1892-9-3.] Prof. Cayley on Uniform Convergence. 205 
n independent of x, say v, such, that \i n>v then E„<a. When tbe 
residue has this peculiarity the series is said to be non-uniformly 
convergent ; and if for a particular value of x, such as ^ = 0 in the 
present example, the number of terms required to secure a given 
degree of approximation to the limit is infinite, the series is said to 
converge infinitely slowly. 
And he thereupon gives the formal definition : If for values of x 
within a given region in Argand’s diagram we can for every value 
of a, however small Mod. a, assign for n an upper limit v inde- 
pendent OP X, such that, when n>v, Mod. E„<Mod. a, then the 
series %f(n,x) is said to be uniformly convergent within the 
region in question. 
The two forms of definition (Jordan and Chrystal) appear to me 
equivalent, and it seems to me that construing the definition 
strictly, and applying it to the above instance (1 -x) + (l -x)x-h 
{1 -x)x‘^-{- . . . , the definition does not in either case indicate 
a breach of uniform convergency at 1, viz. the definition shows 
uniform convergency from x = 0 to x=l - e, e being a positive 
quantity however small, or (as I have before expressed this) uniform 
convergency up to and exclusive of the limit 1 ; and further, it 
shows uniform convergency at the limit 1. For at this limit, the 
series of terms is 0 -f- 0 -l- 0 -f . . . , the residue or sum of the 
{n 4- l)th and subsequent terms is thus also 0 -f 0 -l- 0 -f . . . , and 
we get the value of this residue, not approximately, but exactly, by 
taking a single term of the series. Jordan and Chrystal calculate, 
each of them, the residue from the general expression thereof by 
writing therein for x or z the critical value, and then comparing the 
value thus obtained with the values obtained for the {n+ l)th and 
subsequent terms of the series on substituting therein for x or z the 
critical value, seem to argue that the discrepancy between these two 
values indicates the breach of uniform convergency. 
It may be said that the objection is a verbal one. But it seems to 
me that the Avhole notion of the residue (although very important 
as regards the general theory of convergence) is irrelevant to the 
present question of uniform convergency, and that a better method 
of treating the question is as follows : 
Considering as before the series 
( 0 ), + (!). + ( 2 ), . , . . . . 
