( 635.) 
The greatest of all these values u; is assigned to the parameter 
u of the development (XIII) and in the point « the series for F(z) 
will have the mark wz. 
In nowise will the series obtained in this way have the lowest 
possible mark; the reasoning served only to show that for finite @ 
and for O=|=a; there are series, converging in # with a mark deci- 
dedly lower than unity. | 
This argument does not hold good for those points «, situated exactly 
between the origin and one of the singular points, let us say Ar. 
‘For that case we must return to the development of the series 
(XI). We have now still at our disposal the two parameters w and @, 
and we begin to arrange these in such a manner, that the given point z 
lies inside the region G';;, which by the transformation 
(x, Ay x) 
is deduced from the region G', of fig. 7. For this it is necessary 
to have 
NEEN) Lae 
Further we allow @ to decrease in such a way that the point x 
arrives inside the segments into which the segments of fig. 7 by the 
remaining transformations 
(z, A; 2) 
pass. In other words we put for all singular points, except the one 
point Ax, the inequality 
Nr; 
oe | sin (O—a) |. 
pst ; 
gun —_— 
9, ê 
Very decidedly this inequality is satisfied if @ satifies the condition 
370 . 
sin — < a | sin (O—a;) | « 
cs 
A finite value of « differing from naught is found from all these 
inequalities ; after that we take 
1 
B 
w=1—(1—~) 
rk: 
and we may be quite assured that the series deduced for /(#) out 
of (XI) shall converge in the points « with the mark w. 
So finally it is possible to construct a really convergent series of 
polynomials in each point # of the star. Only on the rays of the 
star the convergence becomes infinity slow and so the development 
becomes worthless. 
