27 
not arrived; but we may already perceive that the result could not 
fill up the gap which, as regards the domains of validity, exists as 
yet between the necessary andthe sufficient condition. First: if a 
function w(x) satisfies the equation of equivalence (24) for a fixed 
value of a on the circle (1,1) and for a certain minimum-exponent 
/,, then, on account of what has been remarked on the expression 
at, immediately after formula (10), that function satisfies the same 
inequality, when the number a varies, in the specified mode, together 
with the argument yw of «. The index /, cannot, however, be dimi- 
nished, because it must at all events be taken for w— —a, if ais 
the argument of the original fved number a. The statement belonging 
to the inequality (9’) informs us, however, that in this case expan- 
sion of w(e) in a binomial series is possible for R(«) >J/,. The 
function 
Eee fg eo 
(5) 
0 
for which we have a= 2, /,=0, affords an illustration of this fact, 
for the expansion is really valid for R (#) > 0, and it is conditionally 
convergent for R(«)<1. Therefore we can never find R (#) > /,+ 4 
as a necessary condition whereas our theorem concerning the sufficient 
condition only says that expansion is possible in the domain defined 
by the last inequality. 
Secondly the last condition only holds in case w(x) has no singu- 
larities in the finite part of the domain A(x) > l, + 4; for otherwise 
for the latter domain the one must be substituted where w(2) is 
regular and that was defined by the inequality A (a) > y. 
Thus the proposition regarding the necessary condition states that 
for points in the domain of absolute convergence of the given 
binomial series we have f(x) >/,-+ 4, but conversely it is not 
true that in the domain determined by this inequality there is 
certainly absolute convergence. A simple example is furnished by 
the function 
LO aes 
For this function 7, = — oo, and yet the function can only in the 
domain of regularity R (7) >>0O be expanded in a binomial series of 
the form (1). 
From these remarks it will be clear that in order to fill up the 
gap existing as yet between the necessary and sufficient conditions 
we must give more specified propositions for both conditions. In 
