Electron necessarily Radiate Energy? 417 



are unit vectors perpendicular to and parallel to E. 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 It, and there is a flux of energy outwards 

 through the surface of amount 





2 e 2 c 2 e 2 

 EH.27rR 2 sina^=^=:^-3V 2 . . (16) 



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, — the 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 

 nucleus 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) docs 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 o£ different radius R/ at the same time t, 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 energy -flow, 

 which are normal to both spheres, are therefore necessarily 

 curved lines, however far out they may be traced, and in 

 this case their curvature is such as to keep the energy 

 permanently in the electronic system. 



In the negative halt' of the field, the energy, although 

 here also it is not travelling outwards 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, # + c£ = 0, we 

 find that it is invariably negative; consequently the boundary, 

 considered as the limit of a thin transverse field separating 

 the electronic field from zero, is always engaged in laving 

 down field energy behind it as it advances, and never, even 

 at £ = — co", 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, 



