193.] On Operators in Physical Mathematics. 119 



When x = 2 we deduce that 



1 4X2 XO'28 T88 



u = 0'28, rj = = ; 



| J 1-28 or 1'49 1'28 or 1'49 



bhe first case being without, and the second with, the smallest term 

 in v. Both results are too great, though the error is less than the last 

 term counted. But this rule breaks down when we pass to x = 3, 

 when we conclude that 



1 0-175 X4X3 3 



= 1'3437 or 



the former case being without, and the latter with, the smallest term 

 in v. But the result is too big, and the error rule just mentioned 

 fails. For if we add on the smallest term a second time, we obtain 

 ri604, which is still too big. . 



40. Since the case n = f is bad, we may expect n = T 9 ^ to be 

 worse. We have 



9.19.29/#\ 3 



.29/#\ 3 



1 ()+..... (Co) 



o \ -ir / * ' 



10 \2 2 \10/ 3 |3 \loy 



_1 f 81 , 81 . 361 , Q 



v _. i Q i l~r rrr~"~rnr/,^^ .TVS ' / v^) 



Here take x = 1, then we conclude that 



1 4'2 



1-0 or 1--81 



But it cannot lie between these limits, being only a little over 

 unity. So add on to v the next term, the third in the v series. This 

 will give 



1 4'2 



which is still too great, and, of course, the error rule is wrong, as we 

 suspected just now. 



Whilst there does not appear to be any departure from numerical 

 equivalence of u and v in the sense used between n = ^ and 

 n = +1, it appears that when n is below ^, there is a tendency for 

 the v series (convergent part) to give too small a result. This ten- 

 dency, which is at first small, becomes pronounced when n is down to 

 T 9 o, at least for small values of x. It is likely that for large values, 

 the rule in question might still hold good. But sinking below T V 

 towards 1 makes the tendency become a marked characteristic, and 

 in the end the rule wholly fails except for an infinite value of x. 



