212 



some quantity divided by C ; and this, again, when substituted in the 

 differential equations for o> 3 will produce a quantity divided by C 2 , 

 which therefore cannot destroy the terms above found. To proceed, 

 then, we have in the first place, 



duo, n B - A 3 a - 2 . 



_J + &c . = ____^l + cos t sin (0-20), 



which will give 



B - A 3 fi 



w, = n + — - — — -„ 1 + cos i sin 2n - 2n 



C 16 r 



and hence 



B - A a 2 . - . 



0 - nt + — 7— — 1 + cos « sm 2w - = nt + P sm 2w - 2w l 



suppose, 



cos 0 = cosn + ^Pcos 3w-2w ! - |Pcosw- 2ft 1 

 multiply this by 



u>! = A l COS ft - w 1 , &c, 



and we have 



cos 0 = . . . + \ A X P cos 2n - n 1 - \ A X P cos 3n - 2n l + &c 



with terms of the same kind for w 2 sin 0, arising from B 2 sin n - n l 



io x cos 0 - a? sin 0 = J - B 2 P (cos 2^ - n l - cos 2n - Sn 1 ) 



t 1 + lA l -B 2 P 



sm 2w - n 1 sm 3w - 2n x 



2n -n 1 2n - 2>n l 



(the terms of the form A 2 -j- B cos 2^ - n\ and may be rejected ; be- 

 cause A 2 + B x contains B - A for a factor ; and as they are multiplied 

 by P, which contains the same factor, they would produce terms mul- 

 tiplied by B - A. Similarly, 



cos 1 i ^ -3 t- I cos 2n - n l cos 2n - Sri 1 



4 1 '\ 2n-n l 2ji-3n l 



with corresponding quantities for yjr. 



There is another way, also, in which terms will arise which must be 

 taken into account, as follows : — The term P sin 2n - n l when intro- 

 duced into the value of 0 in the equations for w l and w 2 will augment 

 the value of A x so as to make it become A x (1 + ^P). Similarly, B 2 

 will become B 2 (1 - jP), &c, and hence the term in 



. j A 1 + B 2 . 



1 i — sm 2n - n 1 



z 2n - n l 



