246 Mr. C. Chree on some Applications of 



The maximum stress-difference and the greatest strain, as 

 given in the table, are both found at the centre. 



The result on the stress-difference theory is nearly inde- 

 pendent of 77, and is more unfavourable 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 com- 

 placency 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 practically 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 tbe 

 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 inde- 

 pendent 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 interpre- 

 tation upon the mathematical expressions answering to most 

 values of r} is such as very forcibly to call attention to the 

 necessity of the limitation (C). Since the gravitational force 

 at an element of a solid sphere depends not only on the total 

 mass which lies nearer the centre than does the element, but 

 also on its absolute distance from the centre, 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 



