204 Proceedings of Royal Soeiety of Edinburgh. [sess. 
value cc=l, but for this value there is a breach of uniform con- 
vergence. 
I remark that Du Bois-Keymond in his paper, “ Notiz liber einen 
Cauchy’schen Satz, die Stetigkeit von Summen unendlicher Keihen 
betreffend,” Math. Ann., t. iv. (1871), pp. 135-137, shows that when 
certain conditions are satisfied, the sum cf>x is a continuous function 
of X, but he does not use the term “ uniform convergence,” nor give 
any actual definition thereof. 
M. Jordan, in his “Cours d’ Analyse de TEcole Poly technique,” 
t. i. (Paris 1882), considers p. 116 the series s = + %2 + W3 + . • • , 
the terms of which are functions of a variable 0, and after remark- 
ing that such a series is convergent for the values of s included 
within a certain interval, if for each of these values and for every 
value of the infinitely small quantity e we can assign a value of 
/^ such that for every value of 2 ^, 
Mod 4- Un +2 + ... + < Mod c, 
e being as small as we please, proceeds — 
“ Le nombre des termes quhl est necessaire de prendre dans la 
serie pour arriver k ce resultat sera en general une function de 0 et 
de e. iN’eanmoins on pourra tres habituellement determiner un 
nombre n function de e seulement telle que la condition suit satisfaite 
pour toute valeur de 2 comprise dans Fintervalle considere. On 
dira dans ce cas que la serie s est uniformement convergente dans 
cette intervalle.” 
And similarly Professor Chrystal in his Algebra, Part II. (Edin- 
burgh, 1889), after considering, p. 130, the series 
rvi /y» /y» 
] + -}- , 
2x+1.3x+l ' ’ ' {n-\)x-^l.nx ->r\ 
for which the critical value is a; = 0, and in which when x = 0 the 
residue of the series or sum of the {n-\- l)th and following terms 
is = — - proceeds as follows : — Now when x has any given value, 
nx 4- i 
we can by making n large enough make — smaller than any 
given positive quantity a. But on the other hand, the smaller x is 
the larger must we take n in order that — — many fall under a ; 
and in general when x is variable there is no finite upper limit for 
