1893.] On Operators in Physical Mathematics. 129 



Therefore, by adding this equation to (88), we obtain 



o == 2 *'[/W] 2 /'(--i) + . (101) 



Now it is a fact that this is true, term by term, when r = 0, 1, 2, 3, 

 &c. But (101) is not true in the same manner generally. Only 

 = 0, that is, when n is a negative integer, do we have 



/ =/'(-!), (102) 



by (93), which is general. Put n = r l to suit (101). But 



(103) 



by (93). Therefore (101) is the same as 



,- ,004 



which does not vanish term by term, except for the special values of 

 r indicated. Integrating (104), we obtain 



_-> x r sin rir . /i ntc\ 



constant = , --- . (105) 



\r TTT 



54. The case r = we have already had, when the constant is 1, 

 so it should be 1 generally. The case r = \ is represented by 



* t^c\K\ 



" ' ( } 



and the following is a verification : The right member is 



= 2 (tan - 1 v -* + tan - 1 V *) = - tan - 



7T 7T 1 - V V 



= - tan- 1 00 = 1. (107) 



Although the validity of this process of evaluation may be doubted, 

 there is no inconsistency exhibited. 



55. The other formula of a similar kind, viz., 



,^i, (108) 



, 



\r \T m 



VOL. L1V. K 



