78 LECTURES ON THE LUNAR THEORY. [LECT. 



u mi, I \ HI' / , 3 , 

 Hence SI =<= + -, 3 (xr--=y + -, 4 a?- - xy ' + .... 





We have tacitly assumed the origin to be at the centre of the Earth ; 

 if we prefer to place it at the centre of gravity of the Earth and Moon, 

 the necessary change is effected by multiplying the last terms, which cor- 

 respond to the Parallactic Inequalities, by (E-M}I(E 



Equating these forces to the accelerations of the Moon parallel to the 

 coordinate axes, we have the equations of motion in the form 



d"x fdy ,., _dfl 



t ,., ATI* --j 11 X j , 



(It at dx 



or, as they may be written, 



. dx dCl 



' 7 -ri/ 2 7/ = T , 



dt dy 



d*x . dii dR 

 _ _ 2?t _ 



dt- dt dx ' 



d~u . dx dR 

 -jJr+2n'-j-= -j-, 



dt- dt dy 



where R n + ;, n'- (ar + y~] 



4 



Now suppose we have found values of x and y which satisfy this pair of 

 equations and which involve two arbitrary constants. This may be ac- 

 complished by taking assumed developments 



x = ta t cos i (t + y), 



= sn* 



substituting in the equations, and equating coefficients of the various terms. 

 The solution found will include the Variation and the Parallactic Inequalities. 

 Let it be required to amend this solution by the introduction of the 

 remaining two arbitrary constants that are required for a complete solution. 



