= h 1 1 +-TO 2 cos 2,0, 



13] STUDIES ON NEWTON'S LUNAR THEORY. 231 



Also let h , h be the double areal velocity in the two orbits ; then 



/i = nor, 



and since ^ = j 1 + m" cos 20 I C , = -^ \ 1 + - - m 2 cos 20, 



?< a = r '{l-2ni 1I cos20 }; 



3 



V 



The force on the Moon perpendicular to the Radius Vector would be, if 

 h e were constant, 



1 dh h* f 3 , . 



-j- = -~ 1 - - m- sm 20, 

 r dt rj ( 2 



But the actual force in this direction is 



- rnVr sin 20 = -^ r ] - m' 2 sin 20 V , approximately. 



w [ 2 J 



The latter minus the former quantity 



= if a {6m*e sin 20 cos (0 CT)} 



if we write ?' = /{! + ecos (^ TO-)}, and neglect e\ If this difference were 

 zero, the elliptic elements, e, w, would not be liable to change in respect 

 to forces perpendicular to the radius vector ; hence it measures Q, the force 

 perpendicular to the radius vector which disturbs the elliptic orbit (r , # ). 



Again, in the direction of the radius vector the force is 

 3 -, = ria - 1 + 2e cos (0 CT) + - m" cos 20 + - mre cos 20 a cos (0 

 if the elements of the elliptic orbit be constant ; but the actual force is 

 _ I n '^r _ n r cos 20 = ^ 2 l + 2m* cos 20 l + 2e cos - 



1 3 



- mVa {1 e cos (^ CT)} - mVa cos 2^ jl - e cos (^ - ra-)}, 



U 2 



putting r a for r, $ for ^ in terms multiplied by m 2 . If 



the terms independent of e agree with the like terms in the expression 

 above, and the latter force minus the former 



= tfa I - m?e cos (0 nr) + 3m?e cos 20 cos (0^ ar) j- . 

 This is the quantity P. 



