164 
SIR G. H. DARWIN ON THE 
“the figures of equilibrium”; the other may be called “Roche’s problem,” and it 
differs only from the former in that one of the two masses of liquid is replaced by a 
particle or by a rigid sphere. However, in the numerical solutions found hereafter, 
Roche’s problem is slightly modified, for the rigid sphere is replaced by a rigid 
ellipsoid of exactly the same form as that assumed by the other mass of liquid in the 
problem of the figures of equilibrium. Thus, with this modification, the two problems 
become identical as regards the shape of the figures; but, as we shall see, they differ 
widely as to the conditions of secular stability. This difference arises from the fact 
that in the one case there are two bodies which may be subject to tidal friction, and 
in the other there is only one. 
If in either problem there is no solution when the angular momentum has less than 
a certain critical value, if for that value there is one solution and for PTeater values 
there are two, then the principle of Poincare shows that the single solution is the 
starting point of a pair of which one has one fewer degrees of instability than the 
other. If, then, one of the two solutions is continuous with a solution which is clearly 
stable, it follows that the determination of minimum angular momentum will give us 
the limiting stability of that solution ; and this is the point of greatest interest in all 
such problems. 
Our two problems differ in the value of the angular momentum of which the 
minimum has to be found. For, if in Roche’s problem the second body is a particle, 
it has only orbital momentum; if the second body is a sphere, it must be deemed to 
have no rotation ; and, finally, in the modified form of the problem, the rotational 
momentum of the rigid body must be omitted from the angular momentum, which 
has to be a minimum for limiting stability. 
It will be useful to make a rough preliminary investigation of the regions in which 
we shall have to look for cases of limiting stability in the two problems. For this 
purpose I consider the case of two spheres as the analogue of the problem of the 
figure of equilibrium, and the case of a sphere and a particle as the analogue of 
Roche’s problem. 
Let p be density, and let the mass of the whole system be f 77-pa 3 ; let the masses oi 
the two spheres be |-7rpa 3 X/(l +\) and |7rpa :! /(l + A), or for Roche’s problem let the 
latter be the mass of the particle. 
Let r be the distance from the centre of one sphere to that of the other, or to the 
particle, as the case may be ; and oj the orbital angular velocity. 
In both cases we have 
The centre of inertia of the two masses is distant r/( 1+A) and \r/(l+A) from their 
respective centres, and we easily find the orbital momentum to be 
4 3 2 
§7rpa cor 
(l+xr 
