and Cosines of Multiples of a variable angle, 267 



As the vanishing factors here are the same a9 in (4.), these all 

 vanish when n > 2m. Therefore 



B,= |^«+ ^)cos.-(<.+ '^),n^2m. . (12.) 



Again, multiplying the first by 2 sin i «, the second by 

 2 sin z(« + ^), the third by 2 sin /(« + 2/^), &c. and summing, 

 it is obvious that the coefficient of Bp will be had by changing 

 p — i into i—p in that of Ap last found, and therefore it will 

 vanish under the circumstances supposed. The coefficient of 

 Bi will be the same as that of A;, and consequently will vanish 

 also. It is to be remembered, that we do not give to p the 

 value p — iy as we always find the terms depending on this 

 value separately. The coefficient of Bq will be 



. ink 

 sm — 



2 sin?( «+(«— 1) — ) 



. ik 

 sm- 



and will therefore vanish. And it is obvious that the coeffi- 

 cients of Ap and A; will be the same as those found in (8.), 

 and will be and n respectively. Therefore 



In (11.), (12.) and (13.), i' has the same values as in the other 

 forms, namely 0, 1,2, .... n—l; and these three formulae give 

 all the coefficients of (C). It will be evident that we may 

 always suppose n as large as we please, and therefore that 

 w > 2»? in (11.). 



If «=0, i'—O, l,....w-l, or if «= — , i'=l, 2, ....n, the 



two sets of formulae coincide. It will be better to make the 

 latter supposition, then we have 



" n "^ \ n / n \ n / \ n / { 



Ai= -2/1 — }s\m[ — ), ?'=1,2, .... 

 n '' \ n / \ n J 



(HO 



We may derive from these others similar to (7.) and (10.). 



TT 



Ifa=-, i' = 0, 1, 2 ....»— 1, n^2m. 



