174 PROFESSOR A. E. H. LOVE ON THE 



gravitating sphere as compounded of two stress-systems : a hydrostatic pressure by 

 which gravitation would be balanced if the sphere were in equilibrium, and an 

 additional stress. He proposed to measure the strain, not from the unattainable 

 state of zero stress, but from the equilibrium state ; and he proposed to take the 

 additional stress to be determined in terms of the strain by those equations which are 

 commonly used in the theory of elasticity. To simplify the problem he proposed to 

 take the material in the equilibrium state to be homogeneous and the elasticity to be 

 isotropic, so that the equations connecting the additional stress and the strain are of 

 the same form as the ordinary stress-strain relations of isotropic elasticity. In 

 justification of the proposed procedure he brought forward theoretical considerations 

 founded upon the general theory of energy, and other evidence drawn from an 

 interpretation of the experimental results in regard to the behaviour of elastic solid 

 bodies. It is not too much to say that all the evidence there is, is just as strong in 

 favour of Lord RAYLEIGH'S proposed method as it is in favour of HOOKE'S law, in the 

 sense in which that law is applied in the ordinary theory. The only objection which 

 can be raised against the method, an objection mentioned by Lord RAYLEIGH himself, 

 is that the body to be treated is certainly not homogeneous, and possibly not isotropic. 

 When the proposed method is adopted, the density and the moduluses of elasticity 

 must be taken to have their mean values. The justification for treating the values of 

 these quantities at any point as equal to the mean values, is that it is advisable in 

 the first instance to work out the simplest case.* 



In the first part of this paper the mathematical problem proposed by Lord RAYLEIGH 

 is worked out ; and the conclusion is drawn that the effective moduluses of elasticity 

 of the Earth, in its present state, are sufficiently great for a homogeneous spherical 

 configuration to be thoroughly stable. The second part of the paper is devoted to 

 developing the consequences of supposing that the elasticity of the material of the 

 Earth was once muclrless than it is at present. 



Statement of the Mathematical Problem. 



2. We have before us a perfectly definite mathematical problem, which may be 

 stated as follows : A sphere of radius a, and of uniform density p , is in equilibrium 

 under its own gravitation, and the stress within it is hydrostatic pressure of amount 

 Po at a distance r from the centre. When any small disturbance takes place, so that 



* It may be observed that the method advocated by Lord RAYLEIGH is the same, except for a slight 

 modification, as that which was used in the second edition of my " Treatise on the Mathematical Theory 

 of Elasticity," Cambridge, 1906, in the discussion of the statical problem of a gravitating sphere held 

 strained by external disturbing forces. The modification consists in the assumption, which was there 

 made, that the material might be treated as incompressible. If this assumption is not made, the analysis 

 becomes much more difficult. An earlier indication of the method will be found in a paper by J. LARMOR 

 Cambridge, 'Proc.. Phil. Soc.,' 9, 1898, p. 183. 



