vapour pressures and the, 441. Isotopic forms of mercury, on the, 818. James (It. W.) on the intensity of reflexion of X-rays by rock-salt, 509. Jeffery (Dr. G. B.) on the gravita- tional field of a particle on Iin- stein’s theory, 823, Jenkins (Prof. Week ) on the deter- mination of H, 454. Joly (Prof. J.) on a quantum theory of vision, 289. 954 Jones (Prof. A. T.) on the motion ot a simple pendulum after the string las become slack, 809. Knott (Dr. C. G.) on earthquake waves, 947. Lamplugh (Prof. G. W.) on the Pleistocene glaciation of England, 158. ‘Latent heat, surface energy, and compressibility, on, 21. -Lattey (R. T.) on the dielectite con- stants of electrolytic solutions, 829. Least common multiple, on the physical significance of the, 707. Levy (Dr. H.) on a method of analy- sis suitable for the differential equations of mathematical physics, 5384. Aiénard (A.) on a method of finding the scalar and vector potentials due to the motion of electric charges, 485. Light, on the weight and fate of, 549; on etner, and matter, 910. Liquids, ou the separation of misci- ble, by distillation, 633; on the discontinuous flow of, past a wedge, 801; on the cvalescence of spheres of, 877. Lodge (Sir O.) on the supposed weight and ultimate fate of ra- diation, 549; on ether, light, and matter, 940; on the Einstein spectral shift, 944. Lyman (Prof. T.) on the spectrum of helium in the extreme ultra- violet, 814. McAulay (Prof. A.) on the inertial frame given by a hyperbolic space- time, 141. McLachlan (Dr. N. W.) on the effective inductance, effective re- sistance, and self-capacity of mag- neto windings, 33. Magnetic field of circular currents, on the, 377; on the determination of the horizontal component of the earth’s, 454. -__— polarity of free electrons, on the, 279. Magneto windings, on the effective inductance, effective resistance, and 'self-capacity of, 33. Magueton, on the mass of the Par- son, 118. _Mallik (Prof. D. N.) on the electric discharge in hydrogen, 304. INDEX. Masson (I.) on the activity of water in sucrose solutions, 484. Masson (Prof. O.) on the constitution of atoms, 281. Matter, on ether, light, and, 940. Mercury, on the separation of, into its isotopic forms by centrifuging, 818. Milner (Dr. S. R.) on the radiation of energy by an accelerated elec- tron, 405. Mixed plates, on the colours of, 338, 860. Molecular diameters, on, 877. Molecule, on the structure of the, and chemical combination, 510. Morgan (J. D.) on impulsive sparking voltage in small gaps, 462, Morton (Prof. W. B.) on times of descent under gravity, 225; on the discontinuous flow of liquid past a wedge, 801. Motion and hyperdimensions, on, 647. Nagaoka (Prof. H.) on the magnetic field of circular currents, 377. Neon, on the critical electron veloci- ties for the production of lumino- sity in, 921. Nitrogen, on the disappearance of, in the electric discharge, 685; on the dissociation of, by electron impacts, 773. Pendulum, on the motion of a simple, after the string has be- come slack, 809. Phosphorus vapour, on the electric discharge in, 692. Photo-electric effect, on the effect of an electric current on the, 916. theory of vision, on the, 347. Physics, on the differential equations of mathematical, 584. Planetary orbit, on the Newton- Einstein, 148. Plates, on the colours of mixed, 338, 860. Platinum wires, on the thermal effeet of a slow current of air flowing past heated, 240. Poisson’s ratio, on the determination of, by the method of flexures, 206. Pol (Dr. B. van der, jr.) on systems with propagated coupling, 826. Polarity, on the magnetic, of free electrons, 279. TANS DERN: mole (J. HW. J.) on the photo- electric theory of vision, 347 ; on tke separation of mercury into its isotopes by centrifuging, 818. Porter (Prof. A. W.) on systems