SECT. II.] DETERMINATION OF COEFFICIENTS. 249 



tions thus multiplied, we eliminate all the unknowns, except 

 that which is required to be determined. The same is the case 

 if we wish to find the value of B i+l ; we must multiply each 

 equation by the multiplier of B i+1 in that equation, and then take 

 the sum of all the equations. It is requisite to prove that by 

 operating in this manner we do in fact make all the unknowns 

 disappear except one only. For this purpose it is sufficient to shew, 

 firstly, that if we multiply term by term the two following series 



sin Qu, sin lu, sin 2u, sin 3u, ... sin (n 1) u, 

 sin Qv, sin lv, sin 2t&amp;gt;, sin 3v, ... sin (n T)v, 

 the sum of the products 



sin Qu sin Oy + sin lu sin lv + sin 2u, sin 2v + &c. 



is nothing, except when the arcs u and v are the same, each 

 of these arcs being otherwise supposed to be a multiple of a part 



of the circumference equal to -- ; secondly, that if we multiply 



term by term the two series 



cos Qu, cos lu, cos 2u, ... cos (n 1) u, 

 cos Qv, cos lv, cos 2v, ... cos (n 1) v, 



the sum of the products is nothing, except in the case when 

 u is equal to v ; thirdly, that if we multiply term by term the two 

 series 



sin Qu, sin lu, sin 2u, sin Su, ... sin (n 1) u, 

 cos Qv, cos lv, cos 2y, cos 3v, ... cos (n 1) v, 

 the sum of the products is always nothing. 



269. Let us denote by q the arc , by pq the arc u, and by 



vq the arc v ; ft and v being positive integers less than n. The 

 product of two terms corresponding to the two first series will 

 be represented by 



sin jpq sin jvq, or - cos j (//, - v) q - ^ cosj (&amp;gt; + v )q, 

 the letter j denoting any term whatever of the series 0, 1, 2, 3... 



