(•. /•. Becker— Proqf of th, Earth" % Rigidity. 





Tnfiuen compressibility. — Ir will In- well to consider 



first iIm' effect of compressibility on rlu> homogeneously strained 

 spheroid already discussed. It'/' is the coefficient of compr 

 sibility (becoming t when the mass i- incompressible) the 

 elongation of a cube of height 2a becomes 



■W + \) 



and the accompanying lateral contraction is 



The elongations and contraction- tor Y and Z are of course of 

 the same form. If these values are introduced into the expr 



os for a, t i, r, it will be found that the terms in & cancel, and 

 that the ellipticity is exactly the same as for the case of incom- 

 preesibility. 



To find the effect of compressibility upon the sphere when 

 the distribution of strains is the real one, it is more convenient 

 to deal with the polar components of the forces or 



F. = ~(a cos' S-l);F, = | r 3 sin S 



cos ^. 



The latter or tangential component will have no effect upon 

 the volume. The forces of this form will form couples tending 

 to increase the angle which each diameter originally made with 



the yz plane. Compression or dilatation will be produced by 

 the component which is directed toward or from the center, or 



* For the very simple method by which those formulas are established, see 

 Thomson aud Tait, Natural Philosophy, §683, or Thomson's article on Elasticity, 

 in the Encycl. Brit. 



Am. Jocr. Sci.— Third Series, Vol. XXXIX. Xo. 233— May, 1890. 

 23 



