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V. On the Vibrations and Stability of a Gravitating Planet 
By J. H. Jeans, B.A., Isaac Newton Student, and Fellow of Trinity College, 
Cambridge. 
Communicated by Professor G. H. Darwin, F.R.S. 
Received November 8,—Read December 4, 1902. 
Introduction. 
§ 1. In a former paper"^" I have considered the effect of gravitation as a factor tending 
towards instability, in the case of a spherical nebula of gas. The object of the jiresent 
paper is to investigate the analogous problem in the case of a spherical planet, the 
planet being supposed composed of solid or fluid matter. The main question at issue 
is the following. 
§ 2. So long as gravitation is neglected there can be no doubt as to the stability of 
an elastic solid; any displacement increases the potential energy, and an unstressed 
configuration of equilibrium is therefore necessarily stable. But when gravitation is 
taken into account, the gravitational energy may be either increased or decreased by 
a displacement from equilibrium, and if a displacement can be found which effects a 
decrease in the gravitational potential energy of amount sufficient to outweigh the 
increase in the potential of the elastic forces, then the equilibrium configuration will 
be unstable. 
Now, in § 2 of the previous paper already referred to, it was shown that for any 
spherical body displacements can be found such that there is a decrease in the 
gravitational potential. This is sufficient to prove that a spherical configuration of 
equilibrium may be unstable. 
In the terminology of PoiNCAREf it appears that on any “linear series” of 
spherical configurations there may be “ jioints of bifurcation.” 
We must, therefore, attempt to settle the position of these points of bifurcation. 
In particular, it will be of interest to examine whether a sphere of the size and 
material of the earth may be regarded as being anywhere in the neighbourhood of a 
point of bifurcation. 
* “ The Stability of a Spherical Nebula,” ‘ Phil. Trans.,’ A, vol. 199, p. 1. 
t ‘ Acta. Math.,’ vol. 7, p. 259. 
VOL. cci.— A 335. 
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