342 
[936 
936. 
NOTE ON UNIFORM CONVERGENCE. 
[From the Proceedings of the Royal Society of Edinburgh, vol. xix. (1893), pp. 203—207. 
Read December 5, 1892.] 
It appears to me that the form in which the definition or condition 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)a: + ( 1)* + (2)3, ... + (ft)a;, ... 
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 of x; but (j>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 <f>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 2 + ..., 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 x = 1 we have the divergent series 1 + 1 + 1 + ..., and this is 
not an instance; but taking, instead, the geometrical series 
(1 — x) + (1 — x)x + (1 — x) x 2 + ..., 
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