FIGURE AND STABILITY OF A LIQUID SATELLITE. 
241 
Tidal friction is a slowly acting cause of instability, and from the point of view of 
cosmical evolution the partial stability of Roche’s ellipsoids is of even greater interest 
than the full secular stability of the system. 
The limiting stability of Roche’s liquid satellite is determined by the consideration 
that the angular momentum of the system, exclusive of the rotational momentum of 
the planet, shall he a minimum. This exclusion of a portion of the momentum of the 
whole system corresponds with the fact that we are to disregard the tidal friction of 
the planet as a cause of instability. If all possible cases of the liquid satellite be 
arranged in order of the corresponding (partial) angular momentum of the system, it 
is clear that for given momentum there will in general be two forms of satellite ; but 
when the momentum is a minimum the two series coalesce. If then we proceed in 
order of increasing momentum, the configuration of minimum is the starting point of 
two series of figures; it is a figure of bifurcation, and one of the two series has one 
fewer degrees of instability than the other. 
One of the two series is continuous with the case of a liquid satellite revolving 
orbitally at an infinite distance from its planet, and this is a stable configuration. 
Moreover, M. Schwarzschild has shown # that the whole series of Roche’s ellipsoids 
does not pass through any other form of bifurcation. Hence we conclude that of the 
two series which start from the configuration of minimum • momentum, one is stable 
and the other unstable. 
The unstable series of solutions is continuous with a quasi-ellipsoidal satellite, 
infinitely elongated along the radius vector of the orbit, and the radius vector itself 
is infinite. Since two portions of matter cannot occupy the same space, the infinite 
elongation of the satellite would be physically impossible, unless the order of infinity 
of the radius vector were greater than that of the longest axis of the satellite. Now 
it appears from the numerical results of § 25 that this is not the case, and that the 
satellite becomes more rapidly elongated than the radius vector increases. Hence if 
the solution of the problem were exact we should reach a stage at which the two 
masses of liquid would overlap. I shall endeavour hereafter to consider the 
interpretation which should be put on this result. 
A series of solutions for Roche’s ellipsoid in limiting stability is tabulated in § 21, 
and the table gives the radius vector and the three semi-axes of each body. The 
unit of length adopted is the radius of a sphere whose volume is equal to the sum of 
the volumes of the two masses. Three of these solutions are illustrated in figs. 2, 3, 4. 
The section shown is that passing through the axis of rotation and the two centres, 
but the places are marked which the extremities of the mean axes would reach if the 
section had been taken at right angles to the axis of rotation. 
The table of § 21 shows that the radius vector at which instability sets in only 
changes from 2'457 to 2'514, whilst X, the ratio of the mass of the satellite to that of 
the planet, changes from zero to unity. The distance between the vertices of the 
* See reference in § 18 above. 
2 I 
YOL. CCVI.—A. 
