422 
for OS Sa and has a finite derivative f (O) at 0, and if f(a) for 
O<a<a converges to fw) and the sequence f (0) is convergent, then 
a necessary and sufficient condition that it should be permitted to 
“differentiate term-wise at 0” i.e. that lim f° (O) should be f'(O), as: 
n= 0 
the quasi-uniform convergence of the sequence of functions 
f@ —f,O 
A) 
U 
at the points of every set «‚‚v,,.... where lima, = 0. 
k= 
i a 
The function f (we) = if = converges everywhere to zero, hence 
MW 
JO) = 9, 
& Leg OSL, WD En 
Nen ’ En NI ars 5 ne 
n (e) Ltn?n? > "0 L+tn?a2*? 1+n?y* 
w (x,y) everywhere converges to a continuous function, 2. €. quasi- 
uniformly in every finite region so that the criterion given in § 3 
is satisfied. Accordingly /' (QO) is finite. However, if we choose 
é<'/, and the number of indices is finite, then for every « which 
is small enough we have for each of these indices Fe (a) S77 ,, 
whilst for «40 we have lim f° (v) = 0. The criterion, enunciated 
n= 
in this § is therefore not satisfied. Accordingly we have 
lans HO = KEE 0). 
„== 00 
The foregoing theorems can be immediately extended to the 
complex plane and are capable of analogous proofs. 
HI. 
5. Let there be given a series of functions f, (2), f,(z),..., each 
of which is analytical in the interior of the circle |z| <{a, and 
convergent within this region. We shall now demonstrate the follow- 
ing theorem: 
A necessary and sufficient condition that the limiting function f 
should be analytical within |\z|<a is the quasi-uniform convergence 
of the sequence of functions 
(ye) fn (a) + (@ —9) fn (2) + (2) fn 9) 
(ez) (y—2) 
Wr (@, yy 2) = (1) 
at the points of every closed and limited set V (a, y, z) having no point 
an common with the sets (@== 2, y Zend De gg: 
