342 G. F Becker— Proof of the Earth's Rigidity. 



First approximation to the earth's bodily tide. — On the hy- 

 pothesis that the earth is homogeneously strained, I have only 

 to find P of the last formula in terms of the forces applied to 

 the earth. This has really been done in a preceding paragraph. 

 Equations (1), (2) and (3) give 



_, 3 v 3 2M a*w 1 _ 



"= TT^ = TT * ^FTs ' ——=—TaW 



2 2 D 3 4 2 



where r=3M/2D 8 is introduced for brevity; and if e s is the 

 ellipticity of the earth regarded as a simple strain ellipsoid 



5 

 e. = raw. 



'20n 



Character of the approximation. — ~No use can be made of 

 the value e s unless it can be shown that it is less than the true 

 value. It has been found on the hypothesis that the stress is 

 uniformly distributed and that the mass is incompressible, and 

 neither supposition is strictly correct. 



If one considers the true distribution of stress, it is almost 

 self-evident that the applied force will be greater near the 

 major axis than near the periphery of the yz section. But 

 this can easily be proved. Suppose a slender cylinder of the 

 elastic spheroid close to its major axis, separated from the sur- 

 rounding mass as if by a diamond drill. Then the force acting 

 upon this cylinder will be its mass into the distance of its cen- 

 ter of inertia from the origin. But this distance is a/2 while 

 the center of inertia of the hemisphere is only Sa/8 from the 

 origin. The cylinder will therefore be much more elongated 

 than it would be if the sphere were homogeneously strained. 



When this axial cylinder is connected with the surrounding 

 mass, it will be held back to some extent and cannot be so 

 much elongated as if it were free. But the fact that, when 

 free, it is elongated by more than the average amount per unit 

 length, proves that even when under constraint there is a ten- 

 dency to greater elongation, which must be operative to some 

 extent. 



If the two lateral compressions due to the forces Z and Y 

 are considered, it is clear that these strains will also be more 

 intense near their respective axes than is assumed on the 

 hypothesis of homogeneous strain. Thus the approximation 

 underestimates the longitudinal extension and underestimates 

 the lateral contraction ; in short, it underestimates the ellipti- 

 city of an incompressible mass. 



The famous result of Sir William Thomson for the case of 

 incompressibility is, say, 



e r = raivzs \'05e. 



19?i 



