Mathematics. — “Theorem on the term by term differentiability 
of a series” By Prof. F. Scnun. (Communicated by Prof. 
D. J. Kortrwne). 
(Communicated in the meeting of June 28, 1919). 
ab 
a. the functions u,(x), u,(a),.... are differentiable in the interval 
a<aS<b, denoted by 2, 
b. the series u’, (a) + u’, (a) +... ts uniformly convergent in the 
interval 2, 
c. the series u, (7) + u, (©) +... ds convergent for a value cof 
vz in the interval 1, 
d. the functions u’, («), u',(x),.... are continuous for the value 
Pp in the interval 7, 
then 
a. the series u, (x) + u, w) +.... is uniformly convergent in the 
interval 1 and 
B. the function u, (w) + u, (w) +... is differentiable for «=p, 
its differential coefficient being u, (p) + wu, (Pp) +. 
Evidently the expression “differentiable” has for «=a or «= b 
to be taken in the sense of differentiable on the right, resp. on 
the left. 
For this theorem (which is usually deduced in a more restricted 
form *) from the term by term integrability) we shall here give a 
simple proof resting on the definition of differential coefficient only. 
2. Proof of a. If we put u @) + Uinpe (2) + ....= F,@), 
we have (« belonging to the interval 2): 
Una (2) + npe (@) HH Unie (©) = Una (€) + tanpa (0) + … + tente (0) + 
+ {ulna (B) + wate (5) +--+. + wine (Bi (@ — 0) = 
= bu (c) + Unto (c) + -.+ + Unk (“) tt Ry (Sj Rn +(E)} (@—c), 
where § =c-+ 6(«—c), hence: 
junta (2) + une (ve) +... + unpe (2)| S 
S Vaan () + urge (0) + e+ + Unk (| +f) Rn (8) + Angel | (6 —4). 
1) ie. on the assumption, that the functions w’; (x), w’, (x), etc. are continuous 
throughout the interval 7. 
