528 Mr. 0. Heaviside. [Feb. 2, 



therefore 



^ ^ + _ 

 n \m \nm \ml \n m + 1 |m + l \n m 1 



+ ...., (90) 

 where m may have any value, and x, y may be exchanged. 



Taylor's Theorem Generalized. 



24. We may also apply the generalized exponential to Taylor's 

 theorem for the expansion of a function in powers of the variable. 

 For this theorem is expressed by 



/(* + *) = **/(*), (91) 



and, if this be true generally, irrespective of the wholeness of the 

 differentiations, we must have 



(92) 



Whether this is true for any function / (x), with the usual limitations, 

 I cannot say. There are probably other necessary limitations. 

 As examples, take/(#) = 1. Then we obtain 



Here put c = h/x ; then, by using (44), we have the result 



, (94) 



(n-l)w 

 where c is to be positive. When n = \, this reduces to 



...., . (95) 



where a is written for c 4 . It is obviously right when a = 1. 



The formula (70) may be derived from (94) by the use of (44). 



Special 'Formulas, for Factorials. 



f ^5. The binomial generalization before given is, of course, a special 

 case of (92), namely, /(a?) = x n l\n. It will be observed that the series 



