( 622 ) 
Now here the question presents itself, if it be possible to bring 
a given point # within the region of convergenee by an appropriate 
choice of the parameter a. It is immediately seen that this question 
is to be answered in the negative, when « is a point of the prolong- 
ation of the radius vector of one of the singular points 4;, i.e. ot 
one of the rays of Mrrrac-LerFLER's star. Furthermore a geometric 
construction can give a complete answer. It is necessary to construct 
ares of circles passing through any pair 4;, Ar of singular points, 
the prolongation ot which passes through the origin. By considering 
this figure it will be seen that a development into a series convergent 
for a given point « is possible or not, according to the radius vector 
of this point intersecting none, or one or more of these ares Aj, Az. 
In such a manner it can be proved that the development (VIII) 
is to be made convergent in any point a, if all the singular points 
A; lie on a right line through the origin. In particular this result 
can be applied to the series (VII) representing tang! x. The para- 
meter a can be determined in such a manner that the double circle- 
segment of fig. 3 includes any point 7 
5. We have but to make a slight modification in the development 
(VIII) in order to find certain other regions of convergence, which 
presented themselves in the investigations of BOREL. 
Given F'(e) by means of a power series, which can be decomposed 
into ” various power series in 2”. We put 
Bij Eet 
These new functions pz (e) ee not only the singular points 
A; of F'(e) in common, now the new singular points 
2mi 4m i 6d 
Aret Are AN, 
also make their appearance. Other singular ane fee are none. 
We again take the function 
a—1 
g (u) == eaf 
(G4 
1 
and consider pz (eg (u)" ) as before we did F(«g(u)). It is only seem- 
1 
ingly that gx (« g(u)") is not holomorphic in u=0; gx (2) being in 
1 
reality a function of 2 we can treat gr(eg(u)" ) as we formerly did 
F(x g(u)). 
For pr (ez) we now find Belen of the Senne (VIII) 
mao | (An 
pr (7) = px (0) at > — = (m—1)j;—1 ee de ) 
=] qm 
gin (a— | yA, 
