Andrade, which forms a mixture of RaB and RaC, might be accounted for by assuming an electron to recombine between circles with angular momenta 4 h 3 h and 6 2π Let more generally an electron pass from a circle with angular momentum n2 h to one with a momentum ni h m 2π If we take into account the variation of mass with velocity, we get ע 1 1 R = (N + 1)'m2 () { 1 + (N+1) (+)}. (28) Ꭱ 2 Putting N=82, m=6, n1 = 3, n=4, we get The numerical agreement is good enough, but I think we must be very careful in drawing conclusions from a single coincidence. I merely put it down as a suggestion which might be worth consideration. Recent experiments of Barkla and Miss White have given indications of a homogeneous J-radiation more penetrating than the K-radiation, which Barkla calls the Y-series. From the absorption coefficient of the rays just hard enough to excite the J-radiation they find for Al a wave-length λ=0.37.10-8 cm. If this series really exists it can hardly be explained by electrons belonging to the external system, but should be produced by the electrons forming part of the nucleus. The equation (28) would give nearly the right frequency when we put m=4, n1=1, and n=2. Physical Institute, Christiania. December 14, 1917. * E. Rutherford and E. N. da C. Andrade, Phil. Mag. xxviii. p. 263 (1914). XXXVI. Relativity and Electrodynamics. By G. W. WALKER, M.A., F.R.S., A.R.C.Sc., formerly Fellow of Trinity College, Cambridge". SIR [Plate X.] IR OLIVER LODGE'S recent papers in the Philosophical Magazine have brought into prominence once more the difference of attitude of the protagonists in "Relativity Doctrine" and "Newtonian Dynamics." That Sir Oliver's equation of motion for a moving planet requires some amplification in order to take full account of the special features of electrical inertia, will be recognized, and Prof. Eddington has suggested a method of dealing with the problem. Unfortunately, Eddington's method introduces an assumption which is frequently made by relativists in dealing with electrical inertia, and which in my opinion is inconsistent with the fundamental equations of electrodynamics. In former papers I have drawn attention to this assumption, which is closely linked with the quasi-stationary principle," and I had not intended to raise the point again. But Sir Oliver has suggested to me that an exposition of my views as to the parting of the ways between the logical development of electrodynamics and the doctrine of relativity would be of value, and I have agreed to his request. My remarks must, however, be confined to electric inertia, and I do not propose to enter on the gravitational and astronomical developments of Einstein's hypothesis. 66 The main point at issue may, I think, be put very concisely. Relativists assume that "the kinetic energy of a moving electrical system is a function of the resultant speed only and is independent of the direction of motion." My thesis is that this assumption is not consistent with the fundamental electromagnetic equations for the æther (supposed immobile), and that "the energy, or preferably the modified Lagrangean function, depends on the acceleration as well as on the speed of the system and involves also the relative direction of these." While the above appears to me to be the main point, there is no doubt that subsidiary considerations arise. Theory and experiment have interacted in a curious way, and I think the discussion should proceed by taking notice of the historical development. Sir Joseph Thomson was the first to *Communicated by Sir Oliver Lodge. prove theoretically that a moving electrified system would possess inertia, which Heaviside showed would depend on the speed with which the system moves. A later calculation by Thomson referred to a particular form of nucleus and to the momentum which it would carry with it in virtue of a uniform translation. It is extremely important to realize that the character of the nucleus determines the manner in which the speed enters in the expression for the momentum or for the energy. It is also vital to realize that while the momentum or the energy can be calculated for a particular form of nucleus moving with a uniform speed, it has not so far been found possible to give a complete solution when the speed is variable. M. Abraham extended Thomson's calculations, and he assumed that while the nucleus was still a sphere it was a perfect conductor, and he consequently obtained a value for the momentum in a state of uniform translation which differed from that found by Thomson when squares of the speed were retained. He emphasized the distinction between the effective inertia for acceleration along and perpendicular to the direction of motion. But finding that he could not obtain the exact solution for a variable speed, Abraham made use of what is called the "quasi-stationary principle," which amounts to saying that if we can calculate the momentum, or if we prefer it the Lagrangean function, for a uniform motion we can infer the equations of motion for a small departure from this state in the ordinary way. My contention is that we can no more do this logically for electromagnetic systems than we can for ordinary dynamical systems. We know quite well that we do not get the correct equations for small departures from a steady state, when the steady motion values are inserted in the Lagrangean function before the differential equations of motion are formed. The steady motion values may be inserted after the equations have been formed from the general Lagrangean function. Abraham calculated expressions