1231 
Therefore the series of LAMBERT F (2) converges absolutely on the 
radius of the point e*5, as soon as s >u, the convergence being 
then independent of @ and uniform on any segment of the radius. 
Supposing z to move continuously along that radius, the value of 
the analytic expression F(z) which for s< 1 is equal to that of 
the funetion , (z) changes also continuously into the value of the 
function , (z) as soon as @ becomes greater than unity. Besides, if 
s is taken sufficiently above the number u, for instance, if we take 
s > 2u—1, the series obtained by differentiating term-by-term the 
series F(z) with regard to o in exactly the same way will give the 
dp,(z) dp,(z) 
dz dz 
order of thought we may ascribe to the functions g, (z) and g, (z) 
a common definite value at the point e?**, though of course that 
point is not an ordinary point. Making wp, (e?**) and g, (e?*%) both 
equal to the finite limit Lim F (ge), we obtain 
eI 
value of or that of according to the value of g. In this 
z 5 cot ans 
DE), C)=—35) + 5 
= ns 
and the series Fat hin 
n==1 ns 
Hence, we have established a certain connexion between the 
functions g, (z) and p,(z) which according to Wererstrass’s theory 
we must regard as essential distinct and in no wise connected. In 
fact, we have shown that in this very special case in which the 
classical continuation by means of power-series is impossible, a new 
kind of continuation, as complete as could be desired, is furnished 
by the series of LAMBERT along the radii of an enumerable infinite set. 
The question arises, whether cases exist in which the continuation 
by means of a series of LAMBERT is effected along the radii of a 
set having the power c of the continuum. The answer is in the 
affirmative, we only want to choose a LAMBERT series the coefficients 
of which are decreasing more rapidly. For instance [ will consider 
the series 
will certainly be convergent, if only s >u. 
n—o | gn 
re fh n=1 nt 1—z" 
Again in this new series the coefficients are positive and zero is 
their common limit, hence according to the proposition mentioned 
in the footnote on p. 1228, the rational points on the circle C are 
singularities of the analytic functions w, (z) and w, (z) defined by 
G (2) inside and outside C. 
