( 200 ) 
ye 
J (os Pp — CO8 yn)! B: ‚(sin 5) 5 rs dp =25 
se 
2y va Pp 
f cos Yn — cos g)¥—! C,, (cos sin % dr, 
"yn 
Yn 
2y—1 
J (os GF — COS Ya)*—! C, ob: (sin =) sngdp=0, 
. 
0 
as 
7. —1 
p 
| (cos yn — cos gp)?! C. of (cos * 5 +) sagpdg=0O0, 
Yn 
. . . . ad 
which relations show that the functions +8 (sin =) and C, (cos 2) 
vanish at least for one value of p within the respective interval 
of integration. This gives rise to the following theorems: 
The smallest among the roots of C (cos x) lying between 0 and 
2y 
is larger than the smallest of the roots of C, ( cos =) ful- 
filling the same conditions and the greatest among the above named 
roots of C (cos x) is smaller than the greatest among the roots of 
2 
C „(cos a belonging to this region. 
The smallest among the roots of C (cos x) lying between 0 and 
Dy 
> is larger than the smallest of the roots of C fs (sin =) ful- 
filling the same conditions, and. the greatest among the above named 
roots of C (cos x) is smaller than the greatest among the roots of 
2v—1 
C (cos = ) belonging to this region: 
Qn+i 2 
By putting in the first proposition v equal to } and by marking 
that 
