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



or | = l| 1+ ^ m4 + 2391 mV2 

 v w*, ^ 



11 / 5 \ 425 e'de' 



m 2 (l--e /a cos(2v-2mv)- -- --m* j- sin (2v - 2mv) 



4 \ 2 / 24 ndt ' 



+ 3m 2 e' cos c'mv + 6m 3 j- sin c'mv 



ndt 



77 595 de' 



- m 2 e' cos (2v 2mv c'mv) + - m 2 j- sin (2v 2mv c'mv) 



IIV ^ V'-'tJ \Jf triuvr \s iivr i i . _ 



+ '-- mre' cos (2v 2mv + c'mv) m" j- sin (2v 2mv + c'mv) I . 



15. Substitute the value before found for a 2 in terms of a/; 



^ * fi _L 9 I 4 + 3 s / _ 3867 4 , 2 

 '' c^~ :a/ \ " 64 * 2 W 64 



11 3 A 5 ,,\ , 425 a e'de' . 



- m 2 ( 1 - e' 2 j cos (2v 2mv) - m 2 -j- sin (2v 2mv) 



+ 3m 2 e' cos c'mv + 6m 3 - --, sin c'mv 

 ndt 



77 595 de 1 



mV' cos (2v 2?nv c'mv) + - _, m 2 r sin (2v 2mv c'mv) 

 8 48 ndt 



11 85 de' .] 



+ m~e cos ( 2v 2mv + c mv) m~ r- sin ( 2v 2mv + c mv) V . 

 8 48 nut ') 



1 a f 1 97 3 3867 1 



1 6. Now, put - = a, 1 1 1 + m- - jj- m 4 + | mV 2 - . ^ mV 2 1 , 



multiply by n, and integrate ; 



f 11 / 5 \ 295 e'de' 



.-. | ndt = v m-(l- e' 2 ) sin (2v 2mv) +~ 9 y m2 --ji cos(2v 2mv) 



+ 3me' sin c'mv + 3 r cos c'mv 

 ndt 



77 2 , . /0 , . 413 , rfe' 



z-g m e sm ( 2 2mv c'mv) m 2 j- cos (2v 2mv c'mv) 



11 59 de' 



+ T^ wV sin (2v - 2mv + c'mv) + m 2 - --,-- cos (2v - 2mv + c'mv). 



-I O 4o 72-W-t 



