379 
1909-10.] Borel’s Integral and ^-Series, 
in which the general term is 
/ _ l\n (1 — g ) ! yni x n+l Q \n(n-\) 
V ' (l-q n b)\ (l-q n x)l * 
The series (3) gives the form 
1 , fa-ll l4l . 
( i-^) +g (i-,)(i-^) +g ’ •’ 
in which the general term is 
J>-l][q-2] . . [a-n] 
(1 - q n x) ! 
As in previous papers, [a] denotes (1 — q a )/(l — q). 
What has been written above with respect to Euler’s transformation 
may be written, mutatis mutandis, with respect to Borel’s integral 
J Tu n = I e X u(x)dx , 
n JO 
where 
u 0 + u i*+u2^+ ■ ■ +“»^ 
also of the integral sum of an asymptotic series 
, a, . a 9 , 
% + — + -f + , 
x x A 
(Of Bromwich, Theory of Infinite Series, pp. 268, 339-40.) 
II. 
In this note I should like to indicate the necessary modification of 
Borel’s method if it is to be available in the case of ^-series. G. H. Hardy,* 
in his papers on non -convergent series, has formulated the following 
principle which lies at the root of Borel’s method : “ If two limiting pro- 
cesses performed in a definite order on a function of two variables lead to a 
definite value X, but when performed in reverse order lead to a meaning- 
less expression Y, we may agree to interpret Y as meaning X.” For example, 
suppose 
fix, n) = <f> Q (x) + <l> 1 (x) + . . . + <f>n( x ) 5 
and that 
&(x) = limit fix , n) , 
n~> oo 
* Trans. Cambridge Phil. Soc., vol. xix., 1904, p. 297. 