for longitudinal and transversal electric inertia by means of the quasi-stationary principle. Experiments on transverse inertia became possible with the discovery of the Becquerel rays, and of the minute negatively charged particles projected from radium with speeds only little short of that of light. The matter was taken up first by W. Kaufmann, and I have a special personal interest in this since I was working side by side with him in the laboratory at Göttingen while his experiments were in progress. Kaufmann deflected the particles by crossed electric and magnetic fields, by which the particles are sifted out according to their speed, so that the ends of their trajectories form a curve on a photographic plate. Kaufmann considered that his measurements proved that Abraham's expression for transverse inertia was correct, and that the inertia of the particles was purely electromagnetic in origin. We must now retrace our steps to consider the important contributions to the theory of moving systems made by Prof. H. A. Lorentz and Sir Joseph Larmor. They proved that a mathematical correlation held between an electrical system at rest and a certain system maintained in uniform translation." If the moving system has a uniform speed ke (where c is the velocity of light) in the direction x, and the linear extent of the moving system is (1-k2) of the linear extent of the fixed system, and the variables time t and distance x in the fixed system are transformed to t' and ' in the moving system by a certain linear transformation involving k, then the state of the fixed system in terms of t and x is the same as that of the moving system in terms of t' and '.' It is reasonable to inquire if the contraction in the proportion (1-2) actually takes place when a system at rest is put into uniform translation, for if so it provides an explanation of the Michelson-Morley experiment. Now it is quite certain that the mathematical transformation is not true when k is variable, and therefore not true at any intermediate stage by which the system at rest might conceivably pass to the correlated system in uniform motion, if it ever does so at all. But relativists have assumed that the correlation proved by Lorentz and Larmor for a uniform translation only is true, and that the change actually takes place, when the speed is variable. It appears to me that if the primary equations are correct, the assumption is not merely not permissible, but is not true; and, on the contrary, if the assumption does represent actual truth, then the primary equations are wrong and must go. proof, which so far has not been offered. We await The longitudinal and transverse inertia of a "contracted" electron have been calculated by the quasi-stationary method, so that there is a double source of error in the result. Experiments by Kaufmann, Bestelmeyer, and others have been offered as experimental proof that the formula for transverse inertia of a contracted electron on relativity doctrine is correct. My contention is that while the experiments do not conflict with the relativity formula, the formula is inconsistent with the electrodynamic equations, and that several other formulæ correctly deduced from the primary equations agree with the experiments equally well. I doubt if many people in this country realize the very meagre character of the experimental results, and I therefore give a full-sized reproduction (Pl. X.) of the photographic plate from which Kaufmann made his measurements. The electric deflexion is across the paper and the magnetic deflexion up the paper, and it may be pointed out that if the inertia of the particles were quite independent of speed, the small curved arcs would be parabolas, and that it is only in so far as these arcs differ from parabolas that any dependence of inertia on speed can be made out at all. Further, the highest speed particles are those for which the deflexion. is least. I now return to the theoretical treatment of electric inertia. In order to avoid the error of the quasi-stationary principle, I developed some time ago a method of obtaining the longitudinal and transversal inertia directly from the primary equations by Newtonian methods. The method is rather tedious, but its correctness has not been called in question. Its application is general, but to get definite results the character of the nucleus must be specified. Various systems may be examined provided they do not violate any fundamental restriction imposed by electrodynamic conditions. In this way I examined the nucleus assumed by Sir Joseph Thomson and was able to confirm his result for transverse inertia, but obtained a different result for longitudinal inertia. On the other hand, with the nucleus assumed by Abraham I was able to confirm his result for longitudinal inertia, but not that for transverse inertia, Again, recently I examined the case of a contracted conducting spheroid which agreed in form with Lorentz's contracted electron for the uniform speed, but did not alter its form when acceleration was imposed. The results for both longitudinal and transverse inertia differ from those adopted by relativists. The differences that arise in these examples only become important when squares and higher terms in the speed are retained, and they arise from the fact that when acceleration is imposed, additional electric forces are set up which have to be allowed for in utilizing the boundary conditions at *The restriction is unnecessary, as I now find that my results are not altered when the surface deforms under acceleration as Lorentz assumes. |