THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
857 
dn _ dW 
dt d% 
= -^-( I ~ 2 P e x 2 {P 8 F x sin 2f s +2P 6 Q 3 G x sin g x + 2P 2 () 6 G x sin g x + Q n F x sin 2f x ] 
-f J 0 2 {4P 4 £ 4 F sin 2 f-f- 2 P 2 Q 2 (P~ — Q ~) 2 G sin g} 
+ 2 Jx 2 { 4 P 4 0 4 (F s sin 2 F + F X sin 2 f x )+ 2 P 3 () 2 (P 3 - Q*f(G* sin g x + G x sin g x )}] (296) 
The first 2 being from iv to i, and the last only for ii and i. 
This is a partial solution for the tidal friction, and corresponds only to the action of 
the moon on her own tides; that of the sun on his tides may be obtained by 
symmetry. 
It is easy to see that for the joint effect of the two bodies we have 
dn 
dt 
1 
(1—t?) 6 (1—F) 
pToJo^P^Fsin 
2f+2P 3 <? 3 (P 2 -(?) 2 G sin g} 
(297) 
From (296-7) and (287-8) the complete solution may be collected. 
In order to find the secular change in the obliquity, we must consider how 1 fs would 
enter in W. 
Now in the development of W, stands for i//, and tt/ stands for 
Tx'—ifj'. Hence from (295) 
Now by (18) 
(2.98) 
. . di dW . dW 
n sin % — — ~i 7 cos \ —— 
dt d% dyjr' 
dW dW 
Then substituting for —- from (296) and for —, from (298), we find 
n~=- (T= ^ r 2 {XE x 2 [P^F x sin 2f*+P 5 £(P 2 +3£ 2 )G x sin g* 
-P() 5 (3P 3 +() 2 )G X sin g x -PQ 7 F x sin 2f x -3P 3 Q s H x sin (xh x )] 
-J 0 3 [ 2 P 3 Q 3 (P 3 -Q 3 )F sin 2 f+P()(P 2 -Q 3 ) 3 G sin g] 
~2J X 2 [2P 3 Q 3 (P 2 -Q 3 )(F X sin 2 f x + F x sin 2 f x ) 
+PQ(P 2 — Q 3 ) 3 (G X sin g x -f G x sin g x )]} . (299) 
