L. Page — Energy of a Moving Electron. 119 



and the gain in energy due to the motion, obtained by sub- 

 tracting from (7) the electrostatic energy of an electron at rest 

 relative to the observer, is given by 



iTf =" , -' ! "[7T i fj.- 1 ][ I + ^] w 



The rate of radiation of energy from an electron is given 

 exactly by 



dt 6ttc 3 K ' 



where <£ is the acceleration of the center of strain* of the elec- 

 tron relative to its own system. 



Second Method. 



Let an electron which is initially at rest relative to the ob- 

 server be given a finite velocity v by imparting to it an infini- 

 tesimal acceleration for an infinite time, and then let the 

 electron be allowed to maintain this velocity forever after. 

 Since the acceleration is infinitesimal, the radiation term will 

 be negligible compared to the other terms in (1). Hence we 

 may write 



j t [~J(JS*+ B') dr^+jF-V dr = (10) 



To find the gain in energy we must calculate the work done 

 by the constant mechanical force which produces the acceler- 

 ation. This force is obviously given by 



*--/>.*= jj=^« (id 



where the acceleration, since it is infinitesimal, can be consid- 

 ered to be the same for all points on the electron. 



Integrating (11) we find for the work done by the mechan- 

 ical force acting on the electron 



^"••'"[vfff- 1 ] ' (I2) 



which is not the same as (8). In fact, if i\ is the velocity of 

 the geometric center, the work done by the mechanical force 

 as calculated above is 



T 2 = -Jy a dt -J}d- 



* By " center of strain" is meant that point inside the electron at which 

 its charge can be considered as concentrated without altering the external 

 aeld. 



