ORAVITATIONAL STABILITY OF THE EARTH. 215 



rigidity of surface rock. Assuming this explanation, we are led to attribute to surface 

 rocks an average rigidity approximately equal to GxlO 11 C.G.S. units, and to the 

 Earth as a whole the much higher mean rigidity 1 '38 x 10 ia C.G.S. units; further, 

 since the ratio of velocities of the first and second set of tremors is approximately 

 2 : 1, we are led to assume for X+2/i the value 5'53x 10" C.G.S. units, and for v, or 

 /x/(X + 2jt), the value ^. By analogy to the " tidal effective rigidity " we may introduce 

 the phrases " seismic effective rigidity " and " seismic effective modulus of compression " ; 

 and the values of these quantities would be 1'38 x 10" and 3'69 x 10 13 C.G.S. units 

 respectively. When the value of X + 2/i for the Earth is taken to be 5'53x 10", the 

 corresponding value of .<r*a' is 0'625. The results of 39 appear to warrant the 

 conclusion that the moduluses of elasticity of the Earth in its present state are 

 sufficiently great to render a spherically symmetrical configuration completely stable. 



41. In obtaining the above values for X + 2/i and p. no account is taken of 

 gravitation or initial stress, and it is possible that the most appropriate values would 

 be a little different from those found atx>ve if gravitation and initial stress, to say 

 nothing of heterogeneity of density, could be taken into account. For this reason, 

 although a complete solution of the problem of wave-propagation in a gravitating 

 planet, even when it is regarded as homogeneous, cannot be obtained, the following 

 argument may not be without value : The equations of vibratory motion of a 

 gravitating sphere in a state of initial pressure have been obtained in 3 alx>ve. 

 From equations (10) and (11) of 3 we can deduce the equation 



(1Q3) 



and the three equations of the type 



at* f 

 where CT T , tsr y , or. denote the components of rotation, so that 



9 3w ov , 1ft ,v 



CT ^ " "" ~~ * ...... * I 1 v *J I 



oy oz 



In a general way we can see that the terms which contain * in these equations are 

 small compared with the remaining terms ; for, if waves of length L are propagated, 

 V 3 A is of the order Lr*A, and s*A is small in comparison with this in the order *!/, 

 which is comparable with L a /a', since **a* is comparable with unity. It would thus 

 appear that the velocities of propagation of the waves are not much affected by 

 gravitation and initial stress when the wave-length is small compared with the radius 

 of the sphere ; and the conclusion would be applicable to superficial waves as well as 

 to waves of dilatation and waves of distortion, because such waves are, in any case, 

 to be investigated by means of equations of the types (103) and (104). 



