ON THE THEORY OP INTEGRAL EQUATIONS. 375 



Put 



b 



f(s) = «(s) - X \K(s,t)a(t)dt 

 and consider the equation 



b 



f(s) = f(s) — \L(s,i)f(t)dt. 



a 



If X„ is not a root of the determinant of this equation it follows that «(s) 

 is the value of f(s) for X = X,,, and so from the definition of X the energy- 

 function \w(\) associated with this equation vanishes for X = X„. But 

 it also vanishes for X =0 and increases continually with X, hence- there is 

 at least one value between and X for which tv(\) and consequently <j>(s) 

 becomes infinite. If X_ and X + are the numerically smallest negative 

 and positive roots of D(A) we must have the inequalities 



1 < i < L 



X_ X x + 



This gives limits for the double integral and reduces to a theorem of 

 Hilbert's in the case when u(s,t) is a definite function and the roots of 

 D(a) consequently all positive. The theorem then takes the form of a 

 problem in the calculus of variations of determining the maximum value 

 of a double integral, the maximum value being provided by a function 

 which satisfies the homogeneous integral equation. 



It is clear that another pair of roots may be used in the inequality 

 instead of X_ and X + , if it is known that «(s) and consequently /(s) 

 satisfies the conditions of type 



ii 



\u(s)<i> n (s)ds = 0, 



a 



which make *(s) finite for X = X _ and A = X + . 



The above derivation of inequalities for the double integral establishes 

 the existence of at least one root of the determinant D(X). The exist- 

 ence theorem is proved by Schmidt and Kneser by showing that the 



series for } ' has a finite radius of convergence. Schmidt also gives 



^ ( A ) , 



a method of obtaining a solution of the homogeneous integral equation 

 by a limiting process, his proof being modelled on a classical one due to 

 Schwarz. 



2 n Hilbert and Schmidt have given a theorem that if 



L(s)f n {s)ds = 



a 



for all the functions </>„(s) which satisfy the homogeneous integral 

 equation 



f K (s) -\ n Us,t)l l (t)dt = 



