120 L. Page — Energy of a Moving Electron. 



whereas the change in energy as given by (8) is 

 A W= [}Ji^ + H*) *■] - [ \Jl E * + H*)cIt^ = -JJf.yd T dt 

 which is not the same as T„ because for the Lorentz electron 

 J\ dtJ}dr=hfjF.\drdt (13) 



This inequality means, physically, that in calculating the 

 work done we cannot replace the forces acting on the elements 

 of the electron by a single force acting at the center or at any 

 other specified point in the electron. This is due, of course, 

 to the fact that the electron is deformable and consequently 

 when it is accelerated, different points on its surface have dif- 

 ferent velocities. In the case of a rigid electron (i. e., one 

 which maintains the same size and shape whatever its velocity 

 relative to the observer), such as Abraham's, the velocities of 

 all points would be the same, and (13) would be an equality. 



It is of interest to examine more closely the right-hand side 

 of (13). If the origin be taken at the center of the electron, 

 and the Z axis in the direction of relative velocity, 



v = »«[l -^(1-/3 S )] (U) 



where <f> is the infinitesimal, constant acceleration of the elec- 

 tron relative to its own system, and primes refer to the 

 electron's system at the instant considered.* Also 



Multiplying (15) by v and integrating over the surface of 

 the electron we get 



vfd'F=- ™° f *v ( 16 ) 



for the rate at which work is done by the electromagnetic 

 forces in resisting the change in motion. Changing the sign 

 of this expression and integrating with respect to the time, we 

 obtain the work done by the mechanical force producing the 

 acceleration as given by (12). 



Multiplying (15) by the second term in (11), we get upon 

 integrating over the surface of the electron 



*See " Kelativity and the Ether," this Journal, xxxviii, p. 1G9, 1914. 



