206 



To find 0 - 9 we have 



d(<j) - yjr) cos *- 1 



<K sin t 



and by proceeding exactly as above we shall find 



(w l sin 0 + ft> 2 cos 0) 



6»! cos 0 + u> 2 sin 0 = i + -Z? 2 ) sin 2n - n 1 + % A x - i? 2 sin n 1 



+ \ A 2 - B x cos 2n - n x + ^ A 2 + B x cos n l 



+ l c -21h {a^ 2 a 2 Tb x -J^ x a^)( — + - x 



8 sm «i \ 2n - n l n l 



sin (2n - 2n x ) 



also if t = «! + 2 where 2 denotes the sum of the quantities which occur 

 in it, we shall have 



cos i - 1 cos i - sin t. 2 - I cos i x - 1 - sin i x 2 



sin i sin «, + cos * 2 . / cos <, . 



suii, 1+- — -2 



sm *, 



cos*! -I ^ f i cos <! - 1 cos *A 

 \ sin 2 *i J 



sm <i 



neglecting higher powers of 2 than the first, since these only are 

 wanted. Multiplying together these factors, we shall have 



d ((b-xlA cos <! - 1 / — - T - . 



- M = — ^ M» + £ * sm 2n ~ » l +> &c - 



dt sm*, 



(^! + ^ 2 ^ 2 -^ -A 2 -B x A x -B 2 ) (l + 



cos*i - 1 cos*, 



sin 2 <! 



cos *i - 1 cos < t \ 2n 



sin a *i ) 2n - n 1 



Integrating and changing the signs on both sides 



, n cos < - 1 1 / Ai + B 2 - - 



(f>- 0 = — : — — — - cos 2n - n l + . . 



sin <i 2 \ 2?a - 



sm 2n - 2w- 



-±(A X +B 2 A 2 + B X - A X -B 2 A 2 -B X ) cos(2w-2w ] ) 



n - n l 2n-n l n 



We have now to substitute these values for t x , &c, in the function 



- - -„ (1 + cos *) 2 sin 20 - 20 

 8 r s v J 



which occurs in the development of N. 



