534 PROCEEDINGS OF THE AMERICAN ACADEMY. 



vanish, there exists a matrix $(a;) with the folloiving properties: 



(1) each clement of ^{x) is analytic except along Ci,..., C^ and at an 

 arbitrary point x = a where the elements may become infinite to finite 

 order; \^{x) \ nowhere vanishes save possibly at x = a; 



(2) the elements of $(x) are continiious and unlimitedly differentiable 

 along each curve Cj from either side, analytic from either side save at 

 points of intersection of the curves, or at those points where an element of 

 Ai{x) fails to be analytic, or at x = a; if a lies on a curve Ci, the matrix 

 (x — aY Ai{x) [or x~^Ai{x) if a = oo ] i^ unlimitedly differentiable along 

 Cifor a suitable l.^ 



(3) if a + and — side of each curve Ci are chosen, then 



x" V * (-^^ = [-""- ^ (^)l ^' ("■') (^ = 1' • • •'^)' 



where the approach to the arbitrary point .r,- of Ci is along the -\- and — 

 side respectively. 



Let us begin by establishing the theorem in the case when a is 

 not a point of Ci, . . . , C^. It has already been established for r = 1 

 (see § 6), with the single notational difference that 4>(.r) was replaced 

 by either of two matrices ^{x) and ^ (x), according as .r was within or 

 without C. To establish the theorem then, we need only show that 

 if it is true for r^k it is also true for r = k -\- 1, when the theorem 

 follows by induction. 



Assume that ^^ (x) is the solution for r = k, and for the matrices 

 Ci, . . . , Ck, Ai{x), . . .,Ak (x), where Ci, . . . , C^+i, and Ai (x), . . ., Ak+i{x) 

 satisfy the requirements of the theorem for r = k -\- 1. Let us sup- 

 pose for the moment that a solution $i_^i (.r) exists with the desired 

 properties. If we write 



(22) ^k+i(x) = U{x)^k{x), 



the following facts are clear from (1), (2), (3) of the theorem: (1') 

 each element U (x) is analytic except along Ci, . . . , C^+i and at the 

 specified point a, where its elements may become infinite to finite 

 order; | U^ (a:) | nowhere vanishes save possibly at x = a; (2') the 

 elements of U(x) are continuous and unlimitedly differentiable along 

 each curve Ci from either side, analytic save at points of intersection 



9 A function will be termed unlimitedly differentiable at x = oo if when we 

 write X = 1/x', the function of x' obtained by the substitution is unlimitedly 

 differentiable at x' = 0. 



