LECTUEE XVIII. 



ON HILL'S METHOD OF TREATING THE LUNAR THEORY, (continued). 



THE equations for Sx, By are 



d'Sx ,d>/ _d"R d 2 R s Y 



= ~ d + ~~ d ~* 



, ,dSx d"R -, d"R 

 2n' -- = --- 



dt dxdy dy- 

 the equations for x, y are 



u dt ~ dx ' 

 dt- dt dy ' 



ft'Y Cf^I 



Multiply the former pair by -=- , -^- respectively, and the latter pair 



Gt-6 Ctt 



dSx dSy 



by T , T^ , and add all together ; we get 

 etc ctt 



dx d-Sx d"x dSx dy d~8y d*y dSy 

 dt ~df Jr ~de~dt *~~di ~dt? + d? ~dt 



_ dx d*R dy\ . / d*R dx tfR cly\ _ 

 = dx* dt + dxdy dt) d f (dxdy dt + dy* dt) y 



dRdSx dRdSy y dx ydy 

 dx dt dy dt dt dt ' 



tfRdx d*R dy = ddR 

 do? dt dxdy dt dt dx ' 



d>R dx d'R dy _ d dR 



T 



dxdy dt dy* dt dt dy ' 

 A. n. 11 



