410 Prof. S. R. Milner : Does an Accelerated 



partly because we cannot say a priori 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 a 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 



. \i<i-w < 7 ) 



Then the total energy of each electron in the system 

 becomes 



H£a-^ (8) 



(5) The electromagnetic momentum parallel to x is 

 given by 



f EHsinfl dY 



J ±7TC 



where 6 is the angle made by the line of force, x — const., 

 with the positive axis of x. On substituting 



rintf= Sl f * Si °* (?) 



cosni/r— cos% 



and integrating with respect to <f> and % as before, but with 

 respect to -v/r up to the surface 



P _ al.l-/3 2 sin 2 y/ ' ' ' ' {W) 



