258 ELECTROSTATICS. [PT. II. CH. VI. 



necessitating 



F=c s , AF = 0, p=Q, 



w n dv 



~ = 0, = 47T(T, 



^v W/fs_/ 



K s dn e 

 the same results as above, writing F for F + V above. 



We may easily show that there is but one equilibrium distribu- 

 tion. For if there be another, F, o-, for which the constant 

 values of F are c s , let us apply Green's theorem to the difference 



u=V-V, 



the volume integral on the right being extended to all space 

 outside of the conductors. But in that space A F = A F = or 

 47r/o', and accordingly AM = 0. Also at the surface of any con- 

 ductor K 8 , 



u = c s c s . 



The integral J(u) therefore becomes 



2* 



Now the surface integral is equal to l/4?r times the dif- 

 ference of the charges of K s in the two distributions. But the 

 charge being originally given this is 0. Accordingly the integral 

 J (u) vanishes, and everywhere 



du da du A TT- rr 



~~ = ~- = ^ = 0, u=V V const. 



ex oy dz 



Since F and F vanish at infinity, the constant is 0, and the 

 distribution <r is the same in either case. Consequently we see 

 that the constant values of the potential on the surface and 

 throughout the substance of the conductors, or as we shall say, the 

 potentials o/the conductors, are determined by the electrifications 

 of D and the total charges of the conductors. 



