THE ELEMENTS OE THE ORBIT OF A SATELLITE. 
803 
To shore hore the equations (224) reduce to those of § 10. 
We now pass to the other extreme, and suppose the solar influence infinitesimal 
compared with that of oblateness. 
Here 
oi—Sb, j8=b , y=g, 8 =d 
k ]l — —(a-j-b), k 3 =0 
Then the equations (224) reduce to 
1 dl x _ , a'b — b'a 
L x dt ^ a(a + b) 
1 dL{ _ , a'b—b'a 
I{ dt ^ b(a + b) 
1^? = 0 o 
Z 2 dt ’ If dt 
Therefore L. 2 and L 2 are constant. Also from the relationship between them 
4 a 
Hence it follows that the two proper planes are identical with one another, and are 
fixed in space. They are, in fact, the invariable plane of the system, as appears as 
follows :— 
If we use the notation of § 10, L l =j, L 1 '=i, and L 1 '/L 1 == — (K l -\-a)/o l =b/a l ; so that 
■di=bj. 
Now a—/ l'tC ; 4, b = rt/n ; and i and j are by hypothesis small, therefore we may write 
the relationship between a, b, i, j in the form 
£ • • 
- sin j = n sin i. 
rC 
This proves that the two coincident planes fixed in space are identical with the 
invariable plane of the system (see 108). 
But the identity of equations (225) with (71) of § 10 and (29) of the paper on 
“ Precession ” remains to be proved. 
If i and^ be treated as small, those equations are in effect 
|=-g(*H) 
1 = *W) 
(or with G and D in place of g and d if the viscosity be not small). 
(225) 
