372 REPORTS ON THE STATE OF SCIENCE. 



while if two functions ^(s,t), h(s,t) are connected by a relation of the 

 type 



U(s,x)f(x,t)dx = \f{s,x)h{x,t)clx 



and no continuous functions exist such that 



b b 



L(x)f(x,t)dx = {f(s,x)iX?)dx = 



then the determinants for the two kernels k(s,t), h(s,t) have the same 

 roots. 



If X„ is a root of the determinantal equation D(\) = 0, then the homo- 

 geneous equations 



<pn{s) = \,Us, t) <j>„(t)dt 



a 



b 



Xn(0 = K\Xn(s) K(s,t)ds 



can be satisfied by continuous functions <p„(s), x»(0 which are not 

 identically zero. 



Conversely, if one of these equations can be satisfied by a continuous 

 function, then X„ is a root x of the determinant D(X) and it follows that 

 the other equation also possesses a solution. 



The roots of the determinant D(A) = are called the roots (Eigenwerte) 

 of the kernel k(s,<) and the aggregate of these roots the spectrum of the 

 kernel for the interval (a, b). 



The functions or sets of functions <j>„{s), x»W which satisfy the homo- 

 geneous equation are called the principal functions associated with the 



root X„. If 



b 



\<P»(s)Xn(s)dS £ 



a 



we can say that (f> H (s) is adjoint to x«( 5 )- 



When the kernel is a symmetric function of s and t the functions </>„(s), 

 X„(s) become the same and form an orthogonal system of functions when 

 the different values of X„ are considered. They may then be called 

 normal functions of the integral equation, the German equivalent being 

 Eigenfunhtion. 



If X„ is a simple root of D(A) = 0, the only solutions of the above 

 equations are constant multiples of (p,,(s), x«W> and the solving function 

 K(s,t) has the form 



K(s,t) = *"(*)Xn(0 + F(M) 



K — x 



where F(s, t) is finite for X = \ n . 



1 These theorems are due to Fredholin. A »ew proof of the last result has 

 been given recently by Sorrel, Bull. Amer. Math. Soc,vo\. xv. (1909), p. 449. Further 

 properties of the homogeneous equation are given in a paper by Schur. Math. Ann. 

 Bd. 67 (1909), p. 306. 



