376 REPORTS ON THE STATE OP SCIENCE. 



for the different values of X„, *(s,t) being a symmetric function, then 



t 



,£(«) = L(s,t)a(fjctt = 0. 



a 



This theorem may be proved by reasoning with the double integral 





K 2 (s,t)a(s)u(t)dsdt 



[Hs)]*d$ 



iii the same way as was done with the double integral with k(s,t) as 

 kernel instead of K 3 (s,t). It is clear from the supposition made with 

 regard to u(s), that if 



f(s) = o(s) - \ L 2 (s,t)u(t)dt 



a 



where 



b b 



= {[n(s)Yds - \ [[Hs)r-ds 



a a 



then the energy function of the integral equation 



b 



f(s) = </,(«) - \L 2 (s,t)f(t)dt 



a 



never becomes infinite. But \w(\) vanishes when X = and X — A„ and 

 increases continually with X. Hence we come to a contradiction unless 

 <£ (s)= 0. The theorem is proved in another way by Schmidt. 



It follows from this theorem that if no continuous function a(t) 

 exists for which 



b 



L(s,t)a(t)dt = 0, 



a 



then there must be an infinite number of functions <f>„(s), and consequently 

 an infinite number of roots X„ of the determinant D(X). This is cer- 

 tainly the case, for instance, if t;(s,t) is a definite function. These 

 theorems are due to Hilbert. 



An analogous set of theorems hold for the integral equation of the 

 first kind l 



M = \{g(s, t) - xh{s, t)]f{t)dt, . . . (i) 



where g(s,t) is definite and h(s,t) symmetrical. 



The energy function is defined as before by the equation 



w ( x ) = \M<Ks)ds 



Most of the properties belong to the integral equation of the second kind 

 from which the homogeneous equation of type (1) is derived. 



Mess Math. 



