THE ELEMENTS OF THE ORBIT OF A SATELLITE 
743 
These results may of course be also obtained when the functions are expressed in 
terms of P, Q, p, q. 
Whence on this hypothesis 
g dj_ r 3 
k sin/ dt g 
sin 4f.^ cos 7 
(64) 
§ 8. Secular change in the mean distance of a satellite, where there is a second 
disturbing body, and where the nodes revolve with sensible uniformity on the fixed 
plane of reference. 
By (11) the equation giving the rate of change of f is 
k' dt ~ de' 
As before, we may drop the accents, except as regards e. 
, cl dW j) 
In § G we wrote <£(e) for the operation tan \j-rp, hence -rr = - </>(e)W, and by 
etc etc (f 
reference to that section the result may be at once written down. We have 
k dt g 
sin 2f x — 2<h 2 F 2 sin 2f, + 2T 1 G 1 sin g x — 
2T 2 G 2 sin g 3 — 2AH sin 2h] . 
(65) 
Where 
$i=l[Py+W+1 GPfrQ 2 q~(Py+ Q 4 f) + 36P i QfiY] 
<& 2 = the same with Q and P interchanged 
T l =2[P 2 Q 2 (Py+QY)+PfP' 2 -3QfiyY+Qf3P z -Q°YpY 
+ 9P~Q°{P°-QYpY] > . 
r 2 = the same with Q and P interchanged 
a = fiPHfy+f) +4P 3 Q 2 (P a - QYpVy+f) 
+pYI(p 2 -QY -2 p 2 Q*Y] 
( 66 ) 
These functions are reducible to the following forms 
2(<t’i-j- < Iq)= 1 — simp sin 4 p — sin 3 7(1 — 2 sin 3 p+ f sin 4 p) 
+ i sin 4 7(1 — 5 sin 3 p +- 8 - sin 4 p) 
2(<I> 1 —<I> 3 ) = cos 7 cosp[l sin 3 p — \ sin 3 7(1 —f sin 3 /)] 
2(r i + r 2 ) = sin 2 /— \ sin 4 p-b sin 3 7(1 — \ sm 3 /+-f sin 4 /) 
—\ sin 4 7(1 — 5 sin 3 /+^- sin 4 /) 
2(r x — r 2 ) = cos 7 cos/[sin 2 / + sin 3 7(1 —| sin 3 /)] 
2A =§ sin 4 / + sin 3 7(§ sin 3 /—y sin, 4 /) 
+ § sin 4 7(1 — 5 sin 3 /+- 8 - sin 1 ;/) 
5 c 2 
y ■ (67) 
