( 307 j 
Now, always supposing 2 to lie inside the region Gp, it is evident 
that Z, (a/z) is related to f(z) in the same manner as 
n 
=o y, 2k a" 
EO (q/z)=> 
is n=0 n 
aes) 
P 
is related to 
7 Ga 
ak f (2) = ty e® 5 
n=O 
hence we may apply equations I and II and conclude that 
Lim e-« BO Gis) ii 
a=+ 0 P 
| ee DÛ (aja) da = kf(@). 
0 
So it appears that we have 
| Pp (a/z) al Oey 7 
or 
ee ee de tE 
r(= +1) 
P 
an equation wholly equivalent to the original formula of BOREL. 
Cases may occur in which the formulae I, II and III established 
in the foregoing have some importance. For, in asking for the 
value of f(z) in a given point z, it may happen that this point 
lies outside Borer’s region of summability G, and that we are able 
by a proper selection of the integer p to find a region Gp, wherein 
z is contained. In that case we can replace BoREL’s equations by 
the formulae II and III, the application of which presents scarcely 
more difficulties than that of the formulae for the region Gj. 
Finally we may remark that equations I and II still hold if p 
is an arbitrarily assigned positive number. For rational non-integer 
values of p however, the extent of the region of summability G, is 
considerably reduced, and for irrational values of p the region G, ulti- 
mately coincides with the circle of convergence, so that the summation- 
formula II is no longer of any use. 
