Mr Orr, On the product J m (x) J n (a?). 97 



equation (14) gives, when x is large and has its argument between 

 — 7r and + ir, the approximate equality 



E *(P)^(0^ (15 c), 



whether n be integral or not*. 

 If n — \ is a positive integer 



UQ—n 



considered as the limit of a function in which n — ^ is not a positive 

 integer, contains a number of terms which are zero, and after these 

 the series begins again, the second set of terms being those of 



x n 



The equation derivable from (12) by changing the sign of n is then 

 to be replaced by 



f (I- n - I n ) = 2 _ w g~ ( W _ n) FQ-n; l-2n; 2x). ..(16), 



wherein the series on the right is not continued beyond the zero 

 terms. Equations (14) and (16) are now identical f. 



We next suppose that in equation (9) m =n. We thus have 



/ »' = 2»{n(n)}' J,(w + * ; n+1 ' 2W+1; ^ (17) ' 



another equation derivable from this by changing the sign of n, 



(18), 



and I n I_ n = ^-^(i; 1+n, 1-n; x 2 ) (19). 



11 (n) 11 (—n) 



The differential equation (1) has now to be replaced by one of 

 the third order of which the series (6) is no lunger a solution. The 



* I would suggest the desirability of making the I function the fundamental 

 one in analysis instead of the J, at least when the variable is complex ; in the case 

 of the former i occurs less frequently explicitly, equation (14) which is fundamental 

 is more easily remembered, and the approximate formulae (15 b ), (15°) are more 

 easily remembered and better adapted for use in physical problems. 



t See Cayley, Messenger of Mathematics, Series i., Vol. v., p. 77 ; Glaisher, Ibid., 

 Vol. vin., p. 20 ; Phil. Trans., 1881. 



X This result was given by Lommel, Math. Annal., in., for the case in which n 

 is an integer. 



