THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
753 
Now 
Therefore 
cW r 
dW 
di 
2<3>F sin 2f and F = 4p 3 (pl 2dJrF sin 2f 
di t 3 _ . „ 
n— -7-- = 2F sm 2f 
dt g 
2PO ^ PQ 37 
On substitution from (44) and (83) for <f> and dt and simplification, we find that if 
f=^{P^(P z -Q z )[(p^-qY-2pYJ+2fq\p 2 -~q 2 Y(P 2 -Qy-2pY(P Q -Q 6 )}(89) 
then 
rl o' n -2 
.(90) 
di t 3 „ „„ . _ . 
w— 4-—= — 2 pF sm 2f sm i „ 
dt g 1 
Pas£ semi-diurnal term. 
di t 3 P dW m , 1 dW m 
n dt ~ g — Q ~7tT 2PQ ~dW 
Since W m is found from W l by writing Q for P, and —P for Q, and — 2f 2 for 2f L , 
therefore in this case ndi/dt is found from its value in the slow semi-diurnal term by 
the like changes, and if 
18P 3 g 3 (P 3 --Q>V+4P*(P 3 -3-py} (91) 
.(92) 
fl 1 rr~ 
n - - 7 - — = — 2 poF., sin 2f, sin i 
dt g - - 
*S7ow diurnal term. 
di r 3 1 
/ }1 -- 1 - — —- 
dt ‘ g 2P0 
'P 3 + 3§ 2 cfWj 
2 <&=' 
d Wj 
•2G, sin g 1 [i w (P 3 -3g a )y 2 3 +18P a ^ s (P 3 --Q 3 )y 2 4 
+3^(3P-- g^spv+4 P 3 gy] 
— = 2G 1 smg 1 .r 1 
Substituting these values and simplifying, it will be found that if 
Gi=i{P 4 (P 3 +3g> 8 +2P 3 (P 3 -3g 3 ) 2 pV-9(P 2 ~g 2 ) 3 pY-2g 3 (3P 3 -g 3 ) 3 y)Y 
-q^phw) 
Then 
di t 2 _ ,, . 
o =2 G i<J i sl,1 Si sma 
(93) 
(94) 
