206 Proceedings of Boycd Society of Edinhurgli. [sess. 
where the functions (0)^, (1)^;. (2)^, . . . are each of them continu- 
ous up to and inclusive of the limit x = a^ and the series has thus a 
definite sum <jix, this sum is 2 ^Smd facie a continuous function of 
ir, and what we have to explain is the manner in which it may come 
to be discontinuous. Suppose that it is continuous up to and ex- 
clusive of the limit x = a^ but that there is a breach of continuity at 
this limit : write x = a- c, where e is a positive quantity as small 
as we please, and consider the two equations 
</)X = (0)^ + (1)^ -f (2)^-f . . . 
(6)«+ (l)a+ (2)(,-p . . . 
then we have 
cf)a — <i>x ~ ^ ^ 
f (O).-(O).. ^ (l).-(l). ^ 
a- X 
a-x ' * / ' 
^ } is a finite magnitude M, not 
indefinitely large for an indefinitely small value of e , we have 
<jf)a - cf)X = eM, which is indefinitely small for e indefinitely small, 
and there is no breach of continuity ; the only way in which a 
breach of continuity can arise is by the series in { } having a 
value indefinitely large for e indefinitely small, viz. if such a 
value is — , 
e 
then cj^a- <jix = e. 
= and as x changes from a - e 
to a, the sum changes abruptly from cf>(a- e) to ^ (a - e) + N. 
The condition for a breach of uniform convergency for the value 
x = a, thus is, that writing x = a- e, € a positive magnitude however 
small, the series 
(O).-(O) . , (1). -(!), , 
a- X {a — x) ’ 
shall have a sum indefinitely large for c indefinitely small, or say 
as before a sum = — . 
€ 
For the foregoing example, where the series is 
(1 - x) -hx(l - x) -hx^(l - x) + . . . 
the critical value is a=l : we have here (?i)^ = x''(l -x), and con- 
sequently 
{n)a - {n)x _ a"'' - x"^ 
a — x a-x {a-x) 
= -I- + . . . -fa;”) 
. - a;” for a - 1 . 
