346 G. F. Becker — Proof of the Earth? s Rigidity. 



If the fluid supposed at first incompressible were to become 

 compressible, it is clear that a would decrease and that </, being 

 inversely proportional to the square of the distance from the 

 earth's center, would increase. Thns the resistance to deforma- 

 tion by the attraction of the moon would increase and the de- 

 formation measured by the ellipticity would decrease. It can 

 also be shown in detail that if the increase in density took place 

 in any manner capable of approximate expression by Laplace's 

 law (that the increase of the square of the density is propor- 

 tional to the increase of the pressure) compressibility would 

 decrease the ellipticity ; but even infinite compressibility would 

 only rednce the ellipticity by a moderate fraction of its value 

 for the case of incompressibility.* Beyond a knowledge of 

 the sense in which compressibility would affect the ellipticity 

 of the fluid sphere it is really not worth while to inquire, for 

 it is not at all probable that the so-called constants of elasticity, 

 n and &, would exhibit invariability if we conld experiment 

 with pressures of hundreds of thousands of pounds to the 

 square inch ; indeed there is good evidence pf great changes in 

 rigidity even at pressures which have been experimentally ap- 

 plied. 



Comparison of results. — In the following table I have col- 

 lected the various ellipticities mentioned above with numerical 

 expressions for spheres of the mean density of the earth, but 

 with the rigidity of glass, brass and steel. Thomson gave fig- 

 ures for glass and steel, and I have added those for brass. I 



* Centrifugal force is equivalent to a simple repulsion from a line, and if w is 

 the angular velocity, the measure of this force is wV. Tide-generating force is 

 equivalent to repulsion from a plane as appears from the manner in which it was 

 deduced above and its measure is 2rx. Hence the formulas for the two cases are 

 convertible by the substitution of one of these forces for the other, bearing in mind 

 that centrifugal spheroids are oblate and that tidal ellipsoids are prolate. See 

 Nat. Phil., ^ 834, page 432. — If E is the mass of the earth, and if / is the ratio 

 which the mean density of a sphere bears to its surface density, the ellipticity of 

 a rotating spheroid in which there is a small increase in density from the surface 

 downward, may be expressed by 



5 w 2 a 3 _ 1 

 e ""TT' 2(1 +4(/- 1)) 



(Cf. Nat. Phil., §§ 824' and 800). In the case of the equilibrium tide, real and 

 apparent gravitation coincide and therefore g = E/a' 2 . Introducing also the value 

 of r this becomes 



5 to, I ♦ 



&= 2 '7'lT4~(/-l) 



Similarly § 824 / (2) gives for the case in which the surface density is infinitesimal 

 as compared with the density at the center 



6 5 aru 6 < 5 aro 



e ~~Tp> T ~~g ~ near J To" 2~ ~g~ 



For the reasons stated in the text these formulas are of no use excepting to show 

 that progressive increase of density decreases the ellipticity of a fluid sphere. 



