188 SOME APPLICATIONS OF 
The maximum stress-difference and the greatest strain, aS given in 
the table, are both found at the center. 
The result on the stress-difference theory is nearly independent of 7, 
and is more unfavorable in every case than that given by the greatest 
strain theory to the view that the material remains perfectly elastic. 
A stress of 16 tons per square inch is not one that an engineer would 
view with complacency in any structure intended to be permanent, but 
it is a low value for the tenacity of good wrought iron. A stress of 
even 33 tons per square inch can easily be borne without rupture by 
good steel, and is perhaps not in excess of the stress under which the 
best steel remains perfectly elastic. The greatest strains are not of 
such a magnitude as to raise any presumption against the linearity of 
the stress-strain relation. Thus, according to all the tests, it is quite 
possible that an originally spherical solid of the earth’s mass but 
devoid of gravitation should remain solid and elastic while assuming 
the form of the earth under rotation. Its material, however, at least 
if homogeneous and isotropic, would require to possess an unusually 
high limit of perfect elasticity. 
The next subject for consideration is how the question is affected by 
the existence of gravitational forces such as are found in the case of the 
earth. The strains and stresses in a slightly oblate spheroid, treated 
as an isotropic elastic solid, all consist of two parts, the first part being 
the same as if the surface were truly spherical, the second depending 
on the eccentricity. It is the second parts that represent the action of 
the gravitational forces in modifying the eccentricity, but these parts 
are in general insignificant so far as the question of the applicability 
of the mathematical theory is concerned. I shall therefore postpone 
consideration of them until an account has been given of the strains 
and stresses which are independent of the eccentricity. 
The mathematical difficulties in applying the ordinary theory to the 
case of a homogeneous solid gravitating sphere are trifling, but the 
difficulty of putting a physical interpretation upon the mathematical 
expressions answering to most values of 7 is such as very forcibly to 
call attention to the necessity of the limitation (C). Since the gravita- 
tional force at an element of a solid sphere depends not only on the to- 
tal mass which lies nearer the center than does the element, but also on 
its absolute distance from the center, we must assume that the equations 
supplied by the ordinary mathematical theory, if they apply at all, hold 
for the position of final equilibrium after the deformations have taken 
place. This seemingly requires that strain should be defined as the 
ratio of the increase of length to the final length, which is not in ae- 
cordance with the usual interpretation of Hooke’s law unless the square 
of the strain be negligible. Supposing the internal equations to refer 
to the final deformed condition, the surface equations will undoubtedly — 
also refer to this condition. Thus, so far as the terms independent of 
the eccentricity are concerned, we may suppose the mathematical theory 
