172, 173J WORK OF STRESS. 455 



By equations 90) the coefficients of du, dv, dw vanish identically, so 

 that, interchanging the order of differentiation and variation, 



121) *W-{x.*& + Y,* + Z.9% + Y, 



or in our later terminology, 



122) 8W= j i C{P x 8s x + P y ds y + P,8s, + 2T x dg x 



+ ZT y dg y + 2T 2 dg 2 }d>t. 



Thus each of the six components of the stress does work on the 

 corresponding component of the strain, and the work per unit volume 

 in any infinitesimal strain is the sum of each stress component by 

 the corresponding strain produced, except that with our terminology 

 the shearing stresses are multiplied by twice the shearing strains or 

 slides. 



173. Relations between Stress and Strain. If a body is 

 perfectly elastic the stresses at any point at any time depend simply 

 upon the strain at the point at the time in question, so that if the 

 elastic properties of the body are known at every point the stress 

 components will be known functions of the strain components, which 

 may differ from one point of the body to another. The stresses will 

 be uniform and continuous functions of the strains and may be 

 developed by Taylor's theorem. If then the strains are small, the 

 terms of the lowest orders will be the most important. The strains 

 dealt with by the ordinary theory of elasticity are so small that it 

 is customary to neglect all terms above those of the first order. The 

 results thus obtained are in good accordance with those obtained by 

 experiment under the proper limitations. The law that for small 

 strains the stresses are linear functions of the strains may be regarded 

 as an extension of the law announced in 1676 by Hooke in the form 



of an anagram, 



ce^^^nosssttuu 



Ut Tensio sic Vis. 



The force varies as the stretch, or in our terminology the stress 

 varies as the strain. Making this assumption we accordingly have 



X = 9>oi 



