THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
781 
Now let 
r “ ( sin 4f i” sin 2gjH- sin 2g) 
G= 
A = 
D= 
4?i 
1 
4 n 
-(sin 4f x — sin 2g x + sin 2g) + — sin 
-(sin 4f x + sin 2g 1 — sin 2g) + — sin 4f— 2^ sin 2g 
^-(sin 4f x + sin 2g 1 — sin 2g)—sin 2g 
>• • (174) 
Then the four equations (139), (149), (158), and (171) may be written 
sin A - 'M 1 
f=-I>'-G»cos(JV-W 
.dxp- 
dt 
di 
dt 
=T>j sin ( N— \Jj) 
=Ai+Dj cos (N—xp) 
(175) 
Also from (151) and (173) 
1 dj 
k dt 
—sin 4fj 
d'H jT 2 . f , T ~ . „ 
—=s— sin 41,+4— sm 41 
dt z a 1 2 a 
(176) 
These six equations (175-6) contain all the secular inequalities in the motions of the 
moon and earth, due to the bodily tides raised by the sun and moon, as far as is 
material for the present investigation. The terms which are omitted only represent 
a very small displacement of the proper planes and of the inclinations of the planes of 
motion of the two parts of the system to those proper planes. 
Then reverting to the earlier notation in which 
y—j sin N, y — i sin xp 
z=j cos N, £—i cos \p 
5 H 
(177) 
MDCCCLXXX. 
