100 Mr Orr, On the product J m (x) J n (x). 



By the aid of (12) and (13), equation (9) may now be written 

 in different forms. 



Should m + 7i be an integer one of the analogues of (9) is to 

 be replaced by 



™— in— n 



I I —I I = 



-*• —m J - — n ■* in - L n — 



2- m - n n(-m)ii(-ra) 



F I , g ; 1-m-n, 1 - m, l-n; ^ 2 j...(30) 



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

 terms. This equation is now identical with (27) or (28) according 

 as m + n is odd or even, the right-hand member of (27) or (28) 

 now terminating. 



Should m — n be an integer a similar equation, obtainable by 

 changing the sign of n, holds. 



Particular cases of the last equations are the well-known 

 relations 



JnJ—n+i 4" J—n"n—i = Sin HIT, 



1TX 



The limiting forms of these various results in case n or m, or 

 both, are positive integers may be obtained in the usual way. 



The function (7) is a multiple of 



\J--n J-n) \J--m J-m)> 



and (8) may be either of the two functions obtained from this by 

 changing the sign of x. 



