( 198 ) 
would not change its sign in the entire region of integration, and 
hence the integral could not be equal to 0. This value certainly 
differs from z, when 
2n > dan +//(1—a*) Y(1—z,?) , 
which can only be the case, if 
ac2z/°—1, 
and this leads, in case « might also be positive, to the supposition 
> 1 
& =e 
n We? 
The entire interval under discussion being a positive one when 
w is taken greater than py/(l—e,°), we find the theorem: 
If zn be a positive root of the function GC: (x) surpassing 1:2 
and a a positive root lying between |\/(1—zn)| and 2 2,2—1, then 
there must be in the interval az,—\/(1—a?) V (1—2n2) to 
wen +V (l—a@*)y (L—2n*) at least one other positive root of this 
function (smaller than zn). 
A corollary of this theorem is the foilowing: 
The smallest positive root of the function C (x) is smaller than 1: v2. 
y ‘ . . 
2. In my paper “Some theorems on the functions C (#)” (“Einige 
n 
Sätze über die Functionen C (x)”’) contained in the 47 vol. of 
“Denkschriften der mathematisch-naturwissenschaftlichen Classe der 
Kais. Akademie der Wissenschaften in Wien” I have given the four 
following equations: 
2yv—1 
Cae ni - ) . 
C (cos x) == fico g — cosa)¥-! 
n x 
ava MI (v—1) sin?v—! 5 0 
