THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
767 
Then 
I = the inclination of the earth’s proper plane to the ecliptic 
J=the inclination of the lunar orbit to its proper plane 
I^the inclination of the equator to the earth’s proper plane 
J^the inclination of the moon’s proper plane to the ecliptic 
J =L lt 1=4', i/=4', J,= —4 
and by (125-6) 
/C| -f- Gi j 0 
~ t- 
a k i + /3 
j=-^-i= 
/ Kc, + a 
/Co + y3y 
(127) 
Thus I and J are the two constants introduced in the integration of the simul- 
taneous differential equations (116). 
It is interesting to examine the physical meaning of these results, and to show how 
the solution degrades into the two limiting cases, viz.: where the planet is spherical, 
and where the sun’s influence is non-existent. 
Let It be the speed of motion of the nodes, when the ellipticity of the planet is zero. 
Let l be the purely lunar precession, or the precession when the solar influence 
is nil. 
Let lit be the ratio of the moment of momentum of the earth’s rotation to that of 
the orbital motion of the two bodies round their common centre of inertia. 
Then 
Then by (121) and (115) we have 
t'z 
«=ml+n, a=ml, /3=l+ -, b=l 
First suppose that It is large compared with l. 
This is the case at present with the earth and moon, because the speed of motion 
of the moon’s nodes is very great compared with the speed of the purely lunar 
precession. 
Then a, /3, b are small compared with a. 
Therefore by (117) 
*1 — k 2— — a- hA k i~\~ k 2 = — a — /3 
and 
/q=—a, k\,= — /3 
5 F 2 
