530 PROCEEDINGS OF THE AMERICAN ACADEMY. 



§ 5. Further Properties of $(.r) and '^(a*). 



It is necessary for us to investigate further the nature of any soki- 

 tion $ (.r), ^ (x) of (17) in the neighborhood of the curve C. In the 

 first place, it is to be observed that at points of C where all of the 

 elements of .'l(.r) are analytic, the same is true of the elements of $(.r) 

 and "^(.r) ; in truth, the equation (17) shows us that analytic extension 

 is possible across the curve at such points. 



We shall prove that the elements of these matrices possess derivatives 

 of all orders, continuous at all points of C. 



Since the elements of ^'l(.r) have line derivatives of all orders along^ 

 C we may write, for t and y upon C, 



(18) A{t) = A{y)+it-y)f^A{y) + ... 



+ ^^^ ^u ^i (2/) + (^ - yf B (t, y), 



where the elements of B{t, y) are continuous functions of t and y 

 along C^ Also by Cauchy's integral formula in matrix form we 

 have from (17) for .r within C 



(M^ (x) ^ {-lf-'{k— 1)1 r ^(t) A(t) 

 dx^-' ~ 2 7r V^Ti Jc {t — .x)^ 



dt {k = 2, 3, . . .)• 



If we substitute the above expression for A (t) in this last equation, we 

 obtain a number of terms of the form (save for a constant multiplier) 



The integral is not altered in value if C is replaced by Co which lies 

 outside of C. Therefore each of these terms represents a function 

 analytic in .r and continuous in y along C. There remains a single 

 term not of the form (19), namely 



(-lY-Hk-l)l Cft-yV- 



27r V— 1 



fJj^J^{t)Bit,y)dt. 



7 When a differentiation or integration sign appears before a matrix, it is 

 understood to apply to each separate element of the matrix. The stated 

 property of B{t, y) comes at once from the explicit formula 



r>.. N 1 ftfz - y\kdk+'Aiz) , 



