THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
757 
obliquity of the ecliptic and on the earth’s precession. In the present configuration of 
the three bodies the problem presents but little difficulty, because the influence of 
oblateness on the moon’s motion is very small compared with the perturbation due to 
the sun; on the other hand, in the case of Jupiter, the influence of oblateness is more 
important than that of solar perturbation. In each of these special cases there is an 
appropriate approximation which leads to the result. In the present problem we have, 
however, to obtain a solution, which shall be applicable to the preponderance of either 
perturbing cause, because we shall have to trace, in retrospect, the evanescence of the 
solar influence, and the increase of the influence of oblateness. 
The lunar orbit will be taken as circular, and the earth or planet as homogeneous 
and of ellipticity C, so that the equation to its surface is 
p=a{l-rf(-^—cos 2 6)} 
The problem will be treated by the method of the disturbing function, and the 
method will be applied so as to give the perturbations both of the moon and earth. 
First consider only the influence of oblateness. 
Let p, 6 be the coordinates of the moon, so that p — c and cos #=M 3 . Then in the 
formula (17) § 2, r=c and - = £(-§-—M 3 3 ), so that the disturbing function 
W=Tt(i-M 3 3 ) 
This function, when suitably developed, will give the perturbation of the moon’s 
motion due to oblateness, and the lunar precession and nutation of the earth. 
Then by ( 21 ) we have 
M 3 =sin i [p 2 sin (/+A T )— (fl sin ( l — N) j+sinj 1 ’ cos i sin l. 
Where l is the moon’s longitude measured from the node, and N is the longitude of 
the ascending node of the lunar orbit measured from the descending node of the 
equator. 
Then as we are only going to find secular inequalities, we may, in developing the 
disturbing function, drop out terms involving l; also we must write N — xp for AT, 
because we cannot now take the autumnal equinox as fixed. 
Then omitting all terms which involve l, 
M 3 2 = sin 2 i [^(p^-bq 4 ')— p~q 2 cos 2 (iV— 1 //)]+-| sin 2 q cos 2 i 
+sin / sin i cos i [p> 2 —q 2 ] cos (N—\p) 
Since p>=cos \ j, q=sin \j, we have 
p A j rf = 1 sin 2 /, prcf—l sin 2 j, qF—q' — cosj 
5 E 
MJDCCCLXXX. 
