85, 86] NEWTONIAN POTENTIAL FUNCTION. 169 



In the limit, then 



(I8) 7 ,--J-ff!(|r + !V-fff -* 



47T JJ r \gfii dnj JJJ^ r 



The volume integral, as before, denotes the potential j M - dr 

 due to the volume distribution, while the surface integral denotes 

 the potential of a surface distribution 1 1 - - , 

 where 



dv 



- 



Hence we get a new proof of Poisson's surface condition, 

 82. 



86. Kelvin and Dirichlet's Principle. We shall now 

 consider a question known on the continent of Europe as 

 Dirichlet's Problem. 



Given the values of a function at all points of a closed surface 

 S is it possible to find a function which, assuming these values 

 on the surface, is, with its parameters, uniform, finite, continuous, 

 and is itself harmonic at all points within 8 ? 



This is the internal problem the external may be stated in 

 like manner, specifying the conditions as to vanishing at infinity. 



Consider the integral 



of a function u throughout the space r within 8. 



J must be positive, for every element is a sum of squares. 



It cannot vanish, unless everywhere ^- = r- = ^- = 0, that is 



dx dy dz 



u = constant. But since u is continuous, unless it is constant on 

 S, this will not be the case. 



Accordingly J(u) > 0. 



Now of the infinite variety of functions u there must be, 

 according to Dirichlet, at least one which makes J less than for 

 any of the others. Call this function v, and call the difference 

 between this and any other hs, so that 



u v + hs, 

 h being constant. 



