494 The Rev. Brice Bronwin on some Definite Integrals. 

 changing n into n + 1, we find 



/*— *(siiuc) m cos«ar= }~ l )l A m {(z-m) n -H(z- mf}. (7.) 



U x nK ' 2 m + 1 P(«— 1) 



The second member of this will not allow us to express 

 particular examples so simply as when n = m —2 i, nor perhaps 

 can we apply summation. 



/*dx ... 

 Let t? = / - — (sin x) m sins x\ we easily find that 2y* 



> n *J X J m,n 



= A.v m _ 1>n ; and therefore by (1.), 



A -<- 1 i,»° gw - ( ip^, 1) A»{( g -m)- 1 g z _ w }. 



Integrating and changing a into z + 1, m into ?w + 1, there 

 results 



•fcj = 2-K-d Ara {(s " M) " _l , *-- ) + c - 



But as one of the quantities m and n must be odd and the 

 other even, the last half of the terms of A OT ( — m) n ~ 1 will reduce 

 to the first, and therefore the last half of A™ {(— m) n ~ l S_ m } 

 to the first with a contrary sign. Consequently 



C= j" 1 *' ■ A m {(-m)"- 1 Q_J = o. 



2 m P(n-l) v ' 



Hence 

 f% (^n *y sin ** = J~^_ {) A» { («-»)- 1 fl,..,}. (C) 



By adding and subtracting -J- A m («— m)"-^ A m ( -w) n_1 

 in this case, 



The last term or correction is zero, except when m = n — 1, as 

 it may in this case be. We may here also change x into 

 (—2), then 



where n—rn—- (2z — 1). If s = or > tw, or if z— or Z_ (— 7w), 



v z =0. 



As particular examples, m > w — 1 ; 



