ON THE THEORY OF INTEGRAL EQUATIONS. 403 



the differential equation when the branch points and group are known. 

 The problem was solved by Schlesinger ' with the aid of Poincare_'s 

 Zetafuchsian functions in the case when all the roots of the characteristic 

 equations, which belong to the fundamental substitutions corresponding 

 to the circuits round the singular points, have their moduli equal to 

 unity. The convergence of the series of Zeta functions is then ensured. 

 Schlesinger - has more recently used a method of continuity to obtain a 

 proof of the solubility of Eiemann's problem. 



A new method of dealing with the problem was invented by Hilbert 3 

 in 1901. He considers first of all Eiemann's general problem 4 of 

 determining, in the interior of a region of the complex plane bounded 

 by a given curve, C, functions of a complex variable when relations are 

 given between the real and imaginary parts of the values which the 

 functions assume on the boundary. Hilbert shows that if 



f(s) = u{x,y) + iv{x,y) 



and u(s), v(s) are the values which the functions u,v take on tho 

 boundary, then the problem of determining f(z) when a relation of the 

 type 



a(s)u(s) + b(s)v(s) + c(s) = 



is given, a(s), b(s), c(s) being periodic functions of the arc s with con- 

 tinuous derivatives, can be reduced with the aid of a Green's function 

 to the solution of an integral equation of the second kind. 5 He then 

 goes on to show that the problem of determining a pair of functions 

 f {z),g (z) which are regular outside C and a pair of functions f { (z), g^(z) 

 which are regular inside C, so that there is a given linear transformation 



fo(s) = c A (s)f l (s) + c,(s)g l (s) 

 9o(s) — *iWffl + e 2 (s)g 2 {s) 



with complex coefficients e,(s), c,(s), c,(s), c 2 (s), each of which possess 

 a continuous second derivative, can be reduced by means of Green's 

 functions to the solution of a pair of integral equations of the second 

 kind, and these may be reduced to a single equation of the second 

 kind by means of Fredholm's artifice. 



These results are then applied to the solution of the problem of the 

 monodromic group by formulating this problem in the following way : 

 Let a closed analytic curve C be drawn so as to pass once through each 

 of the singular points z u z 2 . . . . z m , than we have to find a pair of 

 functions f{z) g(z) which shall behave like regular functions _ of the 

 complex variable at all points of the plane, including the strip of C 

 between z m and z u but which shall exhibit a singular behaviour on the 



1 Comptcs renclus, 1898, t. 126, p. 723-725 ; Math. Ann., 63, pp. 273-276. 



2 Crctle, Bd. cxxx. ; Acta Math., Bd. xxxi. ; Ann. d'Ec. Norm. Sn?> , (3) 20, 

 1D03. 



3 Vorlesungen, 1901-02; Heidelberg Congress, 1904; Gottingcn Aanhr., 1905. 



4 Another treatment of this problem is given by A. C. Dixon, Cambr. I'hil. Trans., 

 vol. xix., part 2, p. 203. 



5 Further investigations on this subject have been made by Haseman, Dissertation 

 Gottingen, 1907. 



