378 VIII. NEWTONIAN POTENTIAL FUNCTION. 



sign of Ji opposite to that of the integral, making the product 

 negative, the whole will be negative. 



The only way to leave the sum always positive is to have the 

 integral vanish. (It will he observed that the above is exactly the 

 process of the calculus of variations. We might put 6 v instead of hs.} 



The condition for a minimum is then 



But by Green's theorem, this is equal to 



Now at the surface the function is given, hence n and v must 

 have the same values, and s = 0. 



Consequently the surface integral vanishes, and 



= 0. 



But since s is arbitrary, the integral can vanish only if everywhere 

 in T, z/v = 0, v is therefore the function which solves the problem. 



The proof of the so-called Existence -theorem, namely, that there 

 is such a function, depends on the assumption that there is a function 

 which makes the integral J a minimum. This assumption has been 

 declared by Weierstrass, Kronecker, and others, to be faulty. The 

 principle of Lord Kelvin and Dirichlet, which declares that there is 

 a function v, has been rigidly proved for a number of special cases, 

 but the above general proof is no longer admitted. It is given here 

 on account of its historical interest. 1 ) 



We can however prove that if there is a function v, satisfying 

 the conditions, it is unique. For, if there is another, v\ put 



1) Thomson, Theorems with reference to the solution of certain Partial 

 Differential Equations, Cambridge and Dublin Math. Journ., Jan. 1848; Reprint 

 of Papers in Electrostatics and Magnetism, XIII. The name Dirichlet' s Prinzip 

 was given by Riemann (Werke, p. 90). For a historical and critical discussion 

 of this matter the student may consult Burkhardt, Potentialtheorie in the 

 Encyklopadie der Mathematik, Bacharach, Abriss der Greschichte der Potential- 

 theorie, as well as Harkness and Morley, Theory of Functions, Chap. IX, Picard, 

 Traite d' Analyse, Tom. II, p. 38. It has been quite recently shown by Hilbert 

 that Riemann's proof given above can be so modified as to be made rigid. 



