538 



PROCEEDINGS OF THE AMERICAN ACADEMY, 



such wise that the combined circuit is reducible to a point, it is clear 

 that 



(25) 



TmT-m-i ... Ti — I, 



a necessary relation between the matrices 7\, . . . , r,„. 



The problem of Riemann is the following: For assigned points 

 ill, . . .,am and assigned matrices Ti,..., Tm for which (25) holds, to 

 construct a matrix Y{x) of functions of determinant not identically 

 zero, with elements analytic save at ch,. . .,0^ where these elements 

 are finite or become infinite to finite order, and undergoing a transfor- 

 mation to Y{x) Ti as X makes a positive circuit of ai(i = 1,. . ., m). 



A solution of this problem has 

 been given by Hilbert and com- 

 pleted by Plemelj (loc. cit.)- It is 

 possible to obtain a more simple 

 solution on the basis of the pre- 

 liminary theorem. 



Let us surround «i, . . . , cim 

 by small non-overlapping simply 

 closed analytic curves Ci, . . . , C^ 

 and let us pass through oi, . . . ,am in 

 cyclical order another closed ana- 

 lytic curve D which meets each 

 curve Ci only twice (Fig. 2), in 4 and m^ say. 



Choose matrices Ai. . .,.-1^, Am+\= Ai of constants such that 



Fig. 2. 



(26) 



Ai+i Ai — li 



(i = 1, . . . ,m). 



Here Ai, for example, may be taken at pleasure and A2, . . . , A^ are 

 then determined. Define a matrix A{x) along D to be equal to Ai 

 on that part of the curve which lies between Cj and Cj+i [C^+i = Ci] 

 and equal to 



li 



(27) 



h 



m,- X 



— Ai-i -j 



Ai {{ = 1, 



,m) 



for X on the part of D within Cj. We will suppose that /,, ???j and D 

 were so chosen that D does not pass through one of the finite numljer 

 of point for which the determinant of any matrix (27) vanishes. 



The matrix A{x) as here defined is continuous along D, analytic 

 save at the points where D intersects Ci, . . . , C^, and of determinant 

 not zero along D. 



