THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
763 
sin 2 j cos iV=sin 2; 0 cos (ac : 
sin 2/ 
j sin iV = sin 2/ 0 sin (Ac^d-nrq) ——qy- sin 27 0 sin («r^+m 2 ) 
/Cj + a 
(Acfi-f-nij)-}- sin 27 0 cos (ac 2 £ +m 2 ) 
(^+m 2 ) 
• r\ m i A?1 “f- a . . 
sm cos i// =-sin 2j 0 cos 
• • • —[— a # 
sin 27 sin i[j =- 1 -sin 2/ 0 sin (Acd-finy) + 
cl 
• • (H9) 
sin 27 0 sin 
From this it follows that 
sin 27 sin 2 \j cos (iV—t//) = — s j n 2 2j Q -— 
ct k<£ "4~ cc 
sin 3 2i r 
+ ( l + sin 27 0 sin 2 j Q cos [_(k l — K.^t+n^— m 3 ] 
K 2 "I a / 
sin 27 sin 2 j sin ( N—xp) = ( l— Kl+ ) sin 27 0 sin 2 j 0 sin [(/<—+ —m 2 ] 
' k 2 + a / 
Now 
Therefore 
(ACi + «)(«- 2 +a)— — (/f 1 -j-a)(«: 1 -f/3)— —ab 
K l -\-K. 2 -\-2a = a — /3 
sin 27 sin 2 j cos (N—\fj)= — ^ ■ — (a sin 3 27 0 —b sin 3 2j 0 
(a — /3) sin 27 0 sin 2 j 0 cos [(a^ — ac 2 )£+m : — m 2 ]} 
K — fC 
sin 2i sin 2/ sin {.V— xp) = — J + sin 2i 0 sin 2J 0 sin |V—.-,)/ +in —in. 
( 120 ) 
I 
J 
From (120) it is clear that the nodes of the lunar orbit will oscillate about the 
equinoctial line, if 
a sin 3 27 0 —b sin 3 2/ 0 be greater than (a — ft) sin 27 0 sin 2 j 0 , 
but will rotate (although not uniformly) if the former be less than the latter. 
With the present configuration of the earth and moon 
a sin 3 27 0 -»~b sin 2 2 j 0 is very small compared with (a ~/3) sin 27 0 sin 2 j 0 , 
and the nodes of the lunar orbit revolve very nearly uniformly on the ecliptic ; also 
the inclination of the orbit varies very slightly, as the nodes revolve. 
In the investigation in Part II. the secular rate of change in the inclination of the 
