458 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY 



R B displacements, so that there appears to be no justification at all for an argument 

 from analogy. 



In each of the neglected displacements, the change in the potential energy will 

 consist of two terms. There will be a change in the elastic energy of the 

 compressible material, and this can be easily shown to involve an increase in the 

 potential energy. There will, in addition, be a change in the gravitational energy, 

 and this can be shown* to involve invariably a decrease in the energy. If W, E, G 

 denote the total, the elastic, and the gravitational potential energies, 



in which SG is invariably negative. The condition for stability is that for every 

 possible displacement E shall be numerically greater than SG. 



It might naturally be thought that by considering a system in which the matter 

 was, so to speak, very gravitational or very little elastic we could have JE small or 

 SG great, and so should have instability of the spherical configuration. But it must 

 be remembered that the gravitation and the elasticity of the matter are not 

 independently at our disposal. The action of the gravitational forces tends to 

 concentrate the matter and so involves that the elasticity becomes large in the 

 equilibrium configuration. If we consider a system in which the elasticity is 

 artificially kept small, as, for instance, by adding an additional repulsive field of force 

 to annul, or partially annul, the gravitational field, we can easily construct systems 

 for which a spherical configuration is unstable,! but, short of this, it appears to be a 

 general law that the elastic and gravitational agencies must march together in such a 

 way that E is always numerically greater than <5G,| so that every natural spherical 

 system is stable. 



The nearest approach in nature to the artificial repulsive field imagined above is 

 found in the influence of rotation. This influence may be represented by the super- 

 position of the usual repulsive field of centrifugal force of potential %w 2 (x 2 + y 2 ). 

 The field is not spherical, and so the figures of equilibrium obtained under its influence 

 cannot be spherical. But it can be regarded as made up of a spherical part wV 

 and a superposed harmonic disturbance fyufPtf*. The first term is certainly a 

 spherical repulsive field, and will, of course, tend to annul the concentrating influence 

 of gravitation. The problem which requires study is that of how far, or in what 

 circumstances, the presence of rotation can disturb the otherwise general law that SIS, 

 is always greater than SG. 



The problem is one of enormous complexity and great generality. It will hardly 

 be expected that the present paper will contain anything approaching a general 



* Of. J. H. JEANS, "The Stability of a Spherical Nebula," 'Phil. Trans.,' A, vol. 199, p. 1. 

 t Of. J. H. JEANS, "On the Vibrations and Stability of a Gravitating Planet," 'Phil. Trans.,' A, 

 vol. 201, p. 157. 



J Cf. below, 11, 22. 



