1892-93.] Prof. Cayley on Uniform Convergence. 
203 
Note on Uniform Convergence. By Professor Cayley. 
(Read December 5, 1892.) 
It appears to me that the form in which the definition or con- 
dition of uniform convergence is usually stated, is (to say the least) 
not easily intelligible. I call to mind the general notion : We may 
have a series, to fix the ideas, say of positive terms 
(0):c +(!):«■•“ • • • 
the successive terms whereof are continuous functions of x, for all 
values of X from some value less than a up to and inclusive of a (or 
from some value greater than a down to and inclusive of a ) : and the 
series may be convergent for all such values of x^ the sum of the 
series (f>x is thus a determinate function (f>x oi x\ but cj^x is not of 
necessity a continuous function ; if it be so, then the series is said 
to be uniformly convergent ; if not, and there is for the value x = a 
a breach of continuity in the function cf)X, then there is for this 
value x = a a breach of uniform convergence in the series. 
Thus if the limits are say from 0 up to the critical value 1, then 
in the geometrical series 1 +x-\-x‘^+ . . . , the successive terms 
are each of them continuous up to and inclusive of the limit 1, but 
the series is only convergent up to and exclusive of this limit, viz. 
for a? = 1 we have the divergent series 1 + 1 -f 1 -e . . . , and this 
is not an instance ; but taking, instead, the geometrical series 
C - x) + {1 - x) x + - x) + . . ., here the terms are each of 
them continuous up to and inclusive of the limit 1, and the series is 
also convergent up to and inclusive of this limit ; in fact, at the 
limit the series isO-bO-fO-i- . . . We have here an instance, 
and there is in fact a discontinuity in the sum, viz. x<\ the 
sum is 
{l-x){l+x + 'j?+ . . .), = (!- 4 = 
whereas for the limiting value 1, the sum isO-fO-1-0-1- . . ., = 0. 
The series is thus uniformly convergent up to and exclusive of the 
