140 SOME APPLICATIONS OF 
more clearly presently, largely reduces the eccentricity which rotation 
would produce in a sphere of given material. Thus the eccentricity 
varying inversely as EH, the value of E answering to a given eccentric- 
ity is necessarily considerably smaller when gravitational forces act 
along with the “ centrifugal” than when the latter act alone. Since the 
surface strata are very variable and of much smaller mean density than 
the earth as a whole, any calculation of the reduction of our estimates 
of E, when gravitational forces are allowed for, which treats the earth 
as of uniform density can not lay claim to great accuracy. For this 
reason, and also because I am specially desirous not to overstate the 
case against the applicationof the mathematical theory, I have, in cal- 
culating the values of s) and wu, in Table rv, ascribed to E the values it 
would possess in the total absence of gravitational forces, viz., the 
values 1020 x 10° for 77 =0 and 1220 x 10° for 77=0°25 in the same units as 
before. The numerical values ascribed to s) and uw, in the table are thus 
essentially minima, which would in reality have to be increased proba- 
bly to a considerable extent. 
It will be seen from the formule and from Table Iv that when 7 is 
zero or is small, the application of the mathematical theory would be 
fully justified on the greatest strain theory, while utterly condemned 
on the stress-difference theory. The principle (C) is in this case en- 
tirely in agreement with the stress-difference theory, and the applica- 
tion of the mathematical theory can in fact be supported only by those 
who reject this principle, and consider it possible for the stress-strain 
relation to remain linear though a solid sphere is reduced to one-fourth 
or less of its original volume. 
Noticing from (1) and (2) that Es/S=2y7, we see that for all values of 
7 less than 0-5 the stress-difference theory is less favorable to the view 
that the mathematical theory is applicable than the greatest-strain 
theory. Ifthere is any truth in either theory, the earth’s material 
can not possibly possess a linear stress-strain relation for values of 7 
such as 0°25 (7. 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 7 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 0:5 a complete 
change comes over the features of the case. The maximum stress 
difference and all the strains diminish, eventually vanishing when 
y='d. Thus none of the objections hitherto encountered can be urged 
against the application of the mathematical theory when 7 equals or 
nearly equals 0-5, To the exact value 0-5 of 7 there is, I admit, a physi- 
cal objection, which would doubtless have been urged by Maxwell, viz, 
that, supposing Young’s modulus to be finite, this implies the mate- 
rial to be absolutely incompressible. There is however no obvious 
“Stewart and Gee, in their HLlementary Practical Physics, vol. 1, p. 192-195, give 
data from which they conclude that india-rubber is such a material. 
