( 201 ) 
1 
C 2 (cos NEREDE 
1 
1 
l mm (2 Lys 
heee Fae 
on sin ZX 
1 2 LO, 
C (sin vy) = (— 1)" sak Sines 4 
On cos x 
and that sin @ increases, cos @ however diminishes with @, we arrive 
at the theorem: 
The positive roots of the ntt spherical function P, (x) lie hetween 
ae es 
v3 2n+1 
contained in n. 
This theorem is a corollary of the one deduced by Bruns in his 
treatise “On the theory of the spherical functions’ (Zur Theorie der 
Kugelfunctionen) published in the 90% vol. of Crelle’s Journal 
and recently proved by MARKOFF!) and STIELTJES”): 
The roots of the spherical functions lie one by one in the intervals 
Qin (2i—1)a 
SS eS OR GO arene ee 
2n-+1 2n+1 
From the preceding theorems we can easily deduce the following: 
C and cos where n, is the greatest even number 
7 
2n+1 
CO 
The difference between the greatest positive root of the function 
0 (x) and unity is less than two times the square of the smallest 
positive root of C, (a). | 
The difference between the greatest positive root of the function 
Br (x) and unity is less than two times the square of the smallest 
2v—1 
positive root of C_— (a). 
nt 
The difference between the greatest positive root of the n'* spherical 
. . c 2 nj AT 
function and unity is less than 2 cos° — — . 
2n+1 
1) „On the roots of certain equations”, (“Sur les racines de certaines équations”) 
Mathem. Annalen, 27th Vol. 
2) „On the roots of the equation Xu 0,” (wSur les racines de l'équation 
Xn = 0”) Acta Mathematica, 1X Vol. “On the polynomia of Legendre”, (/Sur les 
polynomes de Legendre”), Annales de’ la Faculté des Sciences de Toulouse. Vol. IV. 
Markorr and SrreLrses deduce in the cited treatise also the narrower limits 
lx (24 —1l) zr 
n-+-1 ts ee 
