THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
721 
The latter term of this expression will give the perturbing couples ; it is equal to 
CW. 
In the accompanying fig. 1 let X Y Z be axes fixed in space, and (adopting the 
phraseology for the case of the earth) let X Y be the ecliptic; let A B C be axes fixed 
in the planet; let y be the angle A N or B C D; i the obliquity of the ecliptic; xjj the 
longitude of the autumnal equinox from the fixed point X in the ecliptic. 
Fig. 1. 
Now suppose W to be expressed in terms of y, i, \Jj. 
Then the perturbing couples, which act on the planet, are 
dW 
C —tv about N, tending to increase i. 
dW 
C -r— about Z, tending to increase i b. 
f/i/f ’ & T 
dW 
C about C, tending to increase y. 
A 
Now let It, #i, ia be the perturbing couples acting about A, B, C respectively. 
Then must 
dW 
C — It sin i sin y — m sin i cos x+il cos i 
d W 
cos sin y 
Whence 
c f Y=ia 
d X 
1 1 
1 
= -—: cos i — 
C am i 
.d W d\\ r \ . dW 
sm X- di cos x 
M 
i 
d x df) 
. <m dw 
d w . 
f r= ■—d cos I -—- J ] cosy-f —rr sm y 
C sin i \ d x dy]s ) ^ di /v 
U d x 
