374 Prof. Challis on the Theory of the Moon's Motion. 



C 1 



r = a(l—e cos ■f), — (^-fT)=-^r— esinijr, 



A 1 







c* 



and if cn= — , the apsidal motion in one revolution of the moon 



from an apse to the same apse again is 27r(l — c). 



These inferences having been drawn from the mean orbit 

 defined by the equation (D), we may proceed to integrate the 

 equation (C) in the manner I have employed in the Supplement 

 to the Philosophical Magazine for December 1854, the quantity 

 p now representing cnt + e—zr. The resulting value of r to 

 small quantities of the second order is as follows : — 



/ e 2 e 2 \ 



r = ay l — ecosp+ '— — — cos 2p — m 2 cos2j ). 



Also since to the same approximation 



dd h 3n' 2 _ 



dt=^ + ^n- C ° S2q ' 



we have by integration, 



h(„ ax, 2e . , 5e 2 . _ j» 9 _ \ 3m 2 

 6=€ + ^V + ^ t+ m Smp + 4c7i sin2 -?' + ^- co s2?J + -g-cos 2q. 



Hence by the definition of n, 





Also 



^' =/i (i+^- 3 )=Mi+m 2 ) 



because by the former approximation n 2 a s = p, and h = no 1 . Con - 

 sequently to small quantities of the second order, 



«=£=£(H-m 2 ) 



e ~ ^ ~ fj? + 2 

 _ C* _ flC* /j, 1+m 4 



A 4 ( l+ »V0 1+ <? 



