358 REPORTS ON THE STATE OP SCTENCE. 



When H{x,y) vanishes for x = y the results take a different form, and 

 we have the following theorem : — 



y 

 If f 



AV) = <p(x)B.(x,y)dx a> y> o 



where f{y) = y'+Vifj/) 



H(x- y) = 2; atf-if' + 2i x i y n ' x - i L,(a?,y) 



and the quantities a, are constants. 



II fi(y), ~Li(x,y) and their derivatives with regard to y are finite 

 and continuous for (o<x<y) and (o <y<a), and if H(y,y) does not 

 vanish except when ?/ = o, there is one and only one finite and continuous 

 function which satisfies (25) in the case when the roots of the algebraic 

 equation of degree n. 



a,, . «i , a„ 



are finite and different from one another and have their real parts 

 positive. If one or more of the roots have their real part negative and 

 the roots are otherwise finite and distinct, the problem of finding a 

 function <p(.v) which satisfies the integral equation is indeterminate. A 

 useful criterion for determining whetber an equation of the above type 

 has roots with only positive real parts has been given by Hurwitz. 1 



Further investigations connected with the case in which H(x,x) 

 vanishes have been made by Holmgren 2 and Lalesco. 3 The latter con- 

 siders the particular case in which Ii(x,y) is a polynomial of the 

 n"' degree in x, and shows that the integral equation may be reduced to 

 a linear differential equation of the n"' order by differentiation. The 

 algebraic equation for A is then identical with the indicial equation of 

 the adjoint differential equation. The results are then applied to the 

 study of the general case. Further investigations on the connection 

 between linear differential equations and integral equations of Volterra's 

 type have been made by Diui 4 and the author.'' It may be shown with- 

 out difficulty that a linear differential equation may be replaced hy an 

 integral equation of Volterra's type, and then, by applying a method of 

 successive approximations, a series is obtained which represents a solution 

 of the linear differential equation in the whole of the complex plane 

 (excluding singularities). This method of proving the existence theorem 

 is virtually due to Cauchy. G 



When the conditions of Volterra's theorems are not satisfied the 

 solution of the integral equation may not be unique and discontinuous 

 solutions can enter. Thus Bocher 7 gives the example — 



' Math. Ann., Bd. 46, p. 273. 



2 Atti of the Turin Academy, 1900, vol. xxxv., p 570 ; Upsala Memoirs, 1000, vol. iii. 



9 Liouville's Journal, 1908, ser. 6, vol. iv., p. 125. 



4 Annali di Matemntica, 1899, ser. 3, t. ii., pp. 297-321; t. iii., pp. 125-183 

 t. xi., p. 385. 



5 Proe. London Math. Soc, 190C, p. 107 ; Darboux's Bull,, 1906. 



6 Uiuvreg completes, V serie, t. v., p. 394. 



7 Ah Introduction to the Study of Integral /equations, p. 17. 



