262 The Rev. B. Bronwin on the Coefficients of Sines 

 and (1.) will reduce to 



S9{(2^'-H)a}=nBo + SB,f^. . . (2.) 



Ifac=— - and 2w > W2, sin 2ma=0; but sin ?a is not no- 

 2w 



thing. Therefore 



B«=^^4^~j'2''>"'- • • • («•) 



where i' has all the values 0, 1, 2, ....w— 1. This value of 

 Bq requires a value of n only half as large as the supposition 



k= — would require. 



In order to find B„ multiply the first of the assumed equa- 

 tions by 2 cos /a, the second by 2 cos ?(a + ^), the third by 

 2 cos /(« + 2A), &c., and sum as before. Any coefficient B^ 

 will now be multiplied by 



2 cos^acosia= cos {p — i)ci+ cos (p + 2)«, 2 cos^(« + ^) 



cos i{a. + k)= cos {p—i){oi + k)+ cos {p + *)(« + k\ &c. ; 



which will constitute two series, the sums of which must be 

 separately taken. We shall thus have 



. ink 



2t<^{ot. + i'k) cos i(ot + i'k) = 2Bo cos « ( a + (« — 1) ^j 



. , .. nk 



. ik 



sm -— 



2 



SBp cos {p - i) Lt^{n- 1) - j 



k 



2 



+ 2B,cos(p + 0(« + (n-l)|) f . . ('i.) 



sin (^^ + 2)2 



But when ^=/, the coefficient of Bj will be 



. , , sin ink 

 n + cosi{2ci+{n-\)k)-^^jj^. 



We may give to^ all the values 1, 2, .... m, except /; ftnd if 

 n 



