THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
751 
We have 
”^ = T"[ 2 ^ F Sin 2f +>' G sin g] 
«—■—-[2<£F sin 2f+yG sin g] cot i 
It Mali be noticed that in (81) 2 tt has been introduced in the equations instead 
of tt ; this is because in the complete solution of the problem these terms are 
repeated twice, once for the attraction of the moon on the solar tides, and again for 
that of the sun on the lunar tides. 
The case where Diana is identical with the moon must now be considered. This 
will enable us to find the effects of the moon’s attraction on her own tides, and then 
by symmetry those of the sun’s attraction on his tides. 
We will begin with the tidal friction, 
By comparison with (65) 
|;[W i -W iii + 1W 1 -1W 3 ]=2<I> 1 F 1 sin 2f 1 + 2<tuF 3 sin 2f : + F 1 G 1 sin gi + IhGo sin g 3 (82) 
Now when we put N=N' (see (43) and (52)) 
1 
y .( 8i ) 
i 
j 
W n = 2F cos 2f.w n and —-=-- = — 4F sin 2f.w n 
at 
Also 
dWn 
W 2 = 2G cos g,w 3 and —2G sin g.w 2 
Then let 
<E>=2w n = 2P i 'Qf_ (yd—q 2 ) 2 —2p 2 q 2 ] 2 + 8 P~Q~(P i -\- Q^)p z q 2 {p z — q 3 ) 2 + 2p 4, f(P 8 -\-Q 8 ) 
= 2P*Q*(p*-qZf+8py(pZ-qzyP 2 Q\P*+Q±-P2Q2) + 2pY(P 8 +4P±Q i +Q 8 ) (83) 
and let 
ir=w !! =p^ i (r 2 -(3=) ! [(p s -2 ! ) 3 -2p 5 2T 
+[i*(P*-3Qy+<y{sP*-Q*) sr \ P y(p>~q ! ‘r+4P*Q*(P i +Q t )pY 
+ 8 P°~QXP*+Q*-P’-Q*)pY (84) 
And we have 
(Ito 
— ~= —[2^ > 1 F 1 sin 2f 1 -j-2<fiF sin 2f+2d> 3 F 2 sin 2f 2 + F 1 G 1 sin gq 
+ FG sin gd-ToGo sin g 2 ]. . (85) 
5 d 2 
