ON THE THEORY OF INTEGRAL EQUATIONS. 373 



It follows from this result that a solution of the equation 



b 



f(s) = f (*) - X] s («,t)rtt)«B 



a 



becomes infinite when A = X,„ unless 



\ 



f{t) K Jt)dt = 0. 



If this condition is satisfied the equation possesses a finite solution 

 when X = X„, but this solution is not unique since any constant multiple 

 of </>„(s) may be added to it. Plemelj l has obtained an expression for 

 the principal part of the solving function in the vicinity of a value of X, 

 which is a multiple root of the determinant. The most important case 

 is when the roots are all simple poles of ~K(s,t), for then K(s,t) has 

 the form 



K(s,t) = -- 1 -- s *88x«(0 + F(s,0 



A„ — A m 



and the functions f""\s), x' m '(t) satisfy the equation 



n n 



b 



a 



If <t>n(s), X;(0 are functions associated with two different roots X,„ X„ 



we have 



t, 



k^X^rfs = 0. 



a 



The roots of the determinant D(A) = are all real, and are simple 

 poles of the solving function, when «.-(s,<) is a symmetric function 2 of 

 s and t, and also 3 when a definite function g(s,t) can be found, so that 



u 



\g(s,x)i;{x,t)dx = h[s,t) 



is a symmetric function of s and t. The integral equations derived from 

 the two potential problems belong to the second class. It should be 

 noticed that a homogeneous integral equation of this class can be 

 replaced by a homogeneous integral equation of the first kind 



1 Monatshefte f&r Math, in Fhymlt, 1904. Other methods of obtaining the same 

 result are given by Mercer, Camb. Pliil. Trans., vol. xxi., pp. 129-142, and A. C. 

 Dixon, ibid., vol. xix., part 2, p. 203 ; Proc. Lund. Math. Soc, ser. 2, vol. vii., p. 314. 

 See also papers by Bryon Hey wood (Liouville's Journal, 1908) and Goursat (Toulouse 

 Ann., 1908). In obtaining his canonical form Plemelj introduces the idea of an 

 integral congruence. This theory has been further developed by Landsberg (Math. 

 Ann., 1910, Bd. lxix. p. 227). 



2 Hilbert and Plemelj. A simple proof of this result is given by T. Boggio, 

 Comptvs rendus (1907). 



a Mem. of Math. (1908), p. 8. See also J. Marty, Comptes rendus (Paris), Feb. 28 

 and April 25 (1910) ; Mrs. Pell, Bull. Amer. Math. Soc, vol. xvi., July (1910). 



