166 
SIR G. H. DARWIN ON THE 
Thus, in the problem of the figures of equilibrium, if one of the two masses is large 
compared with the other, the two must be far apart to secure secular stability. This 
is exactly what is to be expected from the theory of tidal friction, for limiting 
stability is reached when there is coalescence of the two solutions which correspond 
to the cases where each body always presents the same face to the other.* 
The result when the two masses are equal becomes more easily intelligible when it 
is expressed in terms of the radius of either of them. That radius is a/2 13 , so that 
when X = 1 
r = 1738a = 2-191 (i). 
Thus, in the latter case, limiting stability is reached when the two spheres are 
nearly in contact with one another, for if r were equal to twice the radius of either 
they would be touching. 
When the two bodies are far apart, the solution may be obtained by spherical 
harmonic analysis, and has comparatively little interest. But when the bodies are 
equal or nearly equal in mass, limiting stability for the figure of equilibrium would 
seem, from this preliminary investigation, to occur when they are quite close together. 
Accordingly, in finding numerical solutions hereafter, I have devoted more attention 
to this case than to any other. 
Turning now to the solution of the analogue of Roche’s problem, we see that when 
X = 0, r 2 — 0. This would mean that a very small liquid satellite could be brought 
quite up to its planet without becoming unstable. But we shall see that, when the 
satellite is no longer constrainedly a sphere, instability first occurs through the 
variations in the shape of the satellite. This preliminary solution does not, therefore, 
throw much light on the matter, excepting as indicating that we must consider the 
cases where the satellite is as near to the planet as possible. 
Next, when X = 1, we have 
*=• 21 > = gv- . (A) = i ' 549 (A) 
Thus, when the two masses are equal, their distance apart is only about 1|- radii of 
either, and they will overlap. Here again it would seem as if stability would persist 
up to contact, but, as before, instability first sets in through variations in the shape 
of the satellite. 
Finally, when X is large, r 2 also becomes large. This case is the same in principle 
as that considered in the problem of the figures of equilibrium, for it means that if a 
large liquid body (formerly called the satellite) be attended by a small rigid body 
(formerly called the planet), secular stability will be attained when the small rigid 
body has been repelled by tidal friction to a great distance from the large liquid body. 
* See ‘Roy. Soc. Proe.,’ vol. 29, 1879, p. 168, or Appendix G (b) to Thomson and Tait’s ‘Nat. Phil.’ 
