172 PROFESSOR A. E. H. LOVE ON THE 



alterations of density. In any body the tendency is partially held in check by the 

 elasticity of the body, and, in particular, by the elastic resistance which the body 

 offers to compression. If this resistance is sufficiently great, the body is stable, iu 

 spite of the tendency to instability which arises from gravitation. It is important to 

 determine the conditions of stability for bodies of various forms and constitutions, 

 with various distributions of density. The problem of the stability of spherically 

 symmetrical configurations of a quantity of gravitating gas has been investigated by 

 J. H. JEANS,* and he has drawn from his investigations some interesting conclusions 

 in regard to the course of evolution of stellar and planetary systems. In a subse- 

 quent memoirt he proceeded to investigate a similar problem in regard to gravitating 

 bodies of a. more coherent character. A gravitating solid body, such as a planet may 

 be conceived to be, might exist in a spherical shape with a spherically symmetrical 

 distribution of density. In the absence of gravitation there could be no question of 

 instability. The effect of any local condensation would be to set up vibrations, and 

 the frequency of the vibration of any spherical harmonic type would depend upon the 

 elasticity of the material. If the resistance of the material to compression is suffi- 

 ciently high the stability persists in spite of gravitation. There are thus two 

 competing agencies : gravitation, tending to instability, and the elasticity of the 

 material, tending to stability. In a general way it is clear that, as the elasticity 

 diminishes, the frequency of vibration of any type also diminishes ; and, if the 

 frequency can vanish for sufficiently small elasticity, the planetary body possessing 

 such elasticity cannot continue to exist in the spherically symmetrical configuration. 

 The problem is to determine the conditions as regards elasticity in which the 

 instability occurs. 



A grave difficulty presents itself at the outset. In the equilibrium configuration 

 the gravitating planet is in a state of stress ; and, in a body of such dimensions as the 

 Earth, this stress is so great that the total stress existing in the body when it vibrates 

 cannot be calculated by the ordinary methods of the theory of elasticity. In that 

 theory it is ordinarily assumed that the body under investigation is in a state so little 

 removed from one of zero stress that the strain, measured from this state as a zero ol 

 reckoning, is proportional to the stress existing at any instant. In order that this 

 assumption may be valid, it is necessary that the strain which is calculated by means 

 of it should be so small that its square may be neglected. Now if we apply the 

 equations of the ordinary theory to the problem of a solid sphere strained by its own 

 gravitation, and if we take the sphere to be of the same size and mass as the Earth, 

 and the material of which it is composed to possess moduluses of elasticity as great as 

 those of ordinary steel, we find that the strains may be as great as , and thus the 

 strains are much too great for the assumption to be valid. The initial stress existing 



* " The Stability of a Spherical Nebula," London, 'Phil. Trans. Roy. Soc.,' A, vol. 199 (1902), p. 1. 

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

 Soc.,' A, vol. 201 (1903), p. 157. Quoted below as " JEANS (1903)." 



