Magnetic Energy of two Moving Point Charges. 191 

 The mutual magnetic energy at P per unit volume is 

 /jLe l e. 2 }riU'2 sin 6 sin 0' cos LPM 



Let angle APB = «, PAB=A, and PBA = B ; also, let 

 AB = c. 



By means of a spherical triangle it is easily seen that 



cos a = cos cos 0' + sin sin 0' cos LPM. 



Also, cos — sin A cos cf> and cos #' = sin B cos $, 



where (j> is the angle between the planes APB and LABM. 

 The mutual energy per unit volume is consequently 



fie 1 e 2 WiW 2 (cos a — cos 2 cf> sin A sin ~B)j4:7rr 2 r' 2 . 



Take an element of area dS in the plane APB at P ; then 

 the element of volume at P may be taken 



pd$d$, where p is the perpendicular from P on AB. 



If the triangle APB be rotated about AB through a com- 

 plete revolution, the mutual energy in the ring traced out 

 by dJS is 



4z7T 



1 2 » 1 '2 2 P d^> ( - 71 cos a ~~ s * n ^ sm B 1 cos 2< / ) d<f> ) 

 irvr \ Jo J 



= — * 2 , 2 dS (2p cos a — p sm A sin B). 



Since rr' sin u=pc 

 equal to 



fie^wiw^ 



an 



d sin A sin B=p 2 /rr', this is 



4r 



d& 



[~2 cos a sin 2 ot sin 3 a~] 

 L V c J 



