86, 87] NEWTONIAN POTENTIAL FUNCTION. 171 



Consequently the surface integral vanishes, and 



/// 



But since 5 is arbitrary, the integral can vanish only if every- 

 where in r, Av = 0, v is therefore the function which solves the 

 problem. The proof that there is such a function depends on the 

 assumption that there is a function which makes the integral <7 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*. 



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

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



u = v v. 



On the surface, since v = v', u = 0. In r, since A# and A?/ are 

 zero, Aw = 0. Accordingly J (u) = 0. But, as we have shown, 

 this can only be if u = const. But on 8, u = 0, hence, throughout 

 T, u = and v = v'. 



87. Green's Theorem in Curvilinear Orthogonal Co- 

 ordinates. We shall now consider Green's theorem in terms of 

 any orthogonal coordinates, limiting ourselves to the special case 

 U = const., or the divergence theorem, 35, 



where n s is the outward normal to 8. 



* Thomson, "Theorems with reference to the solution of certain Partial 

 Differential Equations," Cambridge and Dublin Math. Journ., Jan. 1878; Keprint 

 of "Papers in Electrostatics and Magnetism," xm. The name Dirichlet's Princip 

 was given by Eiemann ( WerJce, p. 90). For a historical and critical discussion of 

 this matter the student may consult Bacharach, Abriss der Geschichte der Potential- 

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

 Traite tf Analyse, Tom. u., p. 38. 



