1893.] On Operators in Physical Mathematics. 529 



it gives may be convergent. Thus we may obtain convergent special 

 formulae for \n. Thus, take m = J, n = | in (84). We obtain 



. , 



~~ ~~ 



5.9.13 

 and when x = 1, we have the series 



i-TVa-iKi- ..... (97) 



Similarly, m = f , n- = J, as = 1, gives 



-ACi-iKi-iKi-....; (98) 



and so on. 



Property of the Generalized Exponential. 



26. Notice that the operation v m performed upon the generalized 

 e* reproduces it when m is integral, but gives an equivalent series 

 when m is fractional. If, then, we take the special form of the 

 ordinary stopping series for v w to work upon, we require to imagine 

 that the zero terms are in their places, thus, 



All terms before the 1 are zero, but not their rates of variation with 

 x in the generalized sense, if we are to have harmony with the 

 behaviour of the general form of e*. This is transcendental : and 

 there is much that is transcendental in mathematics. 



The above generalizations are somewhat on one side of our subject 

 of the treatment of operators, though suggested thereby. I propose 

 to continue the main subject in a second paper. 



