64 Mr. S. H. Barbury on Action 



111. ^Ve shall now pass to the following problem. Given a 

 distribution of matter over a closed surface S, known to have 

 constant potential Vq at every point of that surface, to find all 

 the equipotential surfaces due to that same distribution. 



It may be shown, as in Thomson and Tait's ' Natural Phi- 

 losophy,' Appendix A, that the potential of the given distri- 

 bution, having the constant value Yq at every point on S, has 

 that same constant value also at every point within S. For, 

 let V be the potential of the distribution, U any other function 

 whatever of {a;, i/, z) which is equal to zero at every point on S. 

 Then, if the triple integral refer to the whole space within S, 



"°Sm (£'-S)'-(f-f)'-(S-f )'}-*- 



By Green's theorem the last line of the second member of this 

 equation is equal to 



-iTJ>'{l(K2)-|(-'f)-|(K ©}-*-. 



where in the first term the double integration is extended over 



dY 

 the whole surface S, and - ^ is the increase of V per unit 



length of the normal inside 8. But the first term vanishes, 

 because U is by hypothesis zero at every point on S ; and the 

 second term vanishes because of (C). Rejecting then the last 

 line, equation (1) becomes 



I(J-n'{£-S)"-(f-f)HS-S)*}**- 



-j;ij'^{(S)'-(f)"-('ST}-*- 

 -J)^{(f/-(f /^(fj}-'-'- ■ ■ ■ <^) 



This equation holds true whatever U be, ])rovided it be zero at 



