of Stationary States of Motion. 255 



compared with their radii. If the radius of the path, or the 

 distances between the electrons of a stream, were of the order 

 of the electronic radius, the radiation would also be small ; 

 it would of course vanish if the stream of electrons coalesced 

 into an anchor ring revolving about its axis, such as the 

 Parson magneton, but not if the ring moved as a whole with 

 acceleration. 



Thus we see that for any discontinuous distribution of 

 electrons, of the kind with which we are concerned in an 

 atomic model such as is contemplated in Bohr's theory, 

 radiation unavoidably results if we adopt the equations of 

 Maxwell and Hertz for the field at a distance from an elec- 

 tron, together with Poynting's formula for the energy flux, 

 at any rate for electrons moving in coaxal circles. When 

 the paths are not circular, so that there is generally a tan- 

 gential as well as a normal acceleration, we have every 

 reason to suppose that the radiation is increased on that 

 account, and there is little doubt that a formal proof could 

 be given, although it would be much more complicated. 

 There would be no alteration needed as far as § 11 and 

 equation (12), but the approximation used thereafter would 

 no longer apply, because the values of z, vr, z\ m' would no 

 longer be restricted to a small area of the meridian plane for 

 the single electron or stream, or several such areas for a 

 system, but would be spread over a finite area bounded by 

 the extreme values of these coordinates reached during the 

 orbital motion. 



16. Before proceeding to a consideration of the changes 

 needed in the fundamental assumptions in order to remove 

 the contradiction that we have found, we shall consider the 

 case of a uniform spherical electron, which moves in a circle 

 of any finite radius with uniform speed. We shall adopt 

 the conventions of the accepted electron theory and write 

 € = ev, K = 0. Hence we have C x = 6v = '6eco^>l4:7ra d for all 

 points which lie inside the electron at the time t, and C x = 

 for all outside points, whilst all the other components 

 G 2 , C w , K z , K*r, K x vanish. 



In order to express this condition more precisely let us 

 suppose the longitude of the centre of the electron at time t 

 to be cot; then G x is a function of the variables % — w£, z, 

 and ot, and vanishes unless %— cot lies between the limits +«, 

 where a is the least positive angle given by 



a 2 = z 2 + '5J- 2 + p 2 — 2p st cos a, 



I . (21) 

 sinia=v{a 2 - 2 2 -(^-p) 2 }/2v(^). 