with propagated coupling, 482, 827, Positive rays, on the spectrum of hydrogen, 558, 566. Potentials, cn the scalar and vector, * ? ’ due to the motion of electric charges, 483. Prescott (Dr. J.) on the torsion of closed and open tubes, 569. Priestley (Prof. H. J.) on the Kin- stein spectral line effect, 747. Prismatic bars, on the transverse vibrations of, 744. Propagated coupling, on systems with, 432, 826. Quantum theory of vision, on a, Beg. Radiation, on an_ electromagnetic mieony of, 10/; cn. the tem- perature, of gases, 267; on the supposed weight and ultimate fate of, 549. Radium, on the distribution of the active deposit of, in electric fields, 357. Raman (Prof. C. V.) on the colours of mixed plates, 338, 860. Relativity, on electronic energy and, 96. Resistance, on the effective, of mag- neto windings, 33, Rhodopsin, on the liberation of electrons in, 289. Rigidity, on the moduli of, for spruce, 848. Rine electron, on the properties of the, 113. Robinson (J. W. D.) on the Devonian of Ferques, 684. Rock-salt, on the intensity of re- flexion of X-rays by, 309. Rod, on the period of vibration of a tapering, 259. Rontgen rays, on the beta-ray emis- sion from films of elements ex- posed to, 120. Rutherferd (Sir E.) on the collision of alpha particles with hydrogen atoms, 307; on the mass of the long-range particles from thorium U, 570. Saha (Dr. M. N.) on the temperature radiation of gases, 267. 955 Sargent (H. C.) on the chert forma- tions of Derbyshire, 159. Schumann region, on the spectrum of helium in the, 814. Sethi (Prof. N. &.) on Talbot’s bands and the colour sequence in the spectrum, 211. Shafts, on the whirling of, 81. Shaxby (J. H.) on vapour pressures and the isothermals of vapours, 44]. Shear, on the correction for, of the differential equation for vibrations of prismatic bars, 744. Shenstone (A. G.) on the effect. of an electric current cn the photo- electric effect, 916. Simons (L.) on the beta-ray emis- sion from films of elements ex- posed to Rontgen rays, 120. de Sitter’s inertial frame, on a hyper- bolic form of, 141. Slate (Prof. F.) on electronic energy and relativity, 96; on force-transformation, proper time, and Fresnel’s coefficient, 652. Smith (T.) on the accuracy of the in- ternally focussing telescope, 890. Sodium chloride, on the intensity of reflexion of X-rays by, 309. Southwell (R. V.) on a graphical method for determining the fre- quencies of lateral vibration, or whirling speeds, for a rod of non- uniform cross-section, 419. Sparking voltages in small gaps, on impulsive, 462. Spectra, on precision measerements in the X-ray, 675. Spectral line effect, on the [instein, 747, 944, Spectrum, on Talbot’s bands and the colour sequence in the, 211; on the, of hydrogen positive rays, 558, 566; on the, of helium in the extreme ultra-violet, 814. Spruce, on the moduli of rigidity for, 848. Stead (G.) on the design of soft thermionic valves, 470. Sucrose solutions, on the activity of water in, 484. . Surface energy, latent heat, and compressibility, on, 21. Swain (Miss L. M.) on the period of vibration of the gravest mode ofa thin rod in the form of a truncated wedge, 259. 956 Systems with propagated coupling, on, 432, 836. Tacheometry, on the accuracy of the internally focussing telescope in, 890. Talbot’s bands and the colour se- quence in the spectrum, on, 211. Tavani (F.) on motion and hyper- dimensions, 647. Taylor (W.) on the coalescence of guid spheres, 877. Telescope, on the accuracy of the internally focussing, 890. Temperature radiation of gases, on the, 267. Thermionic valves, on the design of soft, 470. Thomas (Dr. J.8. G.) on the thermal effect produced by a slow current of air flowing past a series of fine heated platinum wires, 240; ona null-deflexion constant - current type of hot-wire anemometer, 716. Thomson (G. P.) en the spectrum of hydrogen positive rays, 566. Thomson (Sir J. J.) on the structure of the molecule and chemical com- bination, 510. Thorium active deposit, on the distri- bution of, in electric fields, 357; on long-range particles from, 575. —— C, on the mass of the long- range particles from, 570. Time, on force-transtormation, Fres- nel’s coefficient, and proper, 652. Timoshenko (Prof. S.