865 
found in the fact that the point where, on the circumference of (a), the 
magnitude a, assumes its upper limit value, does not coincide with 
the point where the greatest modulus of the function w and its 
derivatives are found. It will be clear from the considerations in 
N°. 4 that the lack of this coincidence may give rise to cases as 
considered here. 
But it is not the only possible cause; even if the functions «,, (2) 
and the function « with its derivatives have their greatest modulus 
in the same point of a domain («), such a case may sometimes 
arise. Let a transmutation be given for which, « being a real positive 
constant 
Bn Wy == et, if Ten Be A 
ante) == 0, te m == Z2nl 
Here the function «, is equal to the constant c. Consequently the 
upper limit @ of a, in a domain («) is equal to c, and 
gz=a te. 
We take again «—O as centre and consider the function 
DD 
aN es ohms Se he rks ME) 
ro 
0 
which, just as its derivatives, has its greatest modulus in the point 
x—=a of the domain (a), so that, since a, assumes its upper limit 
value c in each point of (a), the coincidence referred to above takes 
place here. The series (1) passes in this case into 
au a en ym 
Tu = Sn ——, m= 227-1, 
4 Is 
1 
Now we have for m == 2?"-1 
| Pd p = Sam 
Um (e) | << | WMO | < De(; Z 
since the series representing the first member of this inequality consists 
of part of the terms of the series in which the last member may 
be developed. 
If in general 
we have for real positive « 
y k al (hi kb)! ) 
kl (Lak ALR 
') By means of the formula of LereNiz we find 
