XLVII XLVIII] InlrniliK-tivn, xxxv 



and tj = 1, 



6-- M, 

 / 1 x (fc + 1) 



, 



, . 

 6 



A 



(p+2s-l)(p + '28) 

 _q-l 



,- *^? 



~p + 2' 



/. 



a s 



M . * _ 

 "~ ^^ 



In applying this method, it is desirable since I x (p, q) = l- /^(fl.p) to select 

 Ix(p,q) or I 1 _ x (q,p) for calculation, according as this or that integral avoids 

 summing through the largest term in the binomial (x + 1 X)P+V+ I (see p. xxxv). 

 The successive convergents are given by P,/Q, where 



One of the advantages of the method is that the true value of I x (p t q) always 



P P 



lies between Q-x? and C j~, so that it is easy to judge the magnitude of the 



V&f Va+i 



error made by stopping at any convergent. 



Illustration. Evaluate 7. 12 (21, 81). 



Let us first find the constant C. We have 



C = ('12) a (-88) 80 T (1 02)/{r (22) T (81)}, 

 log C= 77-221,4199,420 - 101 + 159-974,3250,285 - 19-708,3439,116 



- 118-854,7277,225 



= 237-195,7449,705 - 239-563,07 1C>,:U1 

 = 3-632,6733,364. 



ffff* 



