ON THE THEORY OF INTEGRAL EQUATIONS. 363 



7. The Problem of Dirichlet and (he Method of Successive 



Approximations. 



The so-called problem of Dkichlet, 1 which is to establish the 

 existence of a potential function having given values round a contour, 

 was originally solved by Gauss and Lord Kelvin by means of the 

 calculus of variations, the function V which makes the integral 



-o L 



1©> GST* (£)>** • 



a minimum being the required potential function. This proof was 

 considered sufficient by Eiemann 2 but was rejected by Weierstrass. 3 



In 1870 C. Neumann 4 invented his famous method of the arith- 

 metical mean in which the solution of the problem was really reduced 

 to that of an integral equation of the second kind. 



The reduction of various potential problems to integral equations of 

 the second kind depends upon the fundamental properties of potentials 

 of simple and double layers. 



Let the symbol f{p) be used to denote a function of the co-ordinate 

 or parameters which determine the position of a point P, then if r is the 

 distance between two points A, B and we put 



g(a,b)= log f-j for problems in a plane 

 = -— for problems in space ; 



*M = v 9(s,p) = - 1 ^ in the plane 



fill. «■*■ tv *■ 



dn *™' tt r 



1 COS (I) ■ 



</> being the angle between the radius vector and the normal at a point S 

 on the boundary, then 



V(p) = \9(p,sHs)ds 



w Cp) = \h( s )Hs,p)(^ 



are the potentials of simple and double layers of strength p(s) u(s) 



respectively. ' v ; 



The potentials of a simple layer is continuous over the boundary, but 



' The problem was considered originally by Gauss (1839), Kelvin (1847) and 

 DinchJet (1856). The name Dirichlet's principle was introduced by Riemann Ges 

 M erhe, pp. 35-39, pp. 96-98. ' 



2 Diss. § 16. Fonclions abt'liennes, Avant-Propos, p. 111. 



3 Vober dais pgenaunte Diricfilet'sche Prinzip, Koniol. Altad. der Wist Berlin 

 July 14 (1870); Werke,ii., p. 49(1903). ' erllD> 



4 Lripziger Berichte, 1870 ; Untersuclmngen uber das logarithmitche und 

 Nenton'sche Potential, Leipzig (1877), p. 139. ./«'«««< w«e una 



