207 



In the first place, 



1 cos< ( Ai + B 2 A 2 + B x - A 1 -JB 2 A 2 -B i 



cos t = cos <i+2 + -- 



8 sin < \ (2^-ra 1 ) w l 



sin 2n - 6n l 



^cos^ 1 -i2 2 -sm<! 2-|cos< — — 



1 1 * ' (2n-n l )n l 



sin 2w - 6W 1 



Now, 



^ 2 / 1 aTTBi • i , 1 A 2 +JB, 



2 2 = — sm 2n - n x + - j — cos w, t 1 



\2 2/&- 1 2 n x 



1 A + J^^-A-^^-^i 



4 (2n - n x ) n x 



sm 2n - n x + t. 



cos t = cos <! - sin <! 2 - J cos < x — - — r sin 2n - 2n l 



* (2n-n l )n x 



where P= A x + B 2 , &c. 



1 + cos t = (1 4- cos < x ) a - 2 1 + cos <! sin ; x 2 - i 1 + cos t x cos #, 



P 



(2w - n x ) n 



— — , sin 2n - 2n l + sin * 2 2 



1 \ rn 



= 1 + cos <j -21 + cos * sin ^ 2 + ( ^ sin 2 < — j 1+ cos t x cos ij) 

 also 



(2n-n x )n x 

 sin ( 2n - ti 1 ) 



sin 20 -20 = sin 2H-2* 1 |l-^ C ° g S .^ 1 



cos m - 1 



+ 2 — ~ 008 2^-2^, ^ X -IP - cos 2 (2n-2n x ) 



sin *i 2n — n l n — n x n 



where 2 t is the sum of the terms found in the first approximation. On 

 expanding 22 2 , this becomes 



sin 2n - 2n l [\ -i — - — - — sin 2n - 2n\ +, &c. 



( sin« 2n-n x n x ) 



2cos«-l — — „ /lcosf-l a 1 



sin 2n - 2n x + — . cos 2n - 2n x . 2 X - - 



sin* \4 sin 2 < [2n-w)n x 



1 n \ 



8 2n-n n-n x n 1 



