256 ELECTROSTATICS. [FT. II. CH. VI. 



which being the energy of the distribution &p, Bar may be written 



1 f r f ( /O T7\ 2 /d T7"\ 9. /O T7"\ 9\ 



which is necessarily positive. Consequently the condition for 

 stable equilibrium is that in each conductor the total potential 

 F + V is constant*. 



The integral which takes a minimum in the above investiga- 

 tion is the same one that appears in the demonstration of Kelvin 

 and Dirichlet's principle, 86. We saw that in general there was 

 a doubt as to the existence of a function making the integral a 

 minimum. In the electrical case, experiment shows that there is 

 always an equilibrium distribution, so that the only doubt which 

 may affect the mathematician does not trouble the physicist. 

 Reasoning depending upon such physical facts was frequently 

 made use of by Green, and while not legitimate for purposes of 

 mathematical demonstration is frequently of service to the 

 physicist. 



Since in any conductor V + V = c, 



or there is no force in the substance of a conductor ; further 



A(F+F') = 0. 



But since the distribution causing V lies outside of the con- 

 ductor, A V = in the conductor, and 



(13) AF=0=-4irp. 



Consequently, in every conductor p 0, or the distribution is 

 superficial. Now at the surface distribution <r we have a discon- 



tinuity in the derivative ~ , and 



dV 8F 



But since within the conductor F+ V = c 8) 

 9(F+F')_ 8F 3F'_3i 





* The above demonstration is given by Betti, Teorica delle Forze Newtoniane, 

 p. 164. 



