J* 



uN THE THEORY OP INTEGRAL EQUATIONS! 381 



and the discussion of these depends upon the properties that 



/i(s,ff)ds — 1 



for a regular point of the boundary, and that the kernel 



K (<r,r) = \h(s,>r)g(S,t)ds 



Is a symmetric function of a and r while g(s,r) is definite. 



The first result indicates that the homogeneous integral equation 



xP(t) — X U(s)h(s,t)ds . . . . (1) 



may be satisfied for \ = 1 by i/*(s) = 1, hence the equation for f>(t) or 

 the internal hydrodynamical problem does not possess a finite solution 

 unless 



(f(t)dt = 0. 

 It follows also that the homogeneous equation 



4>(s) = \h(8,t)<l>(t)dt . . . . (2) 



also possesses a solution, and so the equation for fi(t) or the external 

 electrostatic problem does not possess a solution unless a condition of 

 the type 



U(s)f(s)ds = 



J' 



is satisfied. 



It may be shown by use of a theorem relating to the behaviour of 

 the normal derivative of the potential of a double layer l that the 

 homogeneous integral equations do not possess solutions for the case 

 \ = — 1 if the boundary consists of a single closed surface. This 

 proves that the integral equations possess unique solutions for the case 

 A = — 1, and so the internal problem of Dirichlet and the external 

 hydrodynamical problems possess unique solutions. 



Plemelj and Lauricella have shown that the fundamental problem 

 of electrostatics of determining a potential of a simple layer which is 

 such that it is constant over the boundary and 



L(s)ds = M 



has a given value, is solved by means of the function f(s) ; for the 

 function 



ms,r)l>(s)da = ^(r) 



' Careful examinations of the question have been given by Plemelj and 

 Lauricella. A simplified exposition due to Darboux is given in d'Adhemar's book 

 Inequation du Fredholm et les problitnes de Dirichlet et de Neumann, 1909 (Par's). 

 1910. c c 



