SECT. II.] ELIMINATION. 251 



but the quantity J cosj (ft i&amp;gt;) q furnishes equal terms, each of 

 which has the value ^ ; hence the sum of the products term by 

 term of the two first series is i n. 



In the same manner we can find the value of the sum of the 

 products term by term of the two second series, or 



S (cosjvq cosjvq) ; 



in fact, we can substitute for cos jpq cosjvq the quantity 

 J cosj (fj, - v) q + % cosj (fjb + v) q, 



and we then conclude, as in the preceding case, that 2 Jcos j(^+v)q 

 is nothing, and that 2,-J cosj (/it v) q is nothing, except in the case 

 where //, = v. It follows from this that the sum of the products 

 term by term of the two second series, or 2(cosj/j,qcosjvq), is 

 always when the arcs u and v are different, and equal to \n 

 when u = v. It only remains to notice the case in which the arcs 

 fiq and vq are both nothing, when we have as the value of 



S (sinjfjiq sinjvq), 



which denotes the sum of the products term by term of the two 

 first series. 



The same is not the case with the sum 2(cosj/^ cosjvq) taken 

 when /j.q and vq are both nothing ; the sum of the products term 

 by term of the two second series is evidently equal to n. 



As to the sum of the products term by term of the two series 

 sin Ou, s mlu, sin 2u, sin 3u, ... sin (n 1) u, 

 cos OM, cos lu, cos 2u, cos 3u, . . . cos (n 1) u t 



it is nothing in all cases, as may easily be ascertained by the fore 

 going analysis. 



270. The comparison then of these series furnishes the follow 

 ing results. If we divide the circumference 2?r into n equal 

 parts, and take an arc u composed of an integral number p of 

 these parts, and mark the ends of the arcs u, 2u, 3u, ... (n l)u, it 

 follows from the known properties of trigonometrical quantities 

 that the quantities 



sin Qu, sin lu, sin 2u, sin 3w, ... sin (n l)u, 



