250 Mr. C. Chree on some Applications of 



poles. Thus the originally spherical surface will tend to sink 

 at the equator and to rise at the poles ; consequently the 

 difference d between the equatorial and polar semi-axes of the 

 spheroidal surface will diminish, but the diminution is clearly 

 less the smaller the density of the heaped-up material. 



It must be understood that this does not profess to be a 

 complete account of what actually happens ; but it may suffice 

 to show that the gravitational forces tend to reduce the eccen- 

 tricity which the centrifugal forces tend to develop, and also 

 that this reduction may depend largely on the density of the 

 surface layers. Tf the departure of the surface layers from 

 the earth's mean density occurs mainly near the equator, 

 then the action of the gravitational forces in reducing the 

 eccentricity may be much less than it would seem to be on 

 the hypothesis of an earth of uniform density. 



Treating the density as uniform and y) as equal "5, 1 find that, 

 for a given value of E, the existence of the gravitational forces 

 would in such a case as that of the earth reduce the difference 

 between the equatorial and polar diameters called for by the 

 rotation in the ratio of 9 : 40 approximately*. Thus, for a 

 given eccentricity, the value of E when the gravitational 

 forces act is to its value when the centrifugal forces alone 

 exist as 9 : 40. So in the supposed case of the earth, we 

 should have to reduce E from 141 x 10 7 to 32 x 10 7 grammes 

 weight per square centim. The maximum stress-difference 

 reduces to 7*2 tons weight per square inch. The greatest 

 strain remains "0018, as before, but it would answer to a 

 purely longitudinal stress of only 3*6 tons per square inch. 

 Owing to the less density of the surface-strata these reductions 

 may be considerably too great, so that it is advisable to regard 

 32 x 10 7 as essentially a lower limit to the value of E. As 

 stated above, the numerical result for the value of E would 

 be but little altered if we supposed rj slightly less than '5 ; 

 but unless *5— 77 be very small, the terms independent of the 

 eccentricity become of importance in estimating the maximum 

 stress-difference and greatest strain. 



The conclusion to which the previous investigations lead is, 

 that none of the principles at present recognized in the 

 biconstant theory of isotropy are opposed to the hypothesis 

 that the earth possesses in its interior an isotropic elastic 

 solid structure with a linear stress-strain relation, provided 

 its material be very nearly incompressible. But the hypo- 

 thesis that the material in the interior shows an isotropic elastic 

 structure, such as that of the ordinary metals under the 

 ordinary conditions to which they are exposed on the earth's 

 * Cf. (a) formula (21), p. 283, and (5), p. 287. 



