, 284 



" that 



MR. 0. N. WATSON: A THEORY OF ASYMPTOTIC SERIES. 



Z/1 + / + 1)}. 



Now, for all values of , F(/n+l) < Kr(/n+J+l), where K may be taken to be a 

 finite number independent of n. 



Therefore, since \x\>y, assuming that y is not zero, we have 



Comparing this equation with (U), we see that / is a possible grade; in other 

 words, if we are given a number I as an outer grade, and a number k, greater t 

 / as a grade, we may take I as a new possible grade. 



' We also see that if k = I, and we are given a- as an outer radius, and a number p 

 greater than <r as a radius, we may take <r as a new possible radius. 



' 2. Let us now consider the product of two functions f, (x), f, (x) whose expansions 

 (valid over a common range of values of arg x) are 



Let possible constants and characteristics of these expansions be A 1( B 1} k lt Z t , pi, ^ ; 

 A,, Bj, k 2 , l a , pa, <r a respectively. 



We shall show that, for the range of values of arg x for which both these 

 expansions are valid, the product f l (x)/ 2 (x) can be represented by an asymptotic 

 series of which possible characteristics are & . k, PO, <TO 5 where k a , 1 , p , or a denote th 

 greater of the numbers &i, k 2 ; l lt 1 2 ; p 1; p 2 ; <TI, <r 2 respectively. 



By direct multiplication 



where 



S. = a 



X 



X 



so that 



r = 



Now, when* f > 0, T(l + ) is positive and decreases as increases till 

 = 0-461. ..(= ^o, say); when f > ,,r(l + f) increases with f ; and when 0< <1, 



* DE MORGAN'S ' Diflferential and Integral Calculus ' (1842), p. 590. 



