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



When the value of r 2 -p- given by this equation and the fore- 

 going value of r 2 cos 2cf> are substituted in the equation (A), there 

 will be found on the right-hand side of the equation the follow- 

 ing quantity which does not contain the sun's longitude, viz. 



3w' 9 « 2 / 57m 2 15me \ 



After substituting for e cosp its approximate value 1 — , this 



quantity is to be transferred to the left-hand side of the equation. 

 Then putting a + r — a for r in the small terms, and expanding 

 a + i — a so as to include small quantities of the fourth order, the 

 final result is 



3n' 9 a a 



*■* A* V, r ,_ 



(9 \ 5 e 2 



- ^cos{2q+p) +2cos(2q-p)\ + — cos2(?-p) 



+ e 2 (yCos2?-^cos2(?+^)+^-cos2(g-p)j 



/ 7 5 \ 



+ em( — -cos(2<7+j9) + 5 cos(2q—p) — -gcos^q—p)) 



(|cos(2 ? -^)+|cos(2 ? -3p)) -m 2 cos4?l- (C") 



+ - 

 m 



where 



„ a 70 3n' 2 « 4 45w' 2 ma 4 

 h =/i 2 4— 



A t'= / a-2/i' 2 « 3 -^n' 2 ma 3 



n r< o i2 2 45 n' 2 ma 2 171mV 2 « 2 

 C — \j —on a q 1 oni • 



Consequently the equation applicable to the mean orbit is 



"_V +c <=o, 



dv- r z r 

 which gives by integration, 



C* 

 r=a(l— ecos^), — (* + T)=-vJr-esin->Jr, 



_^' 2-1 _^ 