-P.) on the correction for shear of the differ- ential equation for transverse vibrations of prismatic bars, 744. Tobin (T. C.) on times of descent under gravity, 225. ‘Torsion of closed and open tubes, on the, 568. Tubes, on the torsion of closed and open, 568. . Valves, on the design of soft thermi- onic, 470. Vapour pressures and the isother- nals of vapours, on, 44], Vector analysis, on integration theorems of four-dimensional, 389. Vegard (Prof. L.) on the spectrum of hydrogen positive rays, 558. Venturi meter, on the, 286. ENED aeeX Vibrations of beams, on the, 81 ; on the period of, of a tapering rod, 259; on a graphical method for deter mining frequencies of lateral, 419; on the transverse, of pris- jatic bars, 744. Viscous tlow, on, 286. Vision, on a statistical survey of colour, 186; on a quantum theory of, 289; on the photo-electric theory of, 347. Voltages, on impulsive sparking, in small gaps, 462. Walker (Dr. W. J.) on fluid dis- charges as affected by resistance to flow, 286. Warren (8S. H.)-on a natural eoclith factory beneath the Thanet sand, 822. Water, on the activity of, in sucrose solutions, 48-4. Wave propagation - wires, on, 607. Wedge, on the period of vibration of a tr uncated, 259; on the dis- continuous flow of liquid past a, 801. Wells (A. IX.) on the Lower Palz- ozoic rocks of the Arthog-Dolgelly district, 683. Whirling of shafts, on the, 81. speeds, on a graphical method for determining, 419. Wires, on wave propagation over parallel, 607; on heat loss by convection from, 899. Wood (Dr. A. B.) on long-range particles from thorium active de- osit, 570. Wrinch (Dr. D.) on an asymptotic formula for the hypergeometric function ,A,(z), 161; ona general- ized hypergeometric function with nm parameters, 174. X-ray spectra, precision measure- ments in the, 6753; spectral lines, on the interrelation between heta- ray and, 120; on the intensity of reflexion of, by rock-salt, 309; on the scattering and absorption of, in the lightest elements, 733. Young” s modulus, on the deter- mination of, by the method of flexures, 206. over parallel END OF THE FORTY-FIRST VOLUME. Printed by Taytor avd Francis, Red Lion Court, Fleet Street. aol ee Phil. Mag. Ser. 6, Vol. 41. Pl. I. Bonp. Og? ©) “St 2 naiyml | | | ee —— 2 ae ees ie ~attow oer Fig. 14, Phil. Mag. Ser. 6, Vol. 41. Pl IT. + om = ry +a ae _SIMONS. Fig, 4 SILVER £0} a ile ee i ie Oe ==-32 ~_-.__. Barium as Radizton 3 ———___ Silveras Radiator, ers - -- 2.5 a 3S See 250 25) 20 \ fae ee &: = x 1 = ° or a2 o3 0% as of O7 cm. o se a i Actes from ae screen as measured in air at O° C. and 40 ejummuars Ors ae nates, — Sroge(N, ,— Nz), taken from figs. 5 and 6, Fie. 7.—Ordinates (N, — Nz) Plesenting a ae of 2626 in this function and on planes in air, at 0° C ated sereen as marlivalue of —_— = 2 love N, Ni x) is placed on each curve B-ray enersy to be t] < Weomts Naver of Seree of energy crossing planes successively distant from change at the distance indicated. Phil. Mag. Ser. 6, Vol. 41. Pl. III. f sd 4 ” t E e a) . L} ‘ u q + =) rs a § 4 . + , * a y i 1 u 5 4 } b= i - 7 , ; ' . > 4 ~ ¥ , ‘, - a : A 1 { \ ’ iy 7m j ; i = A t f q \ 2 ‘ i } ‘ “ a i ‘ i - at 1 P f oo, ee : / ‘ - i \ A , i] ; 1 : 4 i f . ie . t ‘ } at j $. “ fr f 1 1 . A , " ‘ ‘ f 7 ‘ t 1 Fe 1 a fi: ; \ ; ‘ me i} 5 Fic. 3, =r 2 eens 70 ta Comat 1h t Tenisetion cx fom strn nor oh REP BN 2 2 pas fa ny a ar as & Preserisemef marian Tra. 7. T a - SS es a=. -— Borium as Radiator Silver as Radiaton ROL Conse Mot = fen see a me NaN. Atomts Number of Screen. ing imaginary Vie. 7.