GENERAL THEORY 
53 
2. The following example is instructive as showing that when 
the conditions imposed in 1 are not fulfilled, the relation 1) may 
not hold. 
Example. Since ^ ^ 
I — =+0O, 
x 
there exists, for any 6 n >0, a 0<6 n+1 <5„, such that if we set 
C bn dx n 
) — * x n+1> 
then 
as b n = 0. Let now 
b n+1 37 
0^1 < 0^2 < "• == °°, 
/ = 1 for the rational points in 2Ï = (0,1), 
= - for the irrational. 
x 
ß 
Then 
f /=1 
Let 
tb 
II 
■31« 
*-< 
Let A n denote the points of 5i in (h n , 1) and the irrational points 
in (ô B+1 , 6 n ). 
Then C n 
I > ^n+l = + 30. 
^L A n 
But obviously the set A n is conjugate to On the other hand, 
J> =1 ’ 
lim f f = 
n = 0o 
while 
+ QO. 
56. If the integral 
converges, then 
If 
J 21 
e > 0, o- > 0, i* f 
l-K® 
for any unmixed part $ of 21 such that 
<€ 
(1 
(2