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PHYSICS AND MATHEMATICS TO GEOLOGY. 141 
physical objection to the hypothesis that the material is very nearly 
incompressible, 7. ¢. that 0°5—7 is very small;* and an isotropic sphere 
with such a structure would, according to all our tests, remain per- 
fectly 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 gravita- 
tional forces in reducing the eccentricity was not taken into account. 
If the principles we have laid down as regulating the applicability of 
the mathematical theory be conceded, we need only consider the case 
when 0°5—y is very small; and since the formule show that in this 
case a small variation in the value of 7 is of little consequence, we may 
for simplicity suppose 7,=0-5 exactly. 
Jn order to show the nature of the uncertainty 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 gravitational forces affects the eccen- 
tricity. 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 semiaxes are 
respectively a—2d/3 and a+d/3. Now suppose the heaped-up mate- 
rial to become heavy. Thepressure it exerts on the surface below it 
is greatest in the equator and is zero at the 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 
semiaxes of the sphroidal 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 eccentricity which the centrifu- 
gal forces tend to develop, and also that this reduction may depend 
largely on the density of the surface layers. Ifthe 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 hypoth- 
esis of an earth of uniform density. 
Treating the density as uniform and 7 as equai; 0:5, [ find that, for 
a given value of I, 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. Soin the supposed case of the earth, we 
should have to reduce E from 141 x 10’ to 32 x 107 grams weight per square 
centim. The maximum stress-difference reduces to 7-2 tons weight per 
square inch. The greatest strain remains 0.0018, as before, but it would 
answer to a purely longitudinal stress of only 3:6 tons per square inch. 
*Cf. (a) formula (21), p. 283, and (5), p. 287. 
