FIGURE AND STABILITY OF A LIQUID SATELLITE. 
171 
If we differentiate these with respect to A, substitute in the second equation of 
equilibrium and give to ad its value in terms of r 2 , we find that the result is 
Now 
1 —A 
(1 + A) 3 ’ 
i I 15 ^4-3 = 0 
r 3+ 2 G' r 6 
_ 5 A 2/3 — 1 
3 (1 + A ) 8/3 ’ 
and 
F[ 
G' 
3 1 + A 1/3 +A 2/3 
5 (1+A) 1/3 (1 + A 1/3 )' 
Hence the equation for determining r for a given value of A is 
(3). 
This cubic has three real roots of which one is negative and has no physical 
meaning ; the second gives so small a value to r that the smaller sphere is either 
wholly or partially inside the larger one. The third root is the one required. 
In order to present the result in an easily intelligible form it may be well to express 
it also in terms of the radius of the larger of the two spheres, say a 1} where 
a 3 _9 
2 
1 + A 1/3 + A 2/3 
a 
r 3 2 (1 +A ) 1/3 (1 + A 1/3 ) 
+ 3 = 0 
The following is a table of solutions for various values of A :— 
A 1/3 . 
r/a. 
r/ai. 
r/&i - (1 + A 1/3 ). 
o-o 
1-304 
1-304 
0-304 
o-l 
1-323 
1-323 
0-223 
0-2 
1-368 
1-371 
0-171 
0-3 
1-426 
1-438 
0-138 
0-4 
1-486 
1-517 
0-117 
0-5 
1-543 
1-604 
0-104 
0-6 
1-590 
1-697 
0-097 
0-7 
1-625 
1-793 
0-093 
0-8 
1-649 
1-893 
0-093 
0-9 
1-662 
1 • 995 
0-095 
1-0 
1-666 
2-099 
0-099 
The solution is exhibited in fig. 1 , the larger sphere being kept of constant size and 
the successive smaller circles representing the smaller sphere. Many of the circles 
pass nearly through one point, and it has not been possible to complete them without 
producing confusion. 
The fourth column of the table gives the excess of r above the sum of the two radii 
of the spheres, and it shows what interval of space is unoccupied by matter. It is 
remarkable how nearly constant that interval is throughout a large range in the 
values of A. 
z 2 
