1893.] 



On Operators in Physical Mathematics. 



527 



experience relating to the Bessel function. Using it in. (82) we 

 find 



(83) 



ITO-J-! 

 which by immediate integration gives 



\n 



m m 1 n 





x n-m-l 



+ ..... (84) 



for all values of m. 



The case m = or any integer is that of (82), and m = n is that 

 of (80). The whole series of forms ranges between m = and m = 1, 

 because they recur. When m = J we have 



t 





} 



If in this we take w = 1, we find that the terms can be arranged 

 in pairs, thus 



..... (86) 



or, which is the same, 



The best value of x is obviously 1. 



When two variables x and y are used in the binomial theorem, we 

 have, using v f r d/dx and A for d/dy, 



m 



= 



nl 



? _ 



v - 







We may use any of these forms. Selecting the last but one, and 

 using the generalized exponential, we have 



(89, 



