THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
759 
£ . . clN . . d\Jr , . ,. .. 
k sm ^ ~dt~ n sm sm A lJ rj) 
But £/Jc is proportional to the moment of momentum of the orbital motion, and n is 
proportional to the moment of momentum of the earth’s rotation, and so by the defini¬ 
tion of the invariable plane 
| 
k 
sin j—n sin i 
(108) 
Wherefore and it follows that the two nodes remain coincident. This 
at at 
result is obviously correct. 
In the present case, however, there is another disturbing body, and we must now 
consider 
The perturbing influence of the sun. 
Accented symbols will here refer to the elements of the solar orbit. 
We might of course form the disturbing function, but it is simpler to accept the 
known results of lunar theory; these are that the inclination of the lunar orbit to 
the ecliptic remains constant, whilst the nodes regrede with an angular velocity 
f2V 
4 \/2 ) _ 
M r 
h n 
Now i\n 
n'v 
fi cos j. 
It'. 
/ 2 =-|(f/ 2 /3 ) X—in our notation. 
7- 
Hence I shall write ^ ~ for 
JfTV 
An 
1 _ 3 {A 
s n 
n, although if necessary (in Part IV.) I shall use the more accurate 
formula for numerical calculation. 
For the solar precession and nutation we may obtain the results from (107) by 
putting j =■ 0 , and r for r. 
Thus for the solar effects we have 
dj 
dt 
= 0 
dN . t' 
(109)' 
dt 
di 
dt ° 
. . d -dr . . . . 
n sm j- 7 -=-hC sm 2i 
dt * 
j 
* The following seems worthy of remark. By the last of (109) we have dy(s]df— —r'e cos i/n. 
In this formula t is the precessional constant, because the earth is treated as homogeneous. 
The full expression for the precessional constant is (2C—A—B)/2C, where A, B, C are the three prin¬ 
cipal moments of inertia. 
Now if we regard the earth and moon as being two particles rotating with an angular velocity n about 
5 e 2 
