23] SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. 169 



and 7 may be similarly found by means of the non-periodic terms intro- 

 at/ 



duced into the integral d'R, in the equation 



_- +,-. 



2 df r a j dr 



When all this has been done, and the proper substitutions made, the 

 three expressions for -, are found to agree in giving 



dn die'*} ( 3 3771 4 1 



- V ' J __ _ /)V1- _L ___ AM* L 



ndt~ dt \ 2" 64 "j' 

 which is the result obtained by M. Delaunay and myself. 



de' 

 The supplementary terms in the Moon's coordinates which involve -r- are 



of the order of the disturbing force, and therefore the terms which they 

 introduce into the integrals, 



( t dR , (dR f 



r 2 -j- dv, j- dt, and d'R, 

 j dv J dv J 



will be the order of the square of the disturbing force. 



This is the reason why j . , , , , and -,- are all of the order m 4 . 



J hdt hdt adt 



It may be well to mention, in order to prevent any misapprehension, 

 that in my Memoir of 1853, h has not the same signification as the h of 

 M. Plana's theory. 



It is proved in Art. 11 of the Memoir that 



contains the non-periodic terms 



t ., 



2 r 2 -j dv 

 dv 



f_285 4 495 4 J 

 ' \~ 8 " 64 * e j ' 



=h ,f 1785 



1 (_"' 64 



22 



