ON THE THEORY OF INTEGRAL EQUATIONS. 410 



extended by Plancherel. 1 Hilb also considers the more general equation 

 T / % df du\ , \1t(s) — q(s) 



where g(s), h(s) are analytic functions of s which have the character of 

 integral functions is an interval (o, 1) which includes both limits. 

 Finally g(s), h(s) are real for real values of s, and g(s)>l h(s)>a> o 

 o s<l, h(o) = 1. 



The integral representation is discussed with the aid of a Green's 

 function belonging to the differential equation. 



The general theory of differential equations when the independent 

 variable has an infinite range of values is discussed by Wcyl. He 

 shows that all the integrals of the equation 



d f" , N du~ 

 dsl 13(s) ds_ 



— q(s)u= o 

 tend to finite limits as s->oo if the integral 



is convergent. This condition is necessary and sufficient except when 

 q(s)=o. 



Singular integral equations connected with differential equations of 

 higher orders have been considered by the author. 2 There is an inver- 

 sion formula of the type 



CO 



/(,r) = [j v (xt)YAxt)tf(t)dt 



J (" > - I) 



CO 



Ht) = -2^ t ^J%xt)lxf(x)dx 







for instance, associated with the differential equation of the third order 

 satisfied by xJ'l(xt) — viz., 



dx* \ x l Jdx v V 



Singular integral equations connected with the formula 3 



CO 



J x — t 



have been discussed by Hardy following a remark made by the author. 

 The formula 



1 Math. Ann., 1909, Bd. 67, Heft 4. 



- l'roc. Lond. Math. Soo., 1906-07, scr. 2, vol. iv , p. 483. 



3 This formula has been investigated by Hardy {Tree. Lend. Math. Foe., ser. 2, 

 vol. vii., p. 445). Some remarkable definite integrals are obtained in the course of 

 the work. 



