STABILITY OF A GRAVITATING PLANET. 
177 
planet has thrown off a satellite, the former hypothesis leads to a certain inference as 
to the angular momentum of the system ; the latter to an inference as to the radius 
of the primary. The former hypothesis leads to no inference at all as to the size of 
planets which are to be expected—they are as likely to be of tlie size of billiard balls 
as of the size of the planets of our system—while the latter leads to no inference as 
to the angular momentum of the system, but presupposes it to be small. The 
contention of the present paper is that the inferences which are drawn from the 
former hypothesis are not borne out by observations on the planets of our system, 
while those which are drawn from the latter are borne out as closely as could be 
expected. The true hypothesis must of necessity lie somewhere between the two 
extremes which we are comparing, and our evidence seems to show that it is much 
nearer to the latter (gravitational) than to the former (rotational).'^' 
Stresses and Vibrations in the Earth. 
^31. It has already been seen that in dealing wdth a gravitating sj)here of the size 
of the earth it is necessary to take into account terms which are omitted by Lord 
Kelvin and others—the terms which introduce into our equations the gravitational 
tendency to instability. 
It is of some Importance to kno\v wdiether the existing solution for the vibrations 
and displacements of the earth would be altei'ed to an appreciable extent by the 
inclusion of these terms. The general frequency-e<piation which is given on ]). 1G7 is 
too complicated foi’ manipulation, and is, moreover, ojien to the objection that it does 
not represent tlie facts of the case ; for, inside the earth, the strains caused by 
permanent gravitation cannot legitimately be treated as small.t 
§ 32. Considerations of a general nature aauII, however, give us some insight into 
the question. In an imaginary earth, in w'hich X, p, are infinitely great, the 
gravitational terms will be of no importance in comparison wdtli tliose representing 
the elastic stresses. The true solution will, tlierefore, become identical with the 
classical solution in which the gravitational terms are neglected. Jor smaller values 
of X, jj. tlie error wall become appreciable, and it X, p. continue to decrease this error 
will become infinite as soon as the first point of bifurcation is reached ; for at a point 
of bifurcation the application of an infinitesinially small external force will produce a> 
finite displacement in the solid. For intermediate values of X, p the error wall be 
small if X, p are great compared wath the critical values of X, p at the point ol 
bifurcation, and great if X, p are near to these critical values. 
* In addition to the inference as to the size of the planets, the hypothesis of gravitational instability 
leads to a further inference as to the distances of the fixed stars. This has been discussed in my former 
paper, “On the Stability of a Spherical Nebula” IS), and here also the results seem to agree with 
observation as closely as could reasonably be expected. 
t CiiREE, ‘ Camb. Phil. Soc. Proc.,’ vol. II, or Love, ‘ Elasticity,’ vol. 1, 220. 
VOL. CCI.—A. 
2 A 
