and Cosines of Multiples of a variable angle. 263 



flJc Tile 



sin(p-/) — =0, sin (;? + i)— =:0; 



but 



k k 



are not nothing if w > 2w. Thus the coefficients of Bq and 

 Bp will vanish. Also s\x\ink = Oi but sin e^ is not nothing, 

 and the coefficient of Bj reduces to n. 

 But make /5- = 2a, and (4.) becomes 



22f {(2,'' + lM cosi(2i'+l)« = Bo?^|^ +SB, 



. !^ • (5.) 



sin ( j3 — i) 2na. sin ( /? + 2^*'=* "1 

 .2 sin (j9 — /)« 2 sin {p + i)a. J 



When j!? = /, the coefficient of Bj is 



sin 4 /wa 



n-\ : . 



2 sm 2ia. 



It 



{; 



Make a=— , n:::^m^ and the coefficients of Bq and Bp will 

 vanish; and that of Bj reduces to n. Thus we shall have 



K>/w, B,= ^2f|(2.'+l)£}cosf(2/'+l)^, . (6.) 



where «' has all the values 0, 1, 2, .... w— 1. 



If we had not restricted the value of w, we should have had 



27r 

 a series of terms, as B„_,-, B«, B„+i, &c. when k— — ; and the 



series B2„_<, B^^, Ba^+i when k='lci— -. 



The formulae (3.) and (6".), making 72 > m in both, enable 

 us to determine all the coefficients of (A.) But 



COS (2n-l)- =± cos— , cos {^n-3)^=^+cos—, 



&c. to 



+ cos ^ _ , 

 2w 



if n be even, which it will always be convenient to suppose. 

 Therefore 



2,{(«'+ DfJ cos (.. . i)| = {, (fj ±.(^')} 



2n 



