548 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Thus P{.v) is represented asymptotically in the complete vicinity of 

 X = GO by (39). But we have formally 



(40) ^-^ = (...«|(^-^ + ,^)m..-) + 'M^})- 



If (35) and (40) be used in evaluating the expression (39), it is seen 

 that each element of P{x) is given asymptotically by a power series 

 in descending integral powers of x, with leading term in .r''+^ or lower 

 power of X. It follows that the elements of P{x) are analytic or have a 

 pole of at most the (p+l)th order at x = cc . Hence the elements of 

 P{x) are polynomials of degree at most p + 1 . 



The central problem of the irregular singular point is, for a given 

 choice of pi{x), . . . , p„{x), Vi, . . . , r„ and of Ci, . . . , cy above described, 

 to construct a matrix Y{x) with the above specified properties.^ ^ 



§ 14. Solution of the Problem of § 13. 



In order to solve the problem just stated we make an application of 

 the preliminary theorem of Part I, taking /• = ^72 and for the curves 

 Ci,..., Ci,v fiot the X/2 straight lines formed by the X rays arg x = t^ 

 (vi = 1, . . . , X) but by the A" rays arg x = t'^ (m = 1, . . . , .V), which 

 are obtained from them by a slight rotation € in a clockwise direction 

 in the complex x-plane. It is evident that if e be taken small 

 enough we shall have 



Zm (x) ^ S(x), r'm^ arg.r ^ r'm+b 



for m = 1, . . ., .V. Furthermore we shall have for any m 



9^ (ojm — ctA-,,,) x"-^ > 0, arg X = r'm, 



for m = 1, . . ., iV. This fact is essential to the solution. 



The matrices ^i(.t), . . . , Ai}f(x) which are to be used in the applica- 

 tion of the theorem are defined as follows. Write 



T(x) = (xPj^^'Kv'jdij), 



and then put 



(41) Am (x) = T {x) [I+Cm]T-' (x) (m =1,. ..,X), 



21 Cf. I, § 7. 



