16 
inferred from it, that sometimes not a certain derivative had that 
property. This is not the case; if the characteristic 4’, defined by 
(14), of the coefficients a, in the power-series for the function q(t) 
is finite, this function always has a derivative of a certain order 
— w possessing the property in question and we have 
wd 
Further it is shown by Hapamarp’), that a function g(t), which 
is of order w with regard to the whole circumference, and of lower 
order , on a certain part of it containing the point t=1, can 
always be divided into two functions w‚(f) and ¢,(é), the first of 
which is regular at t—= 1 and of order w on the whole circumference, 
and the other of order w, on the whole circumference. According 
to the theorem proved in the same N°. 6 the integral 
1 
fo (t) 2—! dt, 
0 - 
c.q. an integral with ‚®(t) as subject of integration, can, for 
K(x) > o—1=4', be expanded in a series of factorials. Further 
the integral 
1 
fo. Oet de 
0 
c.q. an integral containing p,@%(t), can be expanded in such a series 
for R(«)>w,. Hence the integral (1), being the sum of the two 
preceding integrals, ¢.q. an integral containing pC (¢), can be expanded _ 
in a series of factorials for 
Klep so, of (2) ta, 
according as w, or 4’ has a greater value. In the first case the 
series converges conditionally for 
Bk (nadeel, 
and in the second for 
Need 
In both cases we may state the following proposition : 
Lf w is the order of p(t) along the whole circle of convergence, 
and therefore 4’ = aw—1 the characteristic of its coefficients, and if 
w, is the order of g(t) at the point t=1 of that circle, then the 
integral (1) (c.g. the integral (8)) can be expanded in a series of 
factorials for such values of « as satisfy at the same time the ine- 
qualities 
R(e) =d and Rl) > o,. 
1) „Essai sur l'étude des fonctions”, Journ. de Math., 1892, p. 172. 
