1893.] On Operators in Physical Mathematics. 525 



in rising powers of x is 



i.w= i+f*+ 2 4+i4+ ..... (72) 



Pat ar/4 = ?/, and it becomes 



Let the differentiator cfycfo/ be called A. Then (73) may be at once 

 written 



+ ....= 6 -'. (74) 



There is nothing hypothetical about this. What is, however, 

 entirely speculative in the absence of trial is whether it is legitimate 

 to substitute the generalized exponential for the ordinary, and there- 

 fore generalized for complete differentiations. But on trial it will be 

 found to work. Thus, using (63) with A" 1 in place of x, we convert 

 (74) to 



_* _m+l A -(m-l) 



+ + ..... 



writing u for the result ; or, which means the same, 



r~-+^-^+ (76) 



where, as before, y = # 2 /4 in connexion with the Bessel function. 



2 



if w = 2m. We have u m u m ~i for all values of m, and %,= T (). 



To test the validity when m is fractional, take m = % or w = 1, 

 then, by (77), we obtain 



This will by numerical calculation be found to give the same value as 

 the series (72) for !<>(). When x is as large as 10 the values agree 

 to the fourth place by the convergent series in (78) alone, the semi- 

 convergent series in (78) having a relatively small value. The value 



