394 JACKSON. 



ks-1 

 Ws (u) = u^'^^ (0) + Z «y w*'^ (0) = 0, s=l,2,...,v-l, 



('«) i:-r 



W, (u) = uik.) (x) + 2: /3,wW (x) + 2: aju^^^ (0) = 0, 

 j=0 ,7=0 



where 



I' - 1 ^ A;i > A;2 > • • • > ^\-i ^0, v - 1 ^ ^^ ^ 0; 



and we shall assume them in this form. It would be possible to 

 simplify them still further, if there were any object in doing so. The 

 coefficients a^-, aj, /?y, are any real or complex constants, and in 

 particular some or all of them may vanish. 



The facts which we shall need concerning the form of solutions of the 

 equation (15), apart from the question of any particular boundary 

 conditions, may be taken from the papers of Birklioff already referred 

 to. Following his notation, we shall use the symbol [a], where a 

 is any constant, to represent a function of p = re^*, or of p and a second 

 variable, which approaches the limit a uniformly with respect to 6, or 

 with respect to d and the other variable, when r becomes infinite; the 

 same symbol will be used to represent as many different functions of 

 this sort as may occur. Furthermore, the roots of the equation 

 w" -\- 1 = are denoted by wi, W2, . . . , i<v. Then it is possible to assign 

 to each value of p a fundamental system of solutions of the equation 

 (15), T/^.T, p), s = 1,2,. . ., V, in such a way that 



(17) ys{x,p) = e'>^s^[l], 

 and at the same time 



(18) £, ys{x,p) = {pwsYe'^-'s- [1], k = \,2,. . .,v - \. 



These solutions have continuous derivatives of all orders with regard 

 to .T, for each value of p. If I is any one of the numbers 0, 1, . . . , 2^ — 1, 

 and p is restricted to the sector Si of the complex plane in which 



lir^ ^ (Z+l)7r 

 — ^ arg p ^ ^^ —^ 



V V 



the choice of this fundamental system may be so made that the func- 

 tions t/g(.T, p) and their derivatives with regard to x shall be analytic 

 functions of p for each value of x, at least throughout the part of the 

 sector exterior to a circle of sufficiently large radius about the origin. 



