212 Mr. W. Steaclman Aldis. 



term is least when r has the largest value, which makes (2r- I) 2 less 

 than Srx. This gives r = q, where q is the integral part of 



Hence the nearest approach of the limits, within which (12) confines 

 the value of K (;z:), is 



l)3 

 ~ 



It is evident that as K (;c) lies between the sum of q terms, and the 

 sum of q + 1 terms, the mean of these two sums is as near an approxi- 

 mation to the actual value of K O (JB) as (12) will give. This mean 

 cannot differ from K () by quite half the quantity (14). 



If x be an integer, the value of q is 2x ; thus, if x = I the third term 

 is the smallest : when x = 5 the eleventh, when x = 8 the seventeenth, 

 and so on. The limit of error, estimated by half the least term, is for 

 x = 1, 0-0162 ; for x = 2, 0'0042 ; f or x = 5, O'OOO 000 022 ; and for 

 larger values of x the limit becomes rapidly smaller. 



For values of x as great as, or greater than, five, K (#) can thus be 

 determined with accuracy to seven or more places of decimals. 



Very similar statements can be made with respect to the determina- 

 tion of K^x) from (13). 



13. From (12) 



i ...... } 



The multipliers, disregarding the sign, by which the coefficients of the 

 powers of x within the bracket are derived, each from the preceding, 

 are 



1 9 25 49 81 121 169 

 8"' T6"' 2T> 7T2> TTI' ~T~~> TIT*' 



Let these numbers be denoted by the symbols m v m. 2 , m 8 , ...... , and 



let (ir/2x)*e~ x be called /3 . Then if a series of quantities ft, /3 2 , /3 3 , ...... , 



be derived by the successive relations 



ft = Wj/V' 1 . ft = m^- 1 , /3 8 = m.f. 2 x~\ ............ (15) 



it is evident that 





The relations (15) are adapted to logarithmic computation. For the 

 value of P Q two logarithms beside that of x are required. These are 



log = 0-4342944819; 

 = 0-0980599325. 



