198 
PKOFESSOR A. SCHUSTER OX SOME DEFINITE INTEGRALS, 
successively the values o-, cr 2, &c. The highest value which X + 2 — p can take 
is -h 1 —and this is negative when p > « -r 1. It follows that 
r+l 
sin pd d/ji — 0 when p > ti I and cr is even. 
J-i 
The restriction that p is even is not necessary, because when p is odd n will be even 
and the factorial in the denominator, according to the above equations, is {X-|-l— p) \ 1 
This again will be infinitely large whenever it is negative, or taking the highest 
value of X, which is now n, whenever p '> n 1. 
We prove exactly in the same way that 
r+l 
Qfj cos p>d dfji = 0 when p > li fi- 1 and cr is odd. 
The value of the integrals in the general case may conveniently be expressed by 
two series and Eo defined by 
(71 — p + 1)!! (n + p + 1) '! (n — cr) ! ^ (n + a) (?i — cr) (n + p + 1) (n — p + 1) 
+ 
■V rz: 1 _ 
(2?i - 1)!! (71 + 1) ! “ 2 . 71 + 1 . n . 2?i - 1 
(n + a) (71 + a —2) (n — cr) (n — cr — 2) (n + p + 1) (n + p— 1) (71 —p + V) (;n — p— \) 
2 .4 
71 + 1 . . -71 — 1 . 71 — 2 
271 - 1 . 271 - 3 
(?1 — p + 1) 1! (71 + p + 1)!: (ti — 0 -)! 
E, =1 
(n + cr — 1) (71 — O' — 1) (ti + p + 1) (71 — ^7 + 1) 
+ 
(2?i — 1) !! 71! p " 2 . 71 . 71 — 1 . 271 — 1 
(71 + cr —I) (n + a — o) (71 —cr —1) ( 71 —cr —3) (7i-|-p + l) (?7 4-p —1) (ti— p + I) (ti— p —1) 
71 . 71 — I . 71 — 2 . 71 — 3 
271 - 1 271 - 3 
Then 
r+l _ «-5+r-i 
sin 2^0 d+ = {— \) " c Ej if p be odd and n + cr is even. 
•'-1 
n —o-T-p+ 1 
= (— 1) " c E^ if 2 -* be even and w + cr is odd. 
= 0 whenever ?i + 2^ “b even. 
I cos pB dp = {— ]) " c E^ if p be even and ti + cr is even. 
^ —1 
» —<r + p+2 
= ( — 1) ' c E^ if p be odd and ii + cr is odd. 
= 0 whenever « + p fi- cr is odd. 
The constant c takes the value 2 or tt, according as p + n is even or odd. The 
series Ej and E^ break off iu all cases ultimately wlien one of the factors of the 
products (71 = cr) . (71 — cr — 2) . . . or (t? — cr — 1) (n — cr — 3) . . . become zero. 
When n — p is odd, the series may break off before it has reached its full number of 
