160 General Analytical Theorems [CH. vn 



The first and last terms together give 4?r x %eV, where e is any element 

 of charge, either of volume-electrification or surface-electrification. Thus the 

 whole equation becomes 



shewing that the energy may be regarded as distributed through the space 



R 2 



outside the conductors, to the amount - - per unit volume the result 



07T 



already obtained in 168. 



184. In Green's Theorem, take 



w = 



Here K is ultimately to be taken to be the inductive capacity, which 

 may vary discontinuously on crossing the boundary between two dielectrics. 

 We accordingly suppose u, v, w to be discontinuous, and use Green's Theorem 

 in the form given in 180. We have then 



, , , , 



K \^ ^ + ^r- -~- + -^- -&-} dccdydz 

 1 dx dx dy dy dz l 



[ f^ 8 



r^1^- 



JJJ [da 



das \ dxj dy \ dy J dz 

 W dW d 



- + m=- + nT 



x dy 



where ^ , r have the meanings assigned to them in 140. 



If we put <I> = 1, "SP = F, in this equation, it reduces, as in 130, to 



1 1 K ^ dS = 4-7T x total charge inside surface, 



JJ on 



so that the result is that of the extension of Gauss' Theorem. Again, if we 

 put <I> = ^ = F, the equation becomes 



f f CKR 2 



1 1 1 dxdydz =^ \ 2<eV, 



and the result is that of 169. 



