ON THE THEORY OF INTEGRAL EQUATIONS. 371 



Plemelj, Lalesco, and Goursat also investigate the order of the 

 integral functions D(A), K(\ ; s, t) and the latter obtains an expression 

 for the determinant of an iterated kernel in terms of the determinant of 

 the original one. If the function /.(s, t) is bounded the order of D(\) 

 is at most equal to two, but this is not the case if k(s, t) is not bounded. 

 The necessary and sufficient condition that the determinant D(\) should 

 have no roots is that 



b b 



[ h a (s,t)dsdt = . n > 1. 



a a 



If \ is not a root of D(\) = the two adjoint integral equations 



b 



g(t) = x(0 - x\x(M*>t)ds 



are solved uniquely by the formulae 



b 



4>(s)=f{s) + \[K(s,t)f(t)dt 



a 

 b 



x(0 = </W + *Us)K(s,t)ds, 



a 



and we have the reciprocal formula 



J /(s) x {s)ds = \<l>(s)g(s)ds, 



a it 



b 



also -^ s) = [K(s,t)f{t)dt. 



or 



It should be noticed that the solving functions for two adjoint integral 

 equations are the same and their determinants are also the same. The 

 determinants are also the same for the functions x 







4M) = | f{s,x)g{x,t)dx . 



a 



and for the functions ^(s,t) ; ^{s,t) if ' ; 



1 See a paper by the author, Cavil). Phil. Trans., 1D0G, vol. xx 



