FIGURE AND STABILITY OF A LIQUID SATELLITE. 
243 
The general effect must he very like what results from the second approximation 
to the pear-shaped figure of equilibrium,* for I found that the ellipsoidal form was 
but slightly changed over the greater part of the periphery, whilst a protrusion 
occurred at one end—in this present case pointing towards the planet. 
In figs. 2, 3, 4, the protrusions at one end and the bluntening at the other, as 
computed from the third, fourth, and fifth harmonics, are indicated by dotted lines. 
It appears from these figures that, at least up to the point when instability sets in, 
the ellipsoid remains surprisingly near to the correct solution. 
For an infinitely small satellite minimum radius vector also gives minimum angular 
momentum, so that the closest possible satellite is also in a state of limiting stability. 
But this is not the case for finite satellites, and there exists an unstable ellipsoidal 
satellite with smaller radius vector than is consistent with stability. Thus for a 
satellite of four-fifths of the mass of the planet the minimum radius vector is 
2‘36, whilst stability ceases at a distance of 2 , 50. Again, for equal masses 
stability ceases at 2'514, whilst the possibility of an ellipsoidal solution extends 
to 2-343. 
If we follow the forms of the more and more elongated satellites, when the radius 
vector has begun to increase again, we find explicitly in the case of equal masses, 
and with practical certainty for all ratios of masses, that the distance between the 
two vertices continues to diminish and finally becomes negative. At this stage the 
two masses overlap, a conclusion which is, of course, physically impossible. But 
the calculation is based on the assumed adequacy of the approximations, and it is 
certain that the harmonic deformations of the ellipsoids increase rapidly, so that 
each body puts out a protrusion towards the other. The two masses of liquid must 
therefore really meet before we reach the stage of overlapping ellipsoids. As far as 
can be seen, the approximation has become very imperfect—perhaps evanescent— 
before the two ellipsoids cross. It will be best to continue the discussion of the 
meaning of this result after we have considered the true secular stability of the two 
masses of liquid. 
If a satellite, being a particle, revolves about a rotating planet, whose tides are 
subject to friction, there are, for given angular momentum, two configurations (if any) 
in which the planet always presents the same face to the satellite. In one of these, 
which is unstable, the satellite is close to the planet; in the other, which is stable, it 
is remote.! If the angular momentum of the system be diminished, the radius vector 
of the stable configuration diminishes and that of the unstable one increases until the 
two coalesce. For yet smaller angular momentum there is no configuration possible 
in which the planet shall always present the same face to the satellite. We see then 
that amongst all possible configurations in which the planet presents the same face to 
* See ‘Stability,’ referred to in the Preface. 
t See ‘Roy. Soc. Proc.,’ No. 197, 1879, or Appendix G (h) to vol. IT. of Thomson and Tait’s ‘Natural 
Philosophy.’ 
2 I 2 
