66 



Rev. F. H. Jackson. [June 15, 



Let /(a) denote 



then, by means of the difference equation, we obtain 

 f(x+l)=f(x). 



If now throughout the infinite products, all expressions of the type 

 p * denote principal values only, then f(x) is a single-valued function 

 with a period unity, and for positive values of x has no singularities. 

 It must therefore be a constant, and we find 



So that finally 



(5), 



S(a) denoting a pseudo-periodic function analogous to sin a. We 

 define 8 (a) as 



When;? = 1, this theorem reduces to the multiplication-theorem of 

 Gauss and Legendre. 



4. In this article a generalisation of the function x n will be formed 

 by means of the generalised factorial T p . 



To deal with simple forms first. In [n] ! we replace p by and 

 obtain 



JP^I ' p* - 1 ' ' ' f^\ == ^ ([ n + *]) ( n positive and integral), 

 80 that 



