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



y(a + 2A') = Bo + Bi cos («+?/•) + B2 cos 2(a + 24) + .... 

 + Ai sin (a + 2Z') + A2sin2(a + 2/^) + .... 



/(« + (w- 1)^) = Bo+ Bi cos (« + (»- 1)A:) + Bg cos 2 

 («+(«-- l)X-) + .... 



+ Ai sin (a + (n— 1)^) + A2sin 2(a+ (« — 1)/^) + .... 



Let B^ or A„ be the last of the coefficients which is sen- 



27r 

 sible. Then if Z'= — , taking the sum of these as before, the 

 n ° 



terms containing the cosines all vanish, as we have found from 



(1.), and the general term of the sines is 



. ink 



Aisin i\ot+{n — \)-j 



sm-^ 



. ik 



sm — 



2 



which vanishes for the same reason, having the same vanish- 

 ing factor. Therefore 



Bo=is/(«+ — ),^>m. . . . (11.) 

 n \ n J ^ 



Now multiply the first by 2 cos ia, the second by 2 cos i 



(« + A:), the third by 2 cos ?(a + 2yt), &c., and sum. The part 



of this sum depending on the cosines is given in (4.) ; and the 



27r 

 result, when k=. — , is the same as there found in the same 

 n 



case. The coefficients of Ap will be 

 sin {p^i)oi.+ sin (p + ?)a, 



sin (p — i) (« + it) + sin {p + i) (a + k\ &c. ; 

 and their sum 



. , .. nk 



(2,-/)(« + (/^-l)|)- ^ +sin(;, + /) 



sm 



sin {p — i) 



2 





nk 

 I' 



The coefficient of Aj will be 



• ^- • ^-z 7\ • -/^ / ,. J. sin ink 



sm 2i«+ sm 2i{oi + k) + ..,.= sm 2(2«+(«— 1)a:) — ; — rr-. 



sm ZfC 



