EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 213 



will be of the same order of magnitude as the squares of the differ- 

 ences in (446). The same will be true with respect to X T , X Z ', Y^, 

 etc., etc. 



It will be interesting to see how the quantities e, /, and h are 

 related to those which most simply represent the elastic properties of 

 isotropic solids. If we denote by V and R the elasticity of volwme 

 and the rigidity* (both determined under the condition of constant 

 temperature and for states of vanishing stress), we shall have as 

 definitions 



V= v-- > when v = r 3 v', (448) 



where p denotes a uniform pressure to which the solid is subjected, 

 v its volume, and v' its volume in the state of reference ; and 



' dx f-,dx\ 2 ' 

 a-j, IM-J/) 

 dy \ dy/ 



___ (449) 



dx'~dy'~~dz'~'' T ^ 



rl/Y> rJ/Y> rJ/ti rl/ii rJ.v. fJ.v. 



and 



dx dy dz 



when - r - / = -^ > = - r -, = r , 



dx dy dz 



dx _dx _dy _dy _dz 

 dy' ~ dz' ~~ dz' ~ dx' ~ dx' 



Now when the solid is subject to uniform pressure on all sides, if 

 we consider so much of it as has the volume unity in the state of 

 reference, we shall have 



r t r t r *, (450) 



and by (444) and (439), 



^ v , = i + 3e<y f + 3/w* + hv. (451) 



Hence, by equation (88), since i/r v , is equivalent to \fr, 



(452) 



. <463) 



and by (448), 



(454) 



To obtain the value of R in accordance with the definition (449), 

 we may suppose the values of E, F, and H given by equations (432), 

 (434), and (437) to be substituted in equation (444). This will give 



for the value of R 



< 



. (455) 



See Thomson and Tail's Natural Philosophy, vol. i, p. 711. 



