18G 
PEOFESSOK A. SCHUSTER OX SOME DEFINITE INTEGRALS, 
To find the values of the coefficients of such a series we put 
cos crcf) = + Aa 2 -fff +2 + • • • A^Tj^ + • • • 
and integrating over unit sphere, after multiplying as usual by Th we find 
Cf A- A = a; f;’| ;;(T:)^’ ^ K 
Substituting on the left-hand side the integral found above we obtain 
1) !! (a - 2) ! ! ^ -A+J - <r - 1) !! (?i - 2)! ■ 
(a + <7)! {ii — cr) '' ('/t + 1) !! 2 (?!- + O') !! -f 1) !! ’ 
. ^ 2?i + 1 (n — cr)! {n -f cr 
A„ = c‘cr-:-. -r- . cr 
where c is 2 or tt accordina; as cr and n are hotli even or both odd 
AVe obtain, for instance, in this way for cr =: 2, 
9 
" ~ 1 . 2,3.4 ^ 
,4.5.0 
^A^ + 5.o'v.-S-'A+.. 
§4. f Q:siidddp, 
* — i 
r +1 
pQn siid^c//x. 
If we put p = cr -f X A 2 in ecpiation (G), and after changing from n + 1 to «, 
integrate both sides, we obtain ; 
• + 1 
siih 6 (A da = 
4? + o- — 1) (n — A. — 2) 
or, ix — 
. _i ^ ^ (n — a) (ii + \ + 1) 
siffi 6 QI _2 clfx 
J -1 
(■« + cr — 1) (« + 0 - — 3) (71 — X — 2) (?^ — X — 4) 
(^)l — O') (?!/ — <T — 2) (7i. + X -j- 1) ('/i. AX — 1) . 
+ 1 
-1 
sird d QAi dp. 
If n ~ & he Olid, a continuation of this process reduces the degree of the tesseral 
function on the right-hand side until it becomes smaller than cr, and tliis will happen 
before anv of the factors m the denominator become zero, so that in that case the 
integral on the left-hand side is zero. 
If 11 -- cr he even, the integral on the right-hand side ultimatelv becomes 
f^‘ sild e Oil clfx = (2a- -1)1! 0 dd = (2g- - 1)! 1 ^ c, 
J-1 Jo ' 
(X A c + 1) ! ! 
where c is 2 or tt according as X A cr is even or odd. 
The integral to he deteiinined now becomes 
+ 1 
I 
- 1 
sin^ 0 da 
u- — a he even) 
« — Ad: [l'‘ - X - 2) - X - 4) ... (cr - X)] (if 
(?t - (7) !; (a A X A 1) :! ^ ^ ^ ^ 
U, if 11 — cr he odd. 