—Ordinates (N,—N,) proportional to the S-ray energy © gina planes in ai, at 0° C. and 76 cm. pressure. at distances from the illumin- : ated screen as marked on the curves, after having adjusted the total B-ray enerzy to be the sane from all the screens. Phil. Mag. Ser. 6, Vol. 41. Pl, IIT. Ne. 4. SS BARIUM AS RADATOR. 15] i Tenlgetion Covet. Proisare ln only murcacy. i 0 a 39 as 40 45 Py is so Irttes thqsgyrmcn Vie. 8. Tie. 9 i T ~ ee SILVER as RADIATOR. 3 from the sereen as qeasured in airat O° C, and Fras, 8 & 9.—For abscissie, distance: taken from figs. 5 und 6, d o = 76 em. pressure. For ordinates, = gloee(S —N a change of 2'626 in this function and on u y.—N,) is place: each curve fiz, 9, 1313. One absolute value of — re Joz.(N,—N-) is placed on each curv each division on fig. 8 representing for reference. : jeune . e , 0 sing lanes suct The rate of diminution of enerey crossing we ee ited the screens suffers a marked change at the distar i ively distant from —_—“-——— Vol, 41, Ph lv. Mag. Ser. 6 il Ph JI. BANER RaMan & poe g 4 re Al aK Pins) : Aira ee a ae “a Raman & BAaneERdII. a RE et Phil. Mag. Ser. 6, Vol. 41, Pl. V. Fre. 12, Car) Sy A Pa ae i ™ f wa ie i¢ iy! Teo ak eer a paar: 2 = ZEW S ==Ze7| Vi aa 'SS20) hokans is NSEEE RIN UA REE : a NSEC SEY 3 ‘iN \NSS SSE E32 2727/7 eb ANN | | a6 UNNSGSEE Ses Cee 27, | | So MENS TEN a : ANION GND ce a Hh = 3 al ue NU ee 7 NUN oe Ty ech SEN aN =—\ Y LON | Zi a vas {¢ j Nees E SSN ZS SIZ nn AA nn NN AN FIST t2FVE-3S}=c on 8 5B) = f Is Phil, Mag, Ser. 6, Vol. 41. Pl. VI. Wie. 2. Fic. 1. NAGAOKA, | + See ASS ZL + Bs \ SO SSRE Ss SAR ZO LIU 2 >a POM & Hi Sire GAUAVAN ES Ue a | : i yj | ZA ~ SS = é F al 3) Nw AEH NSA ey NSIS Mee LEI TALON ALLEL ey ATEN) €} pA z Z Sees 2] * en sieyeee = YP ENAEEN (ZX i yi = \ R : ; i | = LY Tf) \)s| 7 ie LN ° 7 F a = 71 oe & $ | E Bae i lee We <- a |S pa 5 2 i= 3 | J de a A ¥ 8 = | oO pe =} «|i Ee = i As oS 7 = we ra ' oe wy ts | | N We He cs Phil. Mag. Ser. 6. Vol. 41, Pl. VII. SourTHWELL. _ Phil. Mag. Ser. 6, Vol. 41, P). VIII. —= ANODE CURVES GRID CURVES 5 ile} 15 5. 10 [4 03 Hie. 7. Grid voltage Anode voltage Grid current (milliamperes) Grid current (milliamperes) 0 5 10 15 [e) 5 10 Grid volege Grid voltage Fia, 4 Grid voltage 9) ro) nm rece so 2 is} > 8 s x =05 0 F05 15-1005 ros Grid voltage Continuous curves represent anode currents. Broken FA 1, grid currents. “ Oscillating points” shown thus |X - Fie, 5. ANODE CURRENT GRID CURRENT 2 3 1 2 3 a mperes) ° & 2 3 ° 6 Grid current ma) ° rs ° rs Anode current [milliamperes Anode current [oitiia is} nm Grid voltage Pressure of hydrogen='023 mm. x y Ry vie8 vib 2 3 iT 2 3 4 Grid spacing in MM. V'=grid potential. | Pressure=:026 min. Anode voltage Grid current. (ma) Anoge volt Phil. Mag. Ser. 6, Vol. 41, P), VILL. Anoce voltage Percentage incresse> Pressure in MM. Fic. 8, Pressure of Heliam=40lm.m. Pressure of Helium =0:6 m.m. -4 =3 =2 =I i} +I +2 a 3 -2 =I C) +l re eeecret 40 Regior) of iF 35 periaaee ee 30-— Ui 30 Sra, ne \ Yo, SSE? | 5 25 oral, { han Pe ui SS 20 ' Lp \20 ° Noy 15 Sa f = = Als ae v. | i 3 LI KOS t 10 10 : a Nel 8 E i) kK a 5 Ir fr < { T 7 . | a | a a | Oo +1 +2 4-3 eI oO + +2 +3 Grid voltage Grid vokage Continuous curyes represent anode currents. Broken s) ” grid currents, Fie. 9. Fia, 10. 45 Veliurh 40 | @ 3 35 3 ~ 8 30 He g 3 25 s $ 20 oO O15 iit) W520 25 3/0 3:5) ast 2 a > 30 = N 4y as Aza g a eo = 4 20 Te 9 ae Ga 339 25 50 75 Pressure in MM. , ‘ > ~ | | | | : b : > ; ‘ s te nh vl LM we 0 wu > at \ \ nnn i = s imi | ‘XI Id Th TA °9 “19g “Sey Tg iN J 3 9088 01202 5078 | I ———— ——————