GRAVITATIONAL STABILITY OF THE EARTH. 211 



of this quantity. The constant p. denotes the rigidity and X + f/n the modulus of 

 compression. When the harmonic specifying the vibrations is of zero degree, that 

 is to say, when the vibrations are radial, we found that the critical value of .fa? 

 lies between 4 and 5 if v, or /n/(X+2/x), is zero, and that it lies between 3 and 4 if v 

 is \. In the case of vibrations specified by spherical harmonics of the first degree, we 

 found that the critical value of *a a lies between 7 '22 and 8 if v = 0, and it lies 

 between 6'48 and 7'22 if v \. In the cases of vibrations specified by spherical 

 harmonics of the second and third degrees we found that there is no critical value of 

 s 3 a a , or that the sphere is stable, in respect of the corresponding types of displacement, 

 for all values of X + 2/i. It was to be expected that the critical values of s*a' would 

 increase rapidly as the complexity of the type of vibration, specified by the degree of 

 the appropriate harmonic, increases ; and we appear to be justified in concluding that 

 instability cannot occur in respect of displacements specified by spherical harmonics of 

 any degree higher than the first. 



35. The result that the critical values of fa* are lower when v = { than when v = 

 means that a higher value of the constant X+2/^ is required, to secure stability, 

 when there is considerable rigidity than when there is very little rigidity. This 

 result accords with general dynamical principles ; for it is a known result, and one 

 which has been shown to be in accordance with such principles, that the frequency of 

 any mode of vibration, involving compression, of a sphere free from gravitation 

 diminishes as (X + 2/i)//x, diminishes, that is, as v increases.* Consequently, for a given 

 value of y/j u V, the value of yp t a?f(\+2p.) which would be required, in order to reduce 

 the frequency to zero, diminishes as v increases, or the critical value of X+2/x increases 

 as v increases. 



36. The result that the critical value of sW is lower when n = than when n= I 

 means that a higher value of X+2/t would be required, to secure stability, in the case 

 of radial displacements than in the case of displacements specified by spherical 

 harmonics of the first degree. The spherical body of uniform density could l>e stable 

 in respect of all types of displacement except radial displacements. If the value of 

 sV were intermediate between the critical values corresponding to n = and n = 1, 

 this would be the case ; and the body would tend to take up a different configuration, 

 in which the density would be more concentrated towards the centre. The result 

 that, in the case where n = 1 also, there exists a critical value of *', which 

 is not more than twice as great as the value associated with n = 0, the initial 

 state in both cases being one of uniform density, suggests very strongly that there 

 would be a critical value of X + 2/i, in respect of the case n = 1, even if the configuration 

 were such that the body was stable as regards radial displacements. We should then 

 have a body with a spherically symmetrical distribution of density, but with elasticity 

 too small for this configuration to be stable in respect of displacements specified by 

 spherical harmonics of the first degree ; and it may be inferred that the critical mean 



* Cf. H. LAMB, loc. at., ante, p. 173. 

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