FIGURE AND STABILITY OF A LIQUID SATELLITE. 
203 
On putting a — c cos y, b = c cos /3 , and noting that Q = 
2 F 
1+X 
c SUl 
and that 
7 
c 3 cos y cos ft = a 3 , I find that 
a 
3 
2 , F cot v cos Q — 
2 
1 + sec 2 y + sec 2 /3 
3 — cos 2 y + X (1 + cos 2 /3) 
where 
1+X 
1 + sec 2 y + sec 2 /3 
+ § 
(36)* 
S = 
1 + X 
p + e 
_ 3 (17 + e sec 2 /3) 
1 + sec 2 y + sec 2 /3_ 
However, this is not practically the most convenient form from which to compute 
the distance between the two ellipsoids. 
* It is by no means obvious how this formula is consistent with results which we know by other means 
to be true. In the case when X = as we have a liquid planet rotating with the same angular velocity 
as an infinitely small satellite revolving in a circular orbit in its equator. 
Let us first consider the value of f. In the present case the semi-axes A, B, C pertain to the infinitely 
small satellite, and are therefore negligible compared with terms in a, b, c. Since the axis denoted by c is 
that coincident with the satellite’s radius vector, and since the equatorial plane of the planet must have a 
circular section, we have c — b. 
But since b = c cos /3, it follows that fi = 0 or k sin y = 0. Now y does not vanish for a = c cos y, and 
a is the polar semi-radius of the planet; therefore k = 0. 
If we consider the formula (24) for (, expressing, however, the several terms in the form of (15), we see 
that for X = as 
whence 
f + sis' 
i = 
10r- 
3c 2 
56D 
9+ . 
5c 6 . 
Io+ ,n '■>" + 607i sm ' 
(«). 
The factor of correction to Kepler’s law of periodic times for a small satellite revolving about an 
oblate planet, whose equatorial radius is c and whose eccentricity of figure is sin y, is 1 + where £ is 
expressed by the above series (a). 
Now considering the formula (33 bis), we see that for X = oo and /3 = 0 
In (36) we therefore have 
1 + X 
= Ceos 2 /3 = 
1 + X 
= C 
8.1L + e _ — § (*? + £ sec 2 ft) X = 2( [ 1 _ 3 
1 + X L 1 + sec 2 /3 + sec 2 yj ^ L 3 + tan 2 y_ 
^ hen k — 0, the elliptic integral F is equal to y; thus (36) becomes 
2C tan 2 y 
3 + tan 2 y 
2y cot y - 
3 + tan 2 y 
9 _ 
6 
tan 2 y 
3 + tan 2 y ‘ 3 + tan 2 y 
+ 
2 D 2 
