48 LECTURES ON THE LUNAR THEORY. [LECT. 



+ ^ {1 + 3a 2 cos 2i//} {1 + 3e cos (nt - rar)} 



\AJ 



fl 3 } 



-ft' 2 - - + - cos 2t/ - 3 sin 2i/ [2 (1 + &) e sin (ft<-cr)H = 0, 



(2 2 J 



and 



4 (w ft') 2 6 2 sin 2\ji 2 (np}~ (1 + &) e sin (ft< CT) 

 + 4ft (ft ft') a s sin 2/ + 2ft (n p) e sin (nt cr) 



+ 4 (ft ft') (ft p) (!+&) ea . [ sm i 2 '/' '' + HT) + sin (2t/ + ft< w)] 

 + 2 (ft ft') (ft p) efy, [ sin (2\fi nt + CT) + sin (2i/> + ft< or)] 



f3 1 



+ ft' 2 4 - sin 2/> + 3 cos 2i/ [2 ( I + & ) e sin (nt - w)J i- = 0. 



V. y 



It will of course be found that with the values of a 2 , b, of Lecture IV, 

 the terms independent of e vanish identically. 



Equating to zero the coefficients of cos (nt CT) in the first equation 

 and sin (nt cr) in the second, we get 



(ft -pY - 4n (ft -p) ( 1 + 6.) + jj = 0, 



- 2 (ft - p) a ( 1 + 6 ) + 2ft (ft -p) = 0. 



Therefore (ft p) ( 1 + &) = n, 



and (ft ) 2 = 4/r ^ 



a" 



= ft 2 - ft' 2 , approximately, 



u 



f> 3 

 or = -m? = b <> , approximately. 



Now terms have been left outstanding with the arguments Z^i nt + vr, 

 is. These may be removed by assuming 



a 

 [ S ~ =a - cos 2 V + e cos ( nt w) + a 21 e cos (2^ nt + CT) + a w e cos (2/r + ft? ra-), 



^ = nt + e + 6 2 sin 2/> + 2e (1 + &) sin (n - nr) + 6 21 e sin (2^ - ft + vr) 



