23] SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. 167 



If we neglect the eccentricity and inclination of the Moon's orbit, and 

 also omit all powers of m above the fourth, the relations between these 

 several constants and the mean motion n will be expressed as follows : 



!_ f 



1253 ,.T 5593 4 



-' - 



3 288 92 



the sum of the masses of the Earth and Moon being supposed to be unity. 



From these relations we find by differentiation 



dn . dh f/(p") [ 3 2187 4 \ 



ndt~ d hdt^~dT~2 W 128 W ' 



dn . dh d(e'-) f 3 297 



-- -2 m " 32 n ' 



dn 3 da d(e' 2 ) f 3 5337 



___ _ I v ' ' _ VW" J __ _ Wi 



wd~ 2ac^ 4 d { 2 128 



n ' 



having taken care to observe that, since m = and n' is constant, we have 



n 



dm dn 



mdt ndt ' 



If i -j- be neglected in the first of these expressions, we obtain the 



(t i) 



value of y- found in M. Plana's theory, and one of those found by M. 

 de Pontdcoulant. If 77- be neglected in the second, the resulting value 



fj /V\ 



of j- is what would have been found by M. de Ponte"coulant, if he had 

 ndt 



taken his own h to be constant instead of M. Plana's h. 



If in the third expression -y- be neglected, we obtain the value of - j- 



which M. de Ponte"coulant communicated to me as the result which he had 

 found by using t as the independent variable. 



