6%2 Prof. Challis on the Theory of the Moon*s Motion, 



under this restriction. Thus we shall have, 6 being the moon's 

 longitude, 



dz ^ dx^ . dy^ dr' . 7^d&^ 



ft-^' 



dt^ '^ dt^ " dt^ '^ dt^ ' 



gdO dy doc . /^ , ,, u 



and the above integral becomes 

 dr^ r''d4>^ ^^ ,2, 



Also, since 



2fi^2m^r 

 r 'W 



cos 





(a) 



dd dd> ^ , J 



dt 



dt 



it follows that 



dt 



__ m'r sin <^ 



{■-( 



_ d^y d^x 



-^W^dF 



|cos,^J)-*}. (b) 



The equations (a) and (b) between the three variables r, <^ 



and tf will conduct by successive approximations to the moon's 



motion and the form of her orbit. By expanding the trinomial 



13 r 



affected with the indices — tt and —77 to the fourth power of—., 



2 2 ^ a!' 



the following approximate equations, in which »'^ is substituted 

 for -f^ will be found : 



H- -j-f (3cos^ + 5cos3^) 



+ ^2(9+20cob2(^+35cos4<^) 



(A) 



d , r 



(§-') 



37»'V 



dt 



sin2<^ 



2 



3n'V 

 Q ; (sin <^ + 5 sin 3</>) 



(B) 



