1893.] On Operators in Physical Mathematics. 523 



from to x, the other from x to oo. For the first use (60), since 

 Wi = when x = 0; and for the second use (61), since w 2 = when 

 a; = oo. We then get, by (59), 



(62) 

 or, which is the same, 



This is proved when n is greater than 1. But the change of n 

 to n 1 in the series on the right of (63) makes no alteration. We 

 therefore conclude that the series expresses e x for all values of n. 



When n or any integer, positive or negative, we have the usual 

 stopping series for e x . When n is fractional, we obtain semi-con- 

 vergent series. Of course we obtain the whole series of forms by 

 making n pass from to 1. The most interesting case is that of 

 n = -J. This gives 



.5 



where the value of | -J we know to be || \ = JTT*, by (45). 



By means of this series we may pass from one to the other of the 

 two forms of evaluation of Fresnel's integrals, due to Knochenhauer 

 and to Cauchy respectively, which are given in works on Physical 

 Optics. 



21. The function called w above we may obtain in a series of 

 rising powers of x without the exponential factor in the following 

 manner : 



- 



B+1) , (65) 



which is immediately integrable by the binomial expansion ; thus 



