116 Trans. Acad. Set. of St. Louis. 



By Green's equation 



/vAVdv = /v^ds-j-F^dv. (j,^ 



This formula is to be applied to the volume bounded by a 



sphere of very large radius and which includes the electrical 



system. The first term of the second member is to be applied 



to the surface of the sphere, the center of which is at the 



center of gravity of the system. The potential V varies 



dV 

 inversely as r, and the normal force — varies inversely 



as r'^. Since the surface is directly as r^ the total surface 

 integral is inversely as r and becomes nearer zero as r becomes 

 greater. The second member of (14) therefore reduces to its 

 second term and we have for the energy 



w = ^JF^^v. 



(15) 



The energy per unit volume is therefore 



dW _ ]_ 2 



dv ~'Sn^ ' (16) 



To apply this method more specifically let us suppose the 

 electrical charges of the system to be transferred to the spheri- 

 cal surface of large radius which is an equipotential conduct- 

 ing surface of the system. The force outside of the surface 

 will remain unchanged, when the charge comes to equilibrium. 



The force with which an element of surface dS having a 

 charge ^dS is repelled outward by the rest of the electrified 

 surface is 



dp = 2 TT <T^dS (17) 



d p 1 2 



where j-g = ^ F . 



But d S = r2 do) 

 a = ^ 



.*. dp = 5 s da> 



' 8 TT r^ 



