/•". Becker— Proof of th( Earth's Rigidity. 841 



Suppose that the cube were also subjected to a compressive 



force Y acting at right angles to X and to a compressive force 



Z at right angles to each ot the others. Then if the height of 



the cube is ~<i and if a, ,i y are the displacements on the three 



s, the cube will be elongated in the direction of x by 



2aX 2aY 2aZ 



a = + — + 



Sfi 6fi 6w 



and it will be contracted in the direction of y by 



, 2aX laY 2aZ 



/i= — + — — 



(w/ ■'<// on 



and }- will be symmetrical with ft. 



In a Bpecial case let 2Z=2Y=X. Then 



1277 ' ' ~ y ~ ITT' 



w this is precisely the effect which would be produced by 

 simple tensile force, P in the direction of x, unaccompanied 

 bv lateral forces, if 



P - 1 X. (2) 



asider a sphere of radius a inscribed in the cube before 

 the application of the force P. After the mass was strained 

 the sphere would have become an ellipsoid the major semi-axis 

 of which would be 



a / X\ / P \ 



a + — =a( 1+ — =«(1 + — 

 \ 2/1/ V 3/i/ 



and the minor semi-axes would be each 



P 



P 1 



"2/v/l+P 

 3// 



ft / X \ / 

 a = a( 1 ) = ct[ 1 



2 \ W \ 

 The ellipticity of this ellipsoid would be 



c -r.) 



Here P is by supposition a force so small that its square can be 

 elected and thus 



(14-P3»)- , = l-P3/i 



and hence a 



P 



2/1 V ' 



