Energy in the Electromagnetic Field. 151 



For spherical surface charges of radii a x , a 2 , &c, we get 



J' 



2 



£ dv =liw + ^ + ^> 0) 



the integration being over all space external to the charges. 

 Again, 



■+■ ^ IJJ ^T3^ ^ ^+ 'WJJ j rfrl dx dy dz ] ' 



all the other integrals vanishing by symmetry, 



(» 

 ff* *> = |^ lOW* + ^'2)Il + Wl«*If}, • (10) 



where I l5 1 2 denote the same integrals as before. 

 From (4), (10), and (5) we find 



J^T I («i«2 + /3i$> + 7i72 + <7i#}) dr 



= ^ 2 ( Ml Mj + r lV2 + uvco) (21, + 1 2 ) 



_ ^ 1 ^ 2 (m 1 ^2 + ^2 4- ^1^2) 



r 



Adding (2) and (9) for e x and £ 2 , and summing for all 

 charges, we obtain finally 





+ SS5£f(«X+«V«'. + w r«^ • • ■ (11) 



r 



TH 2 + G 2 ) 



The mutual value of 1^ <iu for two circuits is 



2,2, — (u^g "f r^ra + l^Wg) = fc^g 1 a5i«A* 2 , 



