4] THE SECULAE ACCELERATION OF THE MOON'S MEAN MOTION. 121 

 Thus if the variation of e' be taken into account, we must assume 



log-= a + a 2 cos<'+ a 5 cos2+ a s cos(2f <') + a, 



+ So, sin $ + Sa 5 sin 2+ Sa 8 sin (2- <') + Sa 9 sin 

 + 6 2 sin<' + b 5 sin 2f + 6 g sin (2 -<')+ 6 9 sin 

 + S6 2 cos (/>' + 8b, cos 2f + S6 8 cos (2f - <') + S6 9 cos 



where now denotes Jwcfo (7i'i + e'), in order to cancel the terms which 

 are introduced by differentiating the coefficients a, b. Here 8a 2 , 8b.,, 8a 8 , 8^, 



8a 9 , 86 9 are of the first order in e', but 8a 5 , 8& 5 , are of the second 



order. We shall suppose -^r = 0, and we shall ignore terms of the order 



Hence we get 



(log - ) = mna., sin <f>' - (2 2m) na s sin 2 (2 3m) na s sin (i 

 a \ 3 T/ 



(2 m) na 9 sin (2f + <') 



CtCtn r-, CtC6 2 t / I /ri n \ 



+ -. + m?i8a., + - 7 - cos <z> + (2 2m) 

 d* d J L 



tdct "1 

 (2 3m) nSa s + -jf cos (2 <p) 

 at J 



+[< )"*^ 



+ ~ cos 2 



cos 



= n + mn& 3 cos (^' + (2 2m) n6 5 cos 2f + (2 3m) w6 8 cos 

 + (2 m) nb,, cos (2f + <') 



+ [ -mn86 2 +-T 1 ] sin 0'+ | - (2 - 2m) n86. + - sin 5 

 L df J L at -\ 



+ \ (2 3m) w8& 8 + -r^ sin (2 (f>') 



r db ~\ 



+ (2 m) n8b, + ^ sin 



A. II. 16 



