200 
PROFESSOR A. SCHUSTER OX SOME DEFINITE INTEGRALS. 
and p being odd : 
sin pO dp = S''''’ ( 
-1 ’ A=1 
1)" C, f Q; sin‘0 c?,., 
o r+1. 
Qh cos jjd dp = (— !)■ I Q;; sin^d cos 6 dp. 
A = d—1 
Hence from the results of § 4, if p be even and (n — cr) odd : 
J-1^' ..1 ^ (Ji - tr-I):; (it + X+ X - L>);.- 
Tlie constant c is 2 or rt, accordino; as a is odd or even. 
If n be odd and n — cr even, and witii the same meaning: of c : 
cr* 
I sin pO dp = c ( - I ) 0, 2)!!(c7 + X)!! 
J-i ^ ^ x=i ' ^ {n _ cr ) ': (7Z + A + ]) ;: (cr - A - 2) ; ■ 
Similarly if p and n — cr are both even, 
f Q.:: cos pO dp - ^ 1 ^ ^-2)^(cr^)!! 
\=p 
c N-'-’' /„ i\--f n 
\Zo ■ ’ "(a-cr):;(n + A+ l);:(<r-A-2)!:’ 
and if 'pj and ?i cr be both odd, 
cos p9dp = c (- 1)"^^ B, - + —N)iL(.^±^)_lL 
—1 A=o {'ll — cr 
■i):i (7i + A + 2)!! (cr - A - 2):: ■ 
In the two last equations c is ecjual to 2 when cr is even and equal to tt when cr is 
odd. In the interpretation of the summation a little care is required when cr—X—2 
and n ~ X — 2 or ?i — X — 3 are negative, as the alternate factorials are infinite for 
even values of the argument. 
To put the ecjuations into a form useful for numerical calculations, it is convenient 
to designate by separate letters the following six series :— 
(T ~ o . a 
1 . cr + 1 . er + 3 
+ a 
TZ — 0 . -f- 4 
cr — 5.cr~3.cr — l.cr + l.cr + 3.cr + 5 
71 — 3 . ?i — 5 . + 4 . ?i + G 
= B,. 
cr 
1 . cr -fi 1 — Bj 
cr — o . a 
1 . cr “f- 1 . cr T c> 
B= 
?z — 4 . ?(. + 3 
cr — 3. cr — 3. cr— l.cr + l.cr + 3.cr 
7t — 4.71 — G . 71 + 3 . 71 + 7 
T-ff — rf _ O' — 2.cr.cr+2 , ^cr — 4.cr — 2.cr.cr+2.cr + 4 
U,, r=Vio.cr —" Lo r, , o “T Uj 
7Z — 2 . 77 + 3 
77 — 2 . 71 — 4.7Z + 3 . 77 + 3 
