INVERSE METHOD OF DEFINITE INTEGRALS. 373 



the first series may be continued to n terms or infinity indifferently, 

 and the last term in the second series will be ^p^ when n is even, 



2 



and p„_x . cos 9 when n is odd. 



Suppose now that the product 9,u.Sn is decomposed into the sines of 

 the multiples of 9, and that all the multiples higher than the «'" are 

 rejected from this product, the remaining part will evidently be, 



— {aopo—aopi — aip2 — a»-i/Oo} .sin(n — l)0, 



— {aipo + Uopi — aopi —a„.2po}.sm{n — 3)0, • 



— {(hpo + ctipi + aop2 —a„-3po}.sm{n — 5)$, &c. 



the whole of which by the given equations is equal to zero. 



Hence, 



2S.u = A„sm{n + l)9 + B„sin(n + 3).9 + C„sm{n + 5).B, &c. ; 

 .-. 4 cos . S„u = A„ sin (nB) + {A„ + B„) sin in + 2)9 + {B„ + C) sin (« + 4)0, &c. 

 and 2Sn.iU = A„^i sin {n9)+B„.i sin {n + 2).9 + C„_i sin (« + 4) . ; 



.-. 2{2cos9.Sn-j^S,.^} .u = i^A„ + B„-A„.^\.sm{n + 2) . 9, &c. 



Hence it follows that if we put So=po, S^ = po cos 9, 



and u = aoSin9 + a^ sin 3 + a^ sin 5 9 &c. ad inf., then. 



First, Supposing S^.^ and S,n known, form a quantity \„ by dividing 

 the coefficient of sin(/» + l)0 in 2S,„u, by the coefficient of sin(/»0) 

 in S/S'm.i .u. • 



Secondly, Form a quantity S^^^, by the equation 

 -S'„+i = 2 COS0 . /y^ - X^iS*™.! , 

 by which S^, S3 a^^ may be successively formed. 



Then it is obvious that the product 2S„u contains no multiple of 9 

 below the «'\ and therefore the coefficients in S„ must be the required 



quantities po, p^, pi pn-j^ when n is odd, or p^, pi, pt ^p^ when 



2 3 



n is even. _ 



Sc2 



