WEBSTER. — AN ELECTROMAGNETIC THEORY OF GRAVITATION. 567 



hence that the energy in a region T of any distribution of matter acted 

 upon only by gravitational forces is 



/// 



{pc^ {Br^ - 1) _ -L (g. + h'^)} dr. 



While this assumption is not necessarily true for any finite region, it is 

 certainly true in the limit for a sphere whose radius is allowed to be- 

 come infinite. 



Gravitational Radiation. — We may prove the following theorem : 

 If a distribution of matter is affected by forces of which none but 

 those of gravitational origin do any work, the energy, Eg, within any 

 closed surface, S, which neither matter nor electromagnetic energy 

 enters or leaves, and which has a normal at every point, will increase 

 or decrease at such a rate that 



dE. . 1 rr . ^s 



d(Ct) 



= +i^/>'" 



where dS is the element of surface considered as a vector in the direc- 

 tion of the exterior normal. 



The truth of this theorem depends on that of the above assumption 

 about the energy in the region T, but it is certainly true for the in- 

 finite sphere. 



The proof (like that of Poynting's theorem), is as follows : If 7^ is 

 the space within S, 



^' ^ fff^'''^^" ~ ^^ "■ 8^- ^^' + ^'^^ ^' 



dK 



1 / 3g , , ah \ ) - 



-- (g • V xh — h • V xg) - ^.g • B ^ dr. 

 4:7r ) 



