STABILITY OF A GRAVITATING PLANET, 
159 
it is not at present possible to say whether it is more likely to be greater or less 
than unity. 
§ 4. Now, in the case of the earth (Thomson and Ta.it, § 838), we have 
a = 640 X 10® centims,, = 5‘5, 
and the value of y in C.G.S. units is known to he 
y = 648 X 10-1®. 
This gives for yp^o? the value 
=8 X lOii, 
whence it appears that for a sphere of the size and mass of the earth the spherical 
configuration will be unstable unless \ has a value comparable with 8 X lOH. 
Now for steel {cf. Thomson and Tait, p. 435) the values of the elastic constants in 
absolute units are n = p. = 7'7 X lO^, m = \ p = 16'0 X lOH, whence 
\ = 8‘3 X lOii. We therefore see that the critical values of the elastic constants in 
the case of the earth are comparable with those of steel. 
The foregoing calculation is, of course, very rough, but it shows that the critical 
values for the earth are at least in the neighbourhood of what must be supposed to be 
the actual values, so that we are driven to attempting a more accurate determination 
of these values. If the view of the jiresent paper is sound, this approximate ecpiality 
is more than a mere coincidence ; we shall see that it could have been predicted frum 
our hypotheses of planetary evolution. 
We now attempt a rigorous mathematical investigation of certain problems which 
have a bearing upon the astronomical questions in hand. Those readers whose 
interest lies in the application of the results rather than in the processes by which 
they are obtained may be recommended to turn at once to § 22. 
The Stability of a Gravitating Elastic Solid. 
The Equations of Small Vibrations. 
§ 5. We shall begin by discussing the principal vibrations and the frequency 
equation of a spherically symmetrical solid. The case of a non-gravitating sphere 
has been fully discussed by Professor Lamb,* but the inclusion of the gravitational 
terms, as will be seen later, brings about a considerable complication in the analysis. 
The case of a gravitating but incompressible sphere has been considered liy 
Bromwich, t but this has no bearing on the present problem, in which the whole 
“On the Vibrations of an Elastic Sphere,” ‘Proc. Lond. Math. Soc.,’ vol. 13, p. 189. 
t “On the Influence of Gravity on Elastic Waves, Vc.,” ‘Proc. Lond. Math. Soc.,’ vol. 30, p. 98. 
