11)4 
PROFESSOR A. SCHUSTER OX SOME DEFINITE IXTEGRAES. 
it will be negative because cr is the smallest value of X and cr is larger than p. 
l/(/3 — X — 2) !! must therefore be zero, and all the terras of the sum vanish unless 
there is a negative even factorial in the numerator. The only factorial which can 
become negative is (i — X — 2)!!, but n being the highest value of X, it will not be 
negative when i > n. For the special case, % — n, all the terms of the sum vanish 
except the last, for which \ — n. We have in that case 
{i 2)\\ (i -n-2)'.'. _ (-2)!! 
\p- (p-n-2)\ \ ■" (p - n - 2)!' 
and the integral reduces to (— 1) - 
2 (n +_p)J 
2n + 1 (n — cr)! ’ 
as already found in § G. 
The same expression is found when n — cr is odd. 
Another and generally more convenient expression for 
expanding Qf instead of in terms of powers of sin 6. 
n — cr and i — p are both even. 
r+i 
j QnQi’c^/^ is found by 
We thus obtain, when 
dp = cl<^^{-iy 
A=p 
P( n-r-a-l)’:(i + p)]](i + X-l):](n-\-2 )ll 1 _(y + X)\\ 
(n-a)!!(i-p-l)!ld-X)!l(n + A+ 1):' -p)\:{X + p)':! (cr-X-2)::’ 
and when n. — cr and t — p are both odd. 
1 (c r + X )!! 
{n-a-iy.i(i-p)[i{v +x + 2Y:(i-x-i)'’ {x-py.:{x-i-p)':. (a-x-2y:: 
= -1 ^ 2( 7t-X-3)!!(r + X)!: 
A = p 
Writin CT 
o 
* _ {5 + p ) (<r + p — 2 ) . . . {cr — p + 2) ( a — p) 
(2p):] 
A,= 
(g- + p + 2) (cr + p) . . ■ (g- — p) (q- — p — 2) 
2:!(2p + 2)i: ’ 
(cr + p + 4) (g- + p + 2) . . . ( cr — p — 2)(cr — p — 4) 
4'I(2p+’4)!: 
we may put the integral into the following form : 
J if — 0-) even, {i — p) even, and excluding n = i 
^ ( n-p-2)y ^ 
(i — p) ! [ n — a)':l (a + p + 1) ! ! '^’ 
where 
V _ A _ A + P ) : \ (i + p + 1) (i + p + B) (?- p) (t-p-2) 
'-(■/i-p-2) («+p + 3j "^-^(„_p_2)(,i-p-4)(« + p + 3)(7i4-p + 5) 
where 
c = 2, if cr — p is even, and c = tt, if cr — p is odd, 
