THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
755 
Then if these be added and simplified, it will be found that if 
H=|(i>*-2*) [(i> 4 + q l )P 3 Q ! + 2pY(i»- Q-y']. 
Then 
rli T 2 
w—-i-- = — 2 HH sin 2 h sin i. . . . 
dt g 11 
(99) 
( 100 ) 
Then collecting results from the seven equations ( 88 , 90-2-4-6-8, 100 ), 
di r“ 
n~j = ~ sin i{ 2 P 1 F 1 sin 2 f x — 2 pF sin 2 f— 2 P C F 3 sin 2f J -}-20 1 G 1 sin g x 
Ctt> (J 
— 20 G sin g— 2 Q 2 G 3 sin g 3 — 2 HH sin 2 h} . ( 101 ) 
This is only a partial solution, and refers only to the action of the moon on her own 
tides; the part depending on the sun alone may be written down by symmetry. 
The various functions of i and j here introduced admit of reduction to the following 
forms :— 
< 1 >= 4 (i sin 4 i-\-\ sin 3 p(4 sin 3 i— 5 sin 4 sin 4 ./(l — 5 sin 3 sin 4 i )} 
|T = 5 -{ sin 3 i— sin 4 i- j- shr/(l — ^ sin 3 i-\- 5 sin 4 i) 
— sin 4 j (1 — 5 sin 3 sin 4 i )} 
> 
( 102 ) 
Fi + Po cos j{ 1 —f sin 3 i— f sin 3 /(l —| sin 3 i)} 
Fi — p. : cos i {1 — i sin 3 i —2 sin 3 7(1 — f sin 3 «)+-§- sin 4 /(l — sin 3 i) } 
Gl + G-3=¥ cos J 
G1 ~ G 2 := i cos i 1 + 2 sin 3 i— ^ sin 3 j (1 -j- 5 sin 3 i) — f sin 4 ./( 1 — } sin 3 i )} 
p = ^ cos i{ ijr sin 3 i-\- sin 3 ./(l —-§ sin 3 i) — \ sin 4 ./(l — \ sin 3 i) } 
0 = 1 cos i{l — sin 3 i—\ sin 2 j( 1—- 7 - sin 3 i)fi-f sin 4 y(l— sin 3 i )} 
H = i cost'd sin 3 i+1 sin 3 y(l — £ sin 3 i) } 
y ( 103 ) 
$j, $ 0 , Ifi, r 3 are given in equations (67), and <£ and y in equations (80). 
The expressions for p L and p. : are found by symmetry with those for df/ and by 
interchanging i and j; the first of equations (62) then corresponds with the second of 
(103), and vice versd. 
From (103) it follows that 
Fi ■- R-H( G1 ■- G 2) = I cos i( 1 - f sin 3 ./) 
and 
F+iG = i cos 1 ■—f sin 3 ./) 
Also 
Fl+I”3+Gl +Gl+ H— i C0S J 
