Theory of Elasticity. 285 



condition of stress and strain in a structure is not determinable 

 purely from equations of elasticity. If we are considering an 

 existing arch we can only know (since the arch stands) that the 

 line of resistance lies within certain more or less well defiued 

 limits. On the other hand, because in a design we can pass 

 a line of resistance within certain limits it does not follow that 

 in the actual construction it will lie within those limits. This 

 will depend on the nature of the materials and on how the 

 construction is carried out. An arch designed " safe " by 

 drawing a line of resistance within the middle third may in 

 construction have that line pass without the middle third and 

 fail. Consider the additional uncertainties introduced through 

 masonry accessory to the arch band itself and through the man- 

 ner of loading, and it will be seen that the problem actually is 

 an exceedingly indeterminate, complex and at best uncertain 

 one. 



Turning from engineering applications to the domains of 

 speculative science consider the application of the theory of 

 elasticity to the study of the inner condition of the earth. A 

 sphere of homogeneous, isotropic, elastic material, rigidly sub- 

 ject to Hooke's law, stressed only by its own gravitating force, 

 would engender strains, contraction tangentially increasing 

 from six at the surface to eleven at the center, and extension 

 radially of four at the surface changing to contraction radially 

 of eleven at the center. This is based on the value \ for the 

 ratio of lateral contraction to elongation. But this solution* 

 is inadmissible in the case of the earth, not merely because in 

 its case the strains involved are far larger than those to which 

 the ordinary theory of elasticity applies, but because the char- 

 acter of the strains at the surface, contraction in one direction, 

 extension in that at right angles, would necessarily involve 

 rupture in the case of such great strains. As a matter of fact 

 no such stresses and strains are called for. We may have the 

 stresses and strains actually not varying from those of fluid 

 pressure (even though the material be exceedingly -rigid) except 

 in so far as variations in density and departures from the figure 

 of equilibrium under the actual forces (including gravitation, 

 centrifugal force and tidal forces of sun and moon) call for 

 resistance. Thus the material of the earth need be called on 

 only to have sufficient strength to resist the strains due to con- 

 tinental distribution, mountain elevations and tidal phenom- 

 ena. If known materials have ample strength to resist these 

 strains, as Prof. Darwin's investigations would indicate^ 

 then it is shown that known materials are perfectly capable of 



*See Love, Theory of Elasticity, Art. 127, vol. I. 



fSee various articles and also Thomson and Tait, Treatise on Natnral Phi- 

 losophy, vol. ii. 



Am. Jour. Sci.— Fourth Series, Vol. XI, No. 64.— April, 1901. 

 19 



