42 LECTURES ON THE LUNAR THEORY. [LECT. 



Write a for n't is'; then 



cos {2 (6 - n't] - a} = cos (2i/ - a) - (26, sin 2/) sin (2</ - a) 

 = -6,,cosa + cos(2x/> a) + 6 3 cos(4t/ a), 

 sin {2 (0 - n't) - a} = + 6, sin a + sin (2</ - a) + b, sin (4i/ - a), 

 cos ^2 (6 - n't] + a} = - &., cos a + cos (2t/ + a) + b, cos (4i/ + a), 

 sin {2 (9 - n't] + a} = - 1), sin a + sin (2i/ + a) + 6., sin (4/r + a). 



Hence 



X=-^n*(\- 36.,) e' cos a - ^ wV cos (2</ - a) + - nV cos (2i/ + a) 



21 3 



- n'-b.,e' cos (ty a) + - ' 2 6.,e' cos (4i// + a), 



F = 6ra' 2 6 2 e' sin a + ~ V sin (2t/ - a) - | n' 2 e' sin (2i// + a) 



21 3 



+ n'-b.,e' sin (4i/> a) '- n'-bf' sin (4/> + a). 



For our present purpose we shall ignore the small terms in 4t// a 

 and 4t/ + a which are of the sixth order. 



Assume 



81= a^e' cos a + X cos (2t/r a) + c^e' cos (2i/ + a), 



8^ = & s e' sin a + 6/ sin (2t/r - a) + 6 7 e' sin (2i|/ + a). 



Now the terms which arise in the left-hand members of the equations 

 owing to terms a p cos pt in S, and b p sin pt in 8^, will be 



p*a p cospt + 'Za^pcip [cos (pt 2\fi) cos (pt + 2t/)] 



+ ~ a p \ cos j>i + '- a., cos (^>< 2x/>) + - a 2 cos (pt + 2\ft) 



- 2 ( 1 + ') jo6 p cos p - 26 2 p6 p [cos (|) - 2i/) + cos ( pt + 2\fi)] 



+ o W ' 2& P C cos (P i ~ 2 $) ~ cos 



A 



and 



- jt> 2 6 p sin p< + 2a,pb p [ sin (p 2^) + sin (pt + 2\jj)'\ 



') pa p sin ^>< + 2b !1 pa p [sin ( p< - 2i/) + sin 

 respectively, neglecting the very small quantities in 4$. 



