131, 132] PRINCIPLE OF KELVIN -DIRICHLET. 377 



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 -5- = -^ = -= - = 0. that is u = const. 



dx dy dz 



But since u is continuous, unless it is constant on S, this will not 

 be the case. 



Accordingly J(M) > Q 



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. 



The condition for a minimum is that 



J(v) <J(v + hs), 

 for all values of h. 

 Now 



-+ 



'+'+ 69' 



9L dv ds dv ds 

 ^^ 



Integrating, 



23) J (.) .= .7 W + V(.) + 2^///(f: + gg + J g) ^ 

 Now in order that J^) may be a minimum, we must have 



->A\ 72 TV N 07 C C Ci^ V $ S , ^V ds 'd B 8\ -j n 



24) WJ(s} + M- + -r + 3 s dr ' ' 



for aZZ values of ft, positive or negative. But as s is as yet un- 

 limited, we may take h so small that the absolute value of the term 

 in li is greater than that of the term in h 2 , and if we choose the 



