222 
PROCEEDINGS OF SECTION A. 
F or readers desirous of an acquaintance with this idea of infinitely 
slow convergency we refer to Seidel's paper, or to Chrystal’s Algebra 
(Part II, p. 130). J 
The series % 
V° - n 2°° r nx _ ( n — 1) x ~j 
» = i (nop + 1) [( n — 1) x + 1] n = i \nx + 1 (n — 1) x + 1 J 
has been given by P. du Bois-Reymond (Antrittsprogr., p. 25) as an 
instance of infinitely slow convergency for x = 0. The sum of the 
first n terms — 
lim. S, = 1 
n = oo 
nx 
nx + 1 
lim. 
r n > = 1 — S n = 
nx 
nx + 1 
1 
nx + 1 
and we see at once that the smaller x the greater n must be taken to 
have lim. r n > infinitely small. 
The series — 
n= i (1 + x 2 ) n 
presents also the same peculiarity for x = 0. 
In modern language a series 
f ( x ) = u t (, x ) + u 2 (x) + .... + u n (x) + ... . 
convergent for 
a < x < b 
is said to be uniformly convergent in the neighbourhood of x — x oy 
a < x Q < b, when we can take n sufficiently great to have in absolute 
value — 
(«) = U n + 1 W + tin + * (?) + • • • • 
infinitely small, whatever be the value of x between x 0 + c and x 0 — c, 
c being infinitely small. 
.The series is said to be uniformly convergent between a and ft, 
a < a < f3 < b, -when it is uniformly convergent in the neighbourhood 
of any value of x between a and /?. 
The notion of uniform convergency had been clearly introduced — 
as we have seen — in the memoir of Seidel and the note of Cauchy, and 
seemed therefore acquired to science. However, strange to say, it had 
remained unnoticed for years by the majority of mathematicians, when 
"VVcierstrass, without probably any knowledge of Seidel’s paper or 
Cauchy’s note, reintroduced it in his lectures at Berlin. No printed 
communication exists from the pen of Weierstrass, # but one of his 
* See P. du Bois Reymond — Beweis dass die Coefficienten der trigonometrischen 
Reihe, etc. Footnote, p. 119— Abhandlungen der Bayerischen Akademie, Band XII, 
