194 Mutual Magnetic Energy of two Moving Point Charges. 



r^Xot^ + fafa 4- 7l 7 2 ) 



= x 2 (y i v 2 -f ^1^2) + y 2 ( u i u 2 + WM^o) + z 2 (v^v 2 4- u x u 2 ) 



— zy(v 1 ic 2 + m>iV 2 ) — xz(ti 2 w 1 -f- Wito 2 ) — ^j/( w i^2 + w 2^i) 



+ 2 z ( v i w 2 — ^2) + 2 A '( v i w 2— ^1^2) 

 c 2 



— J (^1^2 + ^1^2). 



The total mutual magnetic energy may be found by 



evaluating the integrals I 3 -f— , &c. . . ., but it is not 



J ^ r! r 2 



necessary to do this. Rejecting those which evidently 



vanish, and observing that 



C x 2 dx dy dz _ C z 2 dx dy dz 



it is clear that the total mutual magnetic energy is 



A(2v ± v 2 + u Y u 2 + WilV 2 ) + ~B(uiii 2 + W X W 2 ), 



where A and B are constants to be determined. 



Let ui, u 2 , Wi, iv 2 all vanish, the energy is 2Ai\v 2 . 

 Hence, by the former result for this case, 



A -~2c~' 

 Again, let u u u 2 , v h v 2 all vanish, the energy is 

 (A + B)w l w i , 

 and, therefore, by the former result, 



thus B = 0, and the energy is given by 



l ~^ (2v 1 v 2 + u 1 u 2 + ic 1 w 2 ) i 



or, if Vi and V 2 are velocities of e 1 and e 2 , e the angle 

 between their directions, and 6 1} 6 2 the angles between 

 these directions and AB, by 



/^i^V, (cog g + cog ^ cog Q ^ 



