/•'. //■ her— Proof of ihi Earth's Rigidity. 845 



drawn ae to the rigidity of the earth by taking as a minimum 

 value of tlu 1 distortion 



. two,' 



are perfectly Bafe. 



\\ :;/• — ;>//, as is the ease tor the theoretical, uniconstant, 

 isotropic solid, and at least approximately also tor glass and 



iron, the result reached by the beautiful, hut Laborious, method 

 of Lame and Thomson is, say 



:'(■♦'■)= 





that one is very safe indeed in taking the ellipticity as at 



EUipticity of a fluid sphere. — Having considered the elastic 



ustance of the earth apart from the influence which the mu- 

 tual attraction of its elements produces, it is next necessary to 

 consider the effect of this attraction apart from the rigidity, or 

 if the ma>s were a fluid. Later it will appear how the re- 

 sults are to be combined so as to include both elasticity of form 

 and attraction toward the center. 



It is very easy indeed to prove that if the earth were a homo- 

 leous, incompressible fluid of mean density 5*5, and if the 

 attraction were directed to a single, central point; the ellipti- 

 city of the equilibrium tide would be ar/g, where g is the ac- 

 celeration of pure gravitation. * It is somewhat more difficult 

 to prove that the mutual attraction of the fluid in its deformed 

 state would augment this ellipticity by 5/2. But these are 

 familiar propositions of analytical mechanics, which no one 

 doubts and which geologists are safe in accepting. 



Tlfe ellipticity of an incompressible homogeneous fluid earth 

 would then be, say, 



5 UT 



e,= - = 162X10V. 



2 9 



*The forces acting on a fluid sphere by the equilibrium theory are — ;, — , and 



g 



gravitation which is expressed by - "- where E is the earth's mass. This leads 



a- 



to a value of the major semi-axis expressed by r = a( \ + ~ — ). It is 



\ ED 3 / 



also reai'.ily proved that, for anv ellipsoid closely approaching the sphere, the 



/ 2 \ :i.M'/ 3 



major serai-axis is all +^r e )- Hence e = (In Xat. Phil, s, 804 this 



ellipticity is erroneously stated at twice this quantity). The ordinary meaning of 

 >j is the resultant of gravitation and centrifugal force: but on the equilibrium 

 theory there is no rotation of the globe, and <j is then equal to E/a*. Substituting 

 this and the value of r, the equation becomes e = ar/g. 



