Cte 
between + 1 and — 1 and — apart from the root 0 appearing in 
the case of an uneven n — have in pairs the same absolute value; 
finally that the roots of C (2) and C -@) as well as those of C () 
et: 
C _{@ mutually separate each other. 
n 
In the following lines some new theorems on the roots of these 
functions will be found in a highly simple manner, one of which 
including as a special case a well known theorem of the theory of 
spherical functions. 
1. From the addition-theorem of the functions C (x) arrived 
n 
at by me 
9 11 (n+-v—1) 72 
by Lea +Y (l—a") V (la) cos p] = L(2v—2) ar ae 
a 2v—1 
| 2y+2¢—1 AURA: 
DN NEHAP 6, (x) 6, (zi) se (cos p) , 
p=0 
(L> 2, Oy am 1) 
where the square roots are taken positively and 
i—o) Tv +o-—1 new 
oe et tE a 
2n—p [1 (n+ vy — 1) nk 
Cay ON 
we find the relation 
ah 
| C [ea + f(l—2?) (1e?) cos p] sin?’—! pdg = 
: 9-1 FIT (vy — DP U (n) 
a> ioe AE EE C (x Cle 4 
=D CO oe 
. Nt / 
By putting 2, equal to a positive root <, of the function © (+) 
the equation is transformed into 
kul 
fe [ren + YY (1—2?) (Len) cos p] sin” -lpdp=0 , 
0 
- . v . 
showing that the function C (z) vanishes at least for one value 
of its argument lying between wend (1-2?) (len) and 
wan (1—2?)  (1—en?), as otherwise the function to be integrated 
