5G8 PROCEEDINGS OF THE AMERICAN ACADEMY. 



But pg • B is the increase of kinetic energy per unit volume per unit Ct 

 for a volume element moving with the matter in it ; hence 



d(Ct) 



T 



Q. E. D. 



To find the amount of negative gravitational energy radiated from 

 an accelerated particle when it is at rest, we may make use of the re- 

 sult of the corresponding electrical case. In this case, if the charge on 

 the electron is e, and the accleration is a, and the unit vector in the 

 direction from the electron to the point considered is ii, the electric 

 force due to radiation at a distance r is 



E = - 4- ^p. 



where a^ is the component of a perpendicular to fi, and the magnetic 

 force due to acceleration is 



H = + -^ axri = TixE."' 



The magnitudes of these forces are seen to be equal, and the directions 

 at right angles. For the corresponding gravitational case, we need 

 only to change the signs of these expressions and substitute mk for e, 

 and C for c, and we have 



mk 



"pi 



, mk 



h = -' ^ axri = Tixg. 



We now see that 



. _ m^k% 2 _ m^k^ , .2 



' For proof see Lorentz, " Theory of Electrons," Chapter 1. 



