180 General Analytical Theorems [CH. vn 



In this equation, put 7=-, then, since V 2 F= 4-777?, we nave as the 

 value of the first term, 



And since V 2 7 = 0, the second term vanishes. The equation accordingly 

 becomes 



3n \r y 

 205. Suppose, first, that the surface S is an equipotential. Then 



= 0, 

 so that equation (122) becomes 



Thus the potential of any system of charges is the same at every point 

 outside any selected equipotential which surrounds all the charges, as that 

 of a charge of electricity spread over this equipotential, and having surface 



density -r . Obviously, in fact, if the equipotential is replaced by a 

 conductor, this will be the density on its outer surface. 



206. If the surface S is not an equipotential, the term li^V- (-) dS 



will not vanish. Since, however, u, z- ( - ) is the potential of a doublet of 



r dn \rj 



strength /u, and direction that of the outward normal, it follows that 

 f-j dS is the potential of a system of doublets arranged over the 



surface S, the direction at every point being that of the outward normal, and 

 the total strength of doublets per unit area at any point being F 



Thus the potential V P may be regarded as due to the presence on the 

 surface S of 



(i) a surface density of electricity -j ~ ; 



(ii) a distribution of electric doublets, of strength per unit area, 

 and direction that of the outward normal. 



