20 LECTURES ON THE LUNAR THEORY. [LECT. 



again, -y- = n + 2 (n n') b,. cos 2i/>, 



(//1\ > 

 -77 ) = n' 1 + 4?? (n n') 6 2 cos 2\fi + 2 (n n') 2 6 2 2 [l + cos 4/] , 

 M I 



I + 3a, cos 2i/f + '- a.,- ( 1 + cos 4A) . 

 r 3 a |_ 2 



Also, - = 2 ( - n') a, sin 2i/ - - a,/ sin 4\ji , 



Idrdd ,.r f. l 1 1 



" ji ~rr 2 (re n ) na., sin zw + (n n ) a.,b., - no,.: y sin 4u . 

 r at at \ v 2 " j 



jTj = 4 (n ft.') 3 6., sin 2t//. 



And cos 2 (^ - w'< - e') = cos 2i/ - 6, ( 1 - cos 4t/), 



sin 2 (^ - n'< - e') = sin '2\jj + b., sin 4^. 



Substitute these in the differential equations, and we get, on trans- 

 posing all the terms to the left-hand sides from the first equation 



[1 31 



- a., a + a, cos 2/> - a,,' J cos 4\jj 



- [n- + 2 (n - ft/) 2 /;.; + 4n (i - n') b, cos 2t/ + 2 (ft - ft') 3 6 2 3 cos 4/] 



+ : 1 + - a., 2 + 3a,, cos 2i/> + - a., 3 cos 4i// 

 a 3 L 2 2 r J 



1 3 



- - ft'" - w' 2 [ 6. + cos 2t/> + //, cos 4t/r] , 



and from the second equation 



- 4 (ft ft') 2 &., sin 2t// 



+ 4 (ft n') m,, sin 2\jj + -[ (n ft') a,& 2 fta 2 2 > sin 4t/> 

 + - ft' 2 [sin 2/< + ^ 2 sin 



