380 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
then 
j<E>(x)dx . 
It may happen that the integral on the right is convergent when the series 
on the left is non-convergent. Borel’s integral is the special case in which 
cf>n(x) = e~ X U n - , 
n ! 
where u n is independent of x, and 
4 ’(*)= 1 ™ i te “'{“ 0 + 1<ia:+ ' ' ' 
= e~ x u(x ) . 
We see that if Borel’s method is to be available in the case of any specified 
series ^ 0 +^ 1 + '^ 2 + • . • . we must have considerable prior knowledge of 
the form of 
(in this case supposed convergent, though the condition may subsequently 
be removed). I express this otherwise, by saying, that Borel’s method 
enables us to make use of our knowledge of the form of the limit of an in- 
finite series (convergent for only a finite limited range of values of the 
variable) to perform certain operations and transformations in the case 
where the variable has passed outside the limited region of convergence. 
We must keep in mind that such use of non-convergent series is always 
formal and symbolic. It is interesting in this connection to recall Professor 
Mittag-Lefiler’s remarks : * “ Borel’s idea that he has obtained by his sum- 
mation expression the power series itself, in the case where it diverges, is a 
play upon words, and all the more intemperate because it gives rise to the 
illusion, entirely false, that he has been able to extend the limits of the 
theory of analytic functions beyond those fixed by classic theory.” 
“ As one is able, since the expression of Bor el is convergent, to perform 
the same operation on the divergent series as on the convergent series yp(x), 
Bor el’s scheme implies only the translation for this special case of Weier- 
strass’s theorem, ‘If an analytic relation, however general or however 
special, exists between several different power series or their derivatives, 
this same relation subsists also for the functions in their totality.’ ” 
A consideration of Hardy’s principle will show us why Borel’s integral 
is not available in dealing with g-series. We have, generally speaking, no 
* Bulletin Amer. Math. Soc., vol. xiv., 1908, p. 485 (International Congress, Rome, 1908). 
?[f 
<f) n (x)dx 
