340 G. F. Becker — Proof of the Earth! s Rigidity. 



_2M<k # My 7 __Mz 



These, or equivalent forms, are those used in the theory of the 

 tides and are amply accurate for most problems connected with 

 the subject. 



The total values of these forces acting on the earth are easily 

 found. Consider first the hemisphere to the right of the yz 

 plane. The force acting on each particle is simply propor- 

 tional to its distance from the dividing plane, or to x. Now 

 it is well known that if £ is the distance of the center of 

 inertia of a mass from a given plane,* and if m is the mass of 

 an elementary particle £2 , m=2' (mx). Hence the total force 

 will be proportional to the mass of the hemisphere into the 

 distance of its center of inertia from the origin. For the hemi- 

 sphere £=3a/8 where a is the radius, and if w is the mean 

 density, the whole force acting on the basal plane is say 



._ 2M v/ , 2M 3 2 , 2M na K w ' , . 

 *«X= ^^H=^ 3 .y«.- ?raw= -jj-, — _.' ; (1) 



In precisely the same way one finds fcr the sum of the forces 

 affecting the hemisphere which lies above the xz plane 



M n a* w 



and tlie symmetry of the figure shows that Z=Y. 



The deformation of an elastic sphere of the size and mean 

 density of the earth by a mass so small as the moon (1/83 of 

 the mass of the earth) and at such a distance from it,, is of 

 course very minute. The most probable deformation in fact 

 amounts to an ellipticity of about 1/10,000,000. It may be 

 inferred therefore that the deforming stress on the plane pass- 

 ing through the earth's center of inertia at right angles to the 

 direction of the moon is not very far from uniform. Were 

 this the case the earth would be homogeneously strained, so 

 that each elementary cube would suffer the same distortion. 



Simple strain spheroid. — Suppose a cube of isotropic, homo 

 geneous, incompressible matter (like india rubber) subjected to 

 a uniformly distributed tensile stress in the direction of the x 

 axis. Then by Hooke's law the elongation per unit length 

 will be proportional to the force per unit area, say X ; and if 

 n is the constant called the modulus of rigidity, this elongation 

 is K/Sn. If the mass is but little strained and the volume is 

 constant, this elongation is accompanied by a linear lateral con- 

 traction in each direction which is equal to half the elonga- 

 tion.* 



* If the elongation of the unit cube is a and the linear lateral contraction is (3, 

 while the volume remains coustant, 1=(1 +a) (1 — /5) 2 ; and if a 2 and afi are so 

 small as to be negligible, this reduces to a=2/3. 



