J. IF. Gibbs — iJqnIlibrixiii of Heterogeneous ^Suhstances. 373 



values to '/'v,, etc., when the stresses in the solid vanish. If we 

 denote by r„ the common value of r,, rg, ^3 which will make the 

 stresses vanish at any given temperature, and imagine the true value 

 of il\r, , and also the value given by equation (444) to be expressed in 

 terms of the ascending poM^ers of 



^'i-^o, r^—r^, r^-r^, (446) 



it is evident that the expressions Avill coincide as far as the terms of 

 the second degree Inclusive. That is, the errors of the values of if\, 

 given by equation (444) are of the same order of magnitude as the 

 cubes of the above differences. The errors of the values of 



<7?/v, dtf\, dil\, 



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

 differences. Therefore, since 



di/\, _ dff\, d)\ di/^y, dr^ d4\, dr^ 

 dx dr^ dx dr.^ _,dx dr^ dx ^ *' 



dx' dx' dx' dx' 



whether we regard the true value of ij^, or the value given by equa- 

 tion (444), and since the error in (444) does not affect the values of 



dr^ d)\ dr^ 



dx ' dx ' dx ' 



dx' dx' " dx' 



Avhich we may regard as determined by equations (431), (432), (434), 

 (43V) and (438), the errors in the values of X^, derived from (444) 

 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^,^ ^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 M the elasticity of volume 

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

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

 definitions : 



^~~^'\§v)t' ^^^" v = r^^v', (448) 



* See Thomson and Tait's Natural Philosophy, vol. i, p. 111. 



