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



or 



t/p t £• q C/y 



, /a = e. • = e R 



r/g+1 6, + e, 



where R is the fraction given in the table. If instead of tak- 

 ing Thomson's value for the ellipticity of the elastic spheroid, 

 or e n my rough approximation, e s , is substituted, e simply 

 becomes e g &. In the table R and S are given in decimals in 

 order to show how far they differ. Excepting for this purpose 

 the third decimal is of no value whatever, indeed the second 

 decimal is also nearly if not quite worthless. Now for glass 

 S=0'98 R and consequently is substantially identical with R. 



Both R and S are in fact underestimated ; for the earth is 

 certainly compressible to some extent and, as has been shown 

 above, compressibility increases e r and decreases e g . Hence the 

 deformation of the solid globe as compared with a fluid one of 

 the same mean density is distinctly understated in the above 

 formulas or in other words the globe is really more rigid than 

 the formulas imply.* The extent to which the rigidity of the 

 earth is understated by the formulas however must be quite 

 insignificant, unless indeed the earth contains ellipsoidal shells 

 exhibiting a higher degree of compressibility than is shown by 

 ordinary solids and liquids, f 



Whether R or S is used in the formulas one may thus safely 

 state that, if the earth had the same rigidity as glass, it would 

 yield to the deforming influence of the moon and sun 3/5 as 

 much as if it were fluid and that if it possessed the rigidity of 

 brass it would yield 1/2 as much as if it were fluid. Even in 

 the case of steel S = 097 R and the difference between R and 

 S is far within the limits of error of any knowledge which we 

 can ever hope to reach as to the actual rigidity of the earth. 

 Thus for steel R or S may be taken at about 1/3, and all these 

 values, which follow from the investigations of Lame and 

 Thomson are also deducible from the inequality e > \dxw itself 

 deduced by the most elementary methods. 



* The Rev. Osmond Fisher has attempted to reconcile the hypothesis of a 

 fluid substratum wich the effective rigidity of the earth (Physics of the Earth's 

 Crust, 2d edition, chapter v). If I understand him correctly, he misunderstands 

 Henry's law on which his argument is based. He seems to suppose that the 

 amount of gas which a fluid will dissolve is proportional to the sum of the pres- 

 sures of the gas and the fluid. Henry's law shows the proportionality between 

 the pressure of the gas alone and the amount of gas which the fluid will dissolve. 

 Water from great oceanic depths contains no more gas in solution than that from 

 the surface. 



f Thomson after calculating the effect of the compressibility of the uniconstant 

 solid on the ellipticity of an elastic spheroid, says " we see that the compressi- 

 bility of the elastic solid exercises very little influence on the result." § 837. 



