Pliysics and Mathematics to Geology. 249 



all values of rj less than *5 the stress-difference theory is less 

 favourable to the view that the mathematical theory is appli- 

 cable than the greatest strain theory. If there is any truth in 

 either theory, the earth's material cannot possibly possess a 

 linear stress-strain relation for values of rj such as "25 (i. e. 

 with a structure such as that of the metals) unless it be of a 

 strength compared to which that of steel is insignificant. 

 For such values of t] the strains are also enormously in excess 

 of those which can be admitted according to the principle (C). 

 When, however, 77 approaches the limiting value \5 a com- 

 plete change comes over the features of the case. The 

 maximum stress-difference and all the strains diminish, even- 

 tually vanishing when rj = '5. Thus none of the objections 

 hitherto encountered can be urged against the application of 

 the mathematical theory when r) equals or nearly equals *5. 

 To the exact value * 5 of tj there is, I admit, a physical objec- 

 tion, which would doubtless have been urged by Maxwell, viz. 

 that, supposing Young's modulus to be finite, this implies the 

 material to be absolutely incompressible. There is, however, 

 no obvious physical objection to the hypothesis that the 

 material is very nearly incompressible, i. e. that *5 — rj is very 

 small*; and an isotropic sphere with such a structure would, 

 according to all our tests, remain perfectly elastic when 

 possessed of the earth's mass and exposed to its gravitational 

 forces. 



In our previous estimate of the value of E the action of the 

 gravitational forces in reducing the eccentricity was not taken 

 into account. If the principles we have laid down as regula- 

 ting the applicability of the mathematical theory be conceded, 

 we need only consider the case when *5 — rj is very small ; and 

 since the formulae show that in this case a small variation in 

 the value of rj is of little consequence, we may for simplicity 

 suppose rj= "5 exactly. 



In order to show the nature of the uncertaint}^ that must 

 in reality be attached to the result, it seems desirable to give 

 a general idea of the way in which the existence of gravita- 

 tional forces affects the eccentricity. Let us imagine, then, 

 that over the surface of a perfect sphere weightless matter 

 is piled up, which transforms the surface into that of a slightly 

 oblate spheroid whose polar and equatorial semi-axes are re- 

 spectively a — 2d/3 and a + d/3. Now suppose the heaped-up 

 material to become heavy. The pressure it exerts on the 

 surface below it is greatest in the equator and is zero at the 



* Stewart and Gee. in their ' Elementary Practical Physics/ vol i. 

 pp. 192-5, give data from which they conclude that india-rubber is such 

 a material. 



Phil. Mag. S. 5. Vol. 32. No. 196. Sept. 1891. S 



