378 REPORTS ON THE STATE OF SCIENCE. 



of those belonging to h(s,t). The case in which h(s,t) = has been 

 worked out by Goursat ' and has been extended to infinite values of n 

 by Lebesgue. 2 



In one interesting case the result can be expressed in a very concise 

 form. If in Fredholm's notation 



( s s i s% . . . s n \ 

 ttih . . . tj 

 W = / ,, ,, . . . s n \ 



\t X tj . . . tj ' 



then the solving function G(s,t) and the determinant A(A) for the kernel 

 g(s, t) are given by the formulae 



TT ( S S ' S2 " " ' S "\ 



g m = ,t:t: ' • \\ 



,( g l S 2 ' • ■ S n\ 

 \t[ t<i . . . £„/ 



The function D(X)K (*'* 2 * ' * *A = T>(\ ; S /, 2 " ' ' ']") is called an 



\M f 2 • • • '«/ \ IJ-2 • ■ ■ \nj 



ii"' minor of the determinant D(A) and is of great importance in the 

 study of multiple roots of the determinant. 

 If we apply the formula 



i 



D'(\) 



K( Sl s)ds= - D( ^ 



to the function G (s,t) we obtain the important formula 



H^;t//.^=-JH ; M,:::t> 



a 



The function DIX;. l is, of course, identical with D (\ ; s,t) 

 occurring in Fredholm's formula 



K ^=-%)' - 



When X = X„ is a multiple root of the determinant D (\), it is a minor of 

 type 



DIX • s,Si > s z ■ ' • S ">\ 



which is a solution of the homogeneous integral equation, but the order 

 m of the minor is not necessarily equal to the multiplicity of the root. 

 The properties of these minors are fully worked out by Fredholm and 

 Plemelj. The latter shows that if the solving function has a pole of order 



1 Bull, de la Soc. Math, de France, 1907. 2 Ibid., 1908. 



