12 
On account of the supposed finiteness of s, 4, the limit of the 
first term of the last member is for Zr) > R(c) equal to zero, if / 
increases indefinitely and the series in that member converges abso- 
lutely. Hence the series in the first member converges also and 
we have ; 
bas ml! am A ae Snm T'(e + mt 1) 
>" Il + m -+ 1) im (ee) Dn IME + m + 2) ij ; (45) 
n rn 
by which the lemma is proved. It follows from it that the limit 
between the domain of convergence and the domain of divergence of 
a series of factorials is formed by a straight line parallel to the 
imaginary axis. 
Moreover we may infer from the equality (45) the following 
lemma: 
I]. In any finite part S of the half-plane R@) > Ric) + d, 0 being 
a fixed, but arbitrarily small real positive quantity, the series of 
factorials converges uniformly. 
For nm may be chosen so large, corresponding to any arbitrarily 
given e, that for all m>n 
Sim T(e + m + 1) é 
(«+m TT COEN 
and thus 
nen Ny en F(c +m pated & | a—e | (46) 
| wom (re -+ m + 2) R (wone) © 
Since Mlv—e)>d, the right-hand member of this inequality is, 
for a sufficiently large n-value, which is independent of the value of 
xv in the domain S, less than ¢. This was to be proved. 
From the inequality (46) we may infer the corollary Ha: /f the 
series of factorials (6) converges for a certain value «= c, then it 
converges uniformly on the half-line beginning at x=c and 
having the direction of the positive part of the real axis. 
For on this line R(w—c) = «—c; hence the right-hand member 
of (46) passes into e:nRe-e) and this is less than ¢ independently 
of the place of « on that line. The corollary is an analogon of a 
well-known proposition of Agen on the uniform convergence of a 
power-series along the radius of a certain point of its circle of con- 
vergence, if that power-series converges at that point. 
From lemma II, in connection with the fact that the terms of the 
series of factorials (6) are continuous functions of «, it may be inferred 
that such a series represents a continuous function of 2 in the domain 
