536 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
The conditions of stability of rotating masses of fluid are unfortunately unknown, and 
it is therefore impossible to do more than speculate on the subject. 
The most obvious explanation is similar to that given in Laplace’s nebular hypo¬ 
thesis, namely, that the planet being partly or wholly fluid, contracted, and thus rotated 
taster and faster until the ellipticity became so great that the equilibrium was unstable, 
and then an equatorial ring separated itself, and the ring finally conglomerated into a 
satellite. This theory, however, presents an important difference from the nebular 
hypothesis, in as far as that the ring was not left behind 240,000 miles away from the 
earth, when the planet was a rare gas, but that it was shed only 4,000 or 5,000 miles 
from the present surface of the earth, when the planet was perhaps partly solid and 
partly fluid. 
This view is to some extent confirmed by the ring of Saturn, which would thus be a 
satellite in the course of formation. 
It appears to me, however, that there is a good deal of difficulty in the acceptance 
of this view, when it is considered along with the numerical results of the previous 
investigation. 
At the moment when the ring separated from the planet it must have had the 
same linear velocity as the surface of the planet; and it appears from Section 22 that 
such a ring would not tend to expand from tidal reaction, unless its density varied 
in different parts. Thus w r e should hardly expect the distance from the earth of 
the chain of meteorites to have increased much, until it had agglomerated to a con¬ 
siderable extent. It follows, therefore, that we ought to be able to trace back the 
moon’s path, until she was nearly in contact with the earth’s surface, and was always 
opposite the same face of the earth. Now this is exactly what has been done in the 
previous investigation. But there is one more condition to be satisfied, namely, that 
the common speed of rotation of the two bodies should be so great that the equilibrium 
of the rotating spheroid should be unstable. Although we do not know what is the 
limiting angular velocity of a rotating spheroid consistent with stability, yet it seems 
improbable that a rotation in a little over 5 hours, with an ellipticity of one-twelfth 
would render the system unstable. 
Now notwithstanding that the data of the problem to be solved are to some extent 
uncertain, and notwithstanding the imperfection of the solution of the problem here 
given, yet it hardly seems likely that better data and a more perfect solution would 
largely affect the result, so as to make the common period of revolution of the two 
bodies in the initial configuration very much less than 5 hours A Moreover we obtain 
no help from the hypothesis that the earth has considerably contracted since the shed¬ 
ding of the satellite, but rather the reverse; for it appears from Section 24 that if the 
earth has contracted, then the common period of revolution of the two bodies in the 
* This is illustrated by my paper on “ The Secular Effects of Tidal Friction," ‘ Proc. Roy. Soc.,’ No. 197, 
1879, where it appears that the “ line of momentum ” does not cut the “ curve of rigidity ” at a very small 
angle, so that a small error in the data would not make a very large one in the solution. 
