Forces derivable from a Potential of the Second Degree. 43 

 we get 



= 0, 



or \ a 



(15) 



J 



where p is the perpendicular from the centre on the tangent plane. 



On the " stress-difference " theory of rupture an objection to the 

 application to the earth of the results obtained by applying the 

 elastic solid theory to a perfect sphere, is that the surface values of 



^*L S~*^ ^^N /^ ^ 



nn 00 and nn it would, for ordinary values of 17, be simply enor- 

 mous.* This objection, however, ceases to hold when the earth is 



X^S /*"^N 



treated as incompressible and truly spherical, because tt and 00 then 

 vanish, as well as nn. It is thus important to know what happens 

 in the case of an incompressible material when the surface is slightly 

 spheroidal. To do so, put r\ == ^ in (15), and we find 



"1 

 f 

 J 



nn =. 0, 



it = 



(I 6 )- 



(a2-C*)/285, 



00 = Sirpp^a 2 c 2 )(l-3r 2 /a 2 )/285. 



Over the surface the maximum stress-difference, rf, is the equa- 

 torial value of M 00, and is given by 



S = 8/y 2 (a a c*)/95 (17). 



Substituting for /, a, c values suitable to the case of a homogeneous 

 " earth," we find that approximately 



S = 9'4 tons weight per square inch (18). 



This is large enough to show that even if the earth be supposed 

 incompressible, the consequences of its mutual gravitation cannot 

 safely be ignored. 



The strains and displacements in the general case of gravitation in 

 a nearly spherical ellipsoid may easily be deduced from (6) and (7). 

 From the expressions for the changes in the lengths of the semi-axes 

 we get 



&i 4 TTUP^O? f 2a 2 & 2 c 2 2(l 3/ 3 



(20;. 



* See ' Phil. Mag.,' Sept., 1891, p. 247. 



