21] ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. 145 



~ I 1 ( 1 - 1 e ' 2 ) cos ( 2v ~ 2mv ) +l e> c s (2v - 2mv - cW) 

 ^ a / IA * / 



- - e' cos (2z/ - 2mv + c'mv) laSu 



- ] ( 1 - e' 2 ) sin (2i> 2mi) + - e' sin (2v 2mv - c'mv) 

 * */ VA ' 



, . /r> / \ 



- e' sin (2v 2mv + c'mv) 



' 



1 



- 

 ' } dv 



+ 12 -- I dv I ( 1 - -e' 2 ) sin (2i/- 2mi') + - e'sin (2^- 2mv-c'mv) 

 a , j \\ & I 



- e' sin (2i> 2?ni' + c'mv) laS 

 z J 



3m 2 (<P(a&u) , ) f 7 f / 5 



2 Sm Zv ~ 



7 11 



g e' sin (2v - 2mv - c'mv) --e' sin (2v - 2mv + c'mv) y . 



8. Also, assume 



a8u = m 2 [ 1 - - e /2 J cos (2v - 2mv) + a. M -J- sin (2v - 2mv) 



8 , , de' . 



- rare cos c mv + a ie j- sin c ?^v 



+ - mV cos (2v 2mv c'mv) + a.- r sin (2v 2mv c'mv) 

 2 ndt 



1 de' 



- mV cos (2v 2mv + c'mv) + a M , sin (2v 2mv + c'mv), 



where the coefficients of the terms involving cosines are those given by 

 the ordinary theory, and a w , a w , a w , and a w are numerical quantities to 

 be determined. 



9. In developing the terms of the above equation, by the substitution 

 of this value of a8u, the quantity -j- may be considered constant, and -j- 



must be expressed in terms of it. 



A. 19 



