GRAVITATIONAL STABILITY OF THE EARTH. 173 



iu the gravitating planet, the stress by which the self-attraction of the body IK 

 equilibrated, is much too great to permit of the application of the ordinary theory. 

 The same difficulty presents itself in every problem concerning the elasticity of a 

 gravitating planet, for example, in the problem of tidal deformation or of the stress 

 produced in the interior by the weight of continents. In these problems the difficulty 

 was turned by Lord KELVIN* and Sir G. H. DARWIN! by taking the modulus of 

 compression to be much greater than that of any known material, in other words, by 

 taking the material to lie incompressible. Their object was to determine the degree 

 of rigidity which must l>e assigned to the Earth, and for that object it is permissible 

 to turn the difficulty in this way. When the problem is that of gravitational 

 instability this artifice cannot be adopted, because the whole question is that of the 

 degree of compressibility which is admissible if the gravitating planet is to be stable 

 in a spherically symmetrical configuration. The artifice adopted by JEANS (1903) 

 consisted in annulling the initial stress by introducing an imagined external field of 

 force to equilibrate the self-attraction of the planet. 



The problem thus posed is an artificial one, which may, nevertheless, throw light on 

 the actual problem. When the initial configuration is taken to be one of uniform 

 density, the analysis of the problem is of the same kind as that which presents itselt 

 in the problem of the vibrations of an elastic sphere, a problem which has been worked 

 out very completely by H. LAMB.| The determination of the effect produced by 

 gravitation in lowering the frequencies of the various modes of vibration is reduced to 

 a question of troublesome analytical computation. JEANS worked out the problem on 

 the basis of the ordinary theory of elasticity, using the elastic constants X and p. of 

 I ..\ M f:. The constant /* is the niodulus of rigidity, and the constant X is such that 

 X+f/it is the modulus of compression. In the case of the Earth the values of these 

 constants can be inferred from the observed rates of propagation of the various types 

 of disturlwuice which are perceived as earthquake shocks. He -concluded that, when 

 the proper values are attributed to these constants, the Earth must be held to be in 

 a state far removed from one of gravitational instability ; but he suggested that, if 

 the resistance to compression was at one time considerably smaller than it is now, the 

 spherically symmetrical configuration would then have been unstable ; and he held 

 that there are traces of the instability in the distribution of land and water on the 

 surface of the globe. 



The actual problem differs from this artificial problem in the mode of balancing of 

 the internal gravitation. Lord RAYLEIGH has proposed a method of meeting the 

 difficulty as to initial stress. He proposed to consider the stress in the vibrating 



* See, in particular, KKF.VIN and TAIT'S ' Natural Philosophy,' Part II., S 833-846, Cambridge, 1883. 

 t " On the Stresses caused in the interior of the Earth by the weight of Continents and Mountains," 

 London, 'Phil. Trans. Roy. Soc.,' 173, 1882, p. 187. 



} "On the Vibrations of an Elastic Sphere," London, 'Proc. Math. Soc.,' 13, 1882, p. 189. 



"On the Dilatational Stability of the Earth," London, 'Proc. Roy. Soc.,' A, 77, 1906, p. 486. 



