192 
PROFESSOK A. SCHUSTER ON SOME DEFINITE INTEGRALS, 
§7. C'Q:P,dix, 
J_i 
The integral of the product of a tesseral function and of a power of sin 6 having 
been obtained in § 4, the above integrals are found by expanding the tesseral 
function or the zoned harmonic in a series proceeding by powers of sin 6, 
From the expression of a tesseral function of degree n and type cr, as it is generally 
given, viz.; 
sin"^ 0 + ...], 
it is seen by writing x = sin g = \/' I — x^, and expanding the binomials, that the 
term of lowest power will be sin'" d, and that if n — cr be even, the term of highest 
power is sin" 6. 
The coefficients of the series, which are given by Heine, are most easily obtained 
by going back to the differential equation ; 
o 
(Substituting x = \/1 — p?, and changing variables, this becomes 
n .74+1 
1 — iKY 
q:+ '"f + d 
If the series 
a^x'^ + + . . . <.1,^x1 + + . . . 
satisfies this differential equation, the coefficients and rq must be connected by 
the relation 
cq [a — q) (a -b q -f 1) + cq^^ qy _ q- _p 2) [q -f o- _p 2) = 0, 
as is seen by sulrstitution. 
Tlie first coefficient is determined by the foct that it is equal to the value of 
when g = I, This quantitv is known to be equal to 
(» — 0 -): (ZctF: 
other coefficients may now be determined in terms of this, and we find in this wav 
for Q;^ the series 
I _ fa — (r){n + a -f 1V-rH i ( ~ o')('a — cr — 2)(a -f cr + !)(;/ + cr + 3) Ar 
(71-0-)! (2cr)'’L 1 cr + 1 \2/~‘ 1.2 cr + l.a + 2 * 2 ' 
The series breaks off with the term when n 
holds if — cr is odd. 
The ffictor of .H in the series reduces to 
“ cr is an even number, but also 
