146 ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. [21 



de' ndt de' 



Thus -j- = -5 -- -J-. 



dv dv ndt 



dJ (i _ 11 OT cos (2v - 2mv) - ~ m?e' cos (2v - 2mv - c'niv) 

 ndt 4 



+ - - mV cos (2v 2mv + c'mv) \ . 

 8 J 



Also, integrating by parts, and putting 2 instead of 2 2m, 2 3m, 

 and 2 m in the divisors introduced by integration, since we only want to 



de' 

 find the terms of the lowest order which are multiplied by -5-, we obtain 



_ ^ (dv j(l - - e' 2 ) sin (2v - 2mv) + 7 -e' sin (2v - 2mv - c'mv) 



a , } IV 2 / 



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



= -- [ 1 e' 2 ] cos (2v 2mv) + e'cos (2v 2mv c'mv) 

 2 a, \ 2 / 4 a, 



- - e' cos (2v 

 4 a 



15m 2 f, e'de'ndt . . 21 m* ( 7 de' ndt ln , > 



+ - ay j- -5 cos (2V 2mv) - av j j cos(2v 2mv cmv) 

 2 a f j ndt dv 4 a,J wrf^ rfv 



, 3 m 2 f , c^e' wcZ , , . 



+ T I >* f, ^~ cos (2V 2mv + cmv). 

 4 a, J a cw 



And a' SM' = 3m V sin c'mi/ [ e' sin c'mv] 



--|V, 

 retaining only the term which will be required. 



10. When the proper substitutions are made, the terms involving cosines 

 destroy each other, as in the usual theory, and by equating to zero the 

 terms involving the sines, we obtain 



