( 632 ) 
can always be made to converge with the mark unity in every point 
of the k-plane, if only this point is not situated on one of the pro- 
longations of the radii vectores drawn from the origin to the singular 
points. 
If we wish to include in the region of convergence all the points 
of MirraG-LEFFLER’s star, we must allow ? to approach to naught; 
in this case however we see immediately out of (XII) that all terms 
of this series become infinitesimal, so that in each point x the con- 
vergence with the mark unity becomes infinitely slow. 
7. To this result we will add another and no less important 
one. The supposition 4 = 1 could furnish series only, which were 
convergent with the mark unity; we shall now show that series can 
be formed which converge with a lower mark in a given point 
«= eO, if only G is finite and the argument 9 gives with each of the 
arguments A; = 7; ei a difference with a finite sinus. For this pur- 
pose we make $ approach to naught and obtain in this way the 
auxiliary function 
If it regarded the development of 1:1—z, then the form of the 
regions Gs, G', and Gy would still be about the same as was given 
in fig. 7. In the meanwhile together with / the original quantity 
N=1—(1—w)* has disappeared. The limitation of G' has no longer 
an angular point but now a cusp in «=0. The state is that of fig. 9. 
: ; 5 : : etek 1 
Inside G'; are now two entire circles with the radius — log ee 
1% — u 
and the series (XI) has passed into 
m 7 
1 m= h=m 2 h ae logit 
facade MR ae Goat DEE 
jhe Ties “) it \log(1—p) m ! ( ) 
Putting 
DI logkt 
er == Em» 
m 
then 
i=! 
Em, becomes ee , 
me 
and we have again 
