26 
from which it follows that w («), precisely in the domain (20), satisfies 
the inequality (22) with, for /, the inequality (23). In this manner 
we may prove the same inequality for the intervals (0, ®)\- (1; 2) .+- 
of 2 in succession. 
If w (x) satisfies the inequality (22) for a certain value of /, then, 
evidently, for all greater values. Thus there is a lower limit /, for 
all such values, but this may possibly not be substituted for / in 
(22). Instead of this we may however write 
ale) (ebb) at —G), Soe ee 
with the meaning that (22) holds for any />/,; we may call (24) 
an equation of equivalence and say that w(x) is equivalent to the 
right-hand member of this equation. The exponent /, satisfies the 
condition 
Tae me ARA Rr /2) 
since Ò and d, were arbitrarily small. The proposition relating to 
the necessary condition for a function to be expanded in a binomial- 
series may thus be expressed in the following manner: 
A binomial series of the form (A) represents in any half-plane 
Riv) 54+. 4, differing arbitrarily little from its domain R (x) >4, of 
absolute convergence, a function w (x), which satisfies the equation of 
equivalence (24); the exponent 1, satisfies the inequality (25). 
If, now, this proposition is compared with that relating to the 
sufficient condition, then, to begin with, we find a complete accord 
between the majorant valaes (7) and (22). These majorant-values are, 
therefore, both necessary and sufficient. Further, as regards the 
domains of validity, the inequality (9) here becomes (a) > +}, 
since we had 8=1, or we may also write 
R(w) >1, +4 
if 7, is again the lower limit of the /-values which may be taken 
for the given function. From (25) the same inequality follows with 
regard to the domain of absolute convergence. Since the domain of 
possibly conditional convergence extends at most over a strip of 
unity-breadth on the left of the domain of absolute convergence, the 
investigation performed by us leaves room for the possibility that a 
binomial series sometimes represents a function satisfying the con- 
dition (24) also in a strip determined by 
L—- tke) +3 
or in a certain part of it. In order to come to certainty con- 
cerning this point, we should have to examine how a function repre- 
sented by a binomial series behaves in the domain of conditional 
convergence of that series. To such an investigation we have as yet 
