196 
PEOFESSOE A. SGIIUSTEE ON SOME DEFINITE INTEGEALS, 
[ ^Q/P. = 4, 
£dll = 8 [{n + 4 . ?i - 3) - 3 (i + 2 . { - 1)], 
I Q/Pi dix = 12 [(n + G . n + 4. n — 3 . 71 — 5) — 8 (i~ 1 • + 6 . — 5) 
+ 10 (i + 2 . ^ + 4 . ^ - 1 . 7 - 3)]. 
The equations do not hold for i = n as has already been explained. 
8 . 
+ 1 
r + l 
Q'sin c/p,, Ql cos 1^9 dn. 
•1 .'-1 
These important integrals are obtained in two different forms according as or 
the trigonometrical functions are expressed in terms of the powers of sin 9. In the 
former case the problem reduces itself to the evaluation of integrals of the form 
siiff ^ sin r/p and [ sin^ (9 cos d sin dd. 
It may easily be proved that 
r+i . . r-i ’ 
siiP^sin rj9 da = (—})■ c ^ - ,, - , , when p is odd, 
_i ^ ^ ^ ' (\ +1+p)!:(\ +1-p):i ^ 
= 0 when p is even. 
a: 
I ’’ (A + 1V 
siiOdcospd./p = (-i)-c -—^ ^ ^ 
-1 ^ ^ ’ (A + 1 + p): HA -h 1 - p): 
= 0 when p is odd. 
r+l . . ’■+- 
sin^~' 9 cos 9 sinpd dp = ( — 1) - cp 
■ -1 
= 0 when p is odd. 
sin^“' 9 cos 0 cos2)9 dp = [ — 1) - cp 
- when p is even. 
.... .-when p is even, 
(A -|- 1 + p) • • (A + 1 — p).. 
-1 
(A + 1 + p):!(A i- 1 -p)! 
- wlien p is odd. 
= 0 when p is even. 
In these equations c is equal to 2 or v according as (p + X) is even or odd. 
From the results of § 7 we may now write, if n — cr be even and ^7 odd. 
