1230 
by means of which g, (z), as soon as z along the radius tends to 
ani d: 
e 9 from the outside of the circle, is expressed in p, EE 
The rational points on C thus having been recognized as singula- 
rities of g, (2) and p‚(z), we now must turn our attention to other 
points on the curve, and as such | will consider the points e2/%, 
where § is a root of an irreducible algebraic equation of degree 
u>1 with integer coefficients. Evidently these points e?#ë which I 
will call the algebraic points of order u on C, determine a new 
enumerable set, everywhere-dense on the circle. 
Let z=oe?"%, then it is readily seen that for all values of @ 
Sl zt if cos 22nd << 0, 
——] 
zn 
1 
ee 1) > |sin dang we Mh COS Arnen EE 0; 
zn 
Now in the latter case n& is an irrational number increasing with 
the index n, hence there exists an integer 4, such that |nE—k|< 3: 
But, as cos 2u(ns—k) = cos 2unE > 0, we must have |nS—k|< 4+ 
and sin 2u|n§—k| being the sine of an acute angle is greater than 
2 
the angle itself multiplied by —. 
Jt 
Therefore, if cos 2mn§ >>0, we may write 
|sin21né|=sin 2a | n & —k| 
and 
Now according to LrouvirLe's known theorem about algebraic 
numbers, we have 
ä k 
jn 
n 
where M is a finite number independent of n and only depending 
on the coefficients of the equation of which § is a root. 
In this way we conclude that 
kiten 4 
BD | get 
1 
zs Mnt' 
and consequently that we have for all values of e = | z| 
i zn | 1 
a ve if ‘cos Za nE 0, 
ns 1|—zn 
vi ji, cos peor nd > 0. 
ns 1— zn 
