THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
749 
12. The method of the disturbing function applied to the motion of the planet. 
In the case where there are only two bodies, viz.: the planet and the satellite, the 
problem is already solved in the paper on “ Precession,” and it is only necessary to 
remember that the p and q of that paper are really cos ^(i-\-j), sin \{i-\-j), instead of 
cos hi, sin \i. This will not be reinvestigated, but we will now consider the case of 
two satellites, the nodes of whose orbits revolve with uniform angular velocity on the 
ecliptic. The results may be easily extended to the hypothesis of any number of 
satellites. 
In (18) we have the equations of variation of i, xp, y in terms of W. But as the 
correction to the precession has not much interest, we will only take the two 
equations 
. . di . dW 
n sm i v— cos i —— — 
dt d x 
dW 
d^r' 
dn dW 
dt d X 
y 
i 
j 
(75) 
which give the rate of change of obliquity and the tidal friction. 
In the development of W in § 5, it was assumed that xfj, 'P' were zero, and y, y' did 
not appear, because y was left unaccented in the X'-Y'-Z' functions. 
Longitudes were there measured from the autumnal equinox, but here we must 
conceive the N, N' of previous developments replaced by N — xp, N' — xp'; also LB + e, 
n,'l-\-e' must be replaced by flt-\-e — xp, fl't-\-e — xfj'. 
It will not be necessary to redevelop W for the following reasons. 
fl't-\-e — xp' occurs only in the exponentials, and N' — xp' does not occur there; and 
N' — xp' only occurs in the functions of and k, and fl't-\-e — xp' does not occur there. 
Hence 
dW dW dW 
de' ' dN' 
(76) 
Again, it will be seen by referring to the remarks made as to y, y' in the develop¬ 
ment of W in § 5, that we have the following identities :— 
For semi-diurnal terms, 
f/Wj 
dWj. dW u _ 
_rfW n dW m _ 
d X 
de' 5 df 
di ’ d x ' 
1 
1 
For diurnal terms, 
fW x rfW 3 
dW 3 d\V 3 
^w 3 
d X 
For the fortnightly term, 
2 de' ’ d x ' 
~ dg ’ d x ' - 
2 de' 
- * • 
• ■ • (77) 
8R % 
x\ ^ 
II 
o 
Also 
rfW IT 
dW 9 
V=°> 
^=° 
5 D 
mdccclxxx. 
