Talks for the Solution of an Equation. 203 



" Tables for the Solution of the Equation 



By W. STEADMAN ALDIS, M.A. Communicated by Professor 

 J. J. THOMSON, F.K.S. Eeceived and Bead June 16, 1898. 



1. The object of the present paper is to exhibit the processes of cal- 

 culation of the values of the two solutions of the equation 



for successive values of x in the two cases of n = and n = 1 . 

 That is if 



y = AI n (x)-+EK n (x) 



be the complete integral of (1), where K n () is a function which 

 becomes zero when x is indefinitely increased, our object is to calculate 

 the values of I Q (x), I^x), K (ar), K I (JC) for successive equidistant values 

 of x. 



The values of I Q (x) and I^x) have been calculated and published by 

 a committee of the British Association for the Advancement of Science. 

 To the best of the writer's knowledge, no steps have been taken 

 towards the computation of K () and K 1 (). The Tables I and II, at 

 the end of this paper, give the values of these latter functions for in- 

 tervals of O'l in the argument, to such a large number of decimal 

 places as will make it si mere matter of difference calculation to deter- 

 mine intermediate values of K () and K^a;) to any reasonable degree 

 of accuracy, at any rate for values of x greater than unity ; and also by 

 means of the sequence laws to derive those of K 2 (:r), K 8 (a;) ...... , as 



far as may be requisite. 



It will be convenient to state a number of well-known theorems in 

 regard to the solution of (1). 



2. The function I n (x) is defined by the condition 



x \n+2r 

 o I 



2 / 2. 



For all values of n 



y = AI n (x) (3) 



is a solution of (1), TL(n) being the function defined by Gauss.* 



* 'Werke/vol.3, p. 145. 

 VOL. LXIV. R 



