378 
| (pp ef, eff) 
N, and N,, then it follows from (2) that | sin a Wp (p) | <J 
l 
for |h| <e(h#0), whence (1) may ‘be concluded. 
4. Remarks. In the foregoing proof for the differentiability 
of the function p(x) = u, (v) + u,(@) +.... for e=p use has been 
made of the continuity of the functions u', (2), w'‚ (a), etc. for x= p 
only; therefore it was not necessary to suppose their continuity 
throughout an interval. Also there was no need to assume the inte- 
grability of the functions u’, (a), w'‚ (x), ete., so that the usual proof 
(where the series w', (©) + u',(v) +... is integrated term by term) 
does not apply to the more general formulation as given in N°. 1. 
5. When the convergence of the series u, (x) + u,(@)+...... 
throughout the whole interval 7 is assumed, then it is sufficient, in 
order to establish 8, to assume the semi-uniform convergence (simple- 
uniform convergence of Dini) of the series u’, (7) + wu’, (@) +..... 
in the interval ¢. This semi-uniform convergence namely is in the 
first place sufficient to establish the continuity of yp (wv) for «= p. 
The determination of the numbers N, and A, presents further no 
difficulty, after which it is possible to attribute to 2 such a value 
> N, and > N,, that the inequality |B, (2)| <i0 holds for every 
x in the interval z. It is immaterial whether this inequality also is 
satisfied for every greater value of n. 
In his “Teorica delle funzioni di variabili reali’? Dint has demon- 
A Jan h) — p (p) 
strated 8 by a more complicate transformation o 
h 
thereby omitting the assumption d and assuming the series u, (#) + 
+u,(v)+.... to be convergent throughout the interval 2. 
6. In order to demonstrate the differentiability of pw) vt was 
not necessary to assume the absolute convergence of the series 
ul, (a) Hu, w) +... By supposing |u',(@)| << Cn (Cn independent 
of wv) and the series c, + c, + .... convergent (which includes the 
uniform absolute convergence of the series u, (a) + w',(v) +...) 
Porter (Ann. of Math. ser. 2, vol. 3, 1901, p. 19) has proved the 
differentiability of g (wv) for «=p in a very simple way without 
any supposition concerning the continuity of the functions w', (2) 
viz. by making use of the equality 
(p hn yl h — U, 
+ Zus (p + 6; h) — & uy (p). 
n—+-1 n+1 
