J. W. Gihhs — Equilibrium of Heterofjeneous Siibstances. 855 



(2), if two oi t\\^ principal tractions A'x, ^x, ^i. are equal, when the 

 surface is perpendicular to the ])lane containiug the two correspond- 

 ing axes, (in this case the traction across any such surface is equal to 

 the common vahie of the two principal tractions) ; 



(3), if the principal tractions are all equal, the traction is normal and 

 constant for all surfaces. 



It will he observed that in the second and third cases the position of 

 the ]>rincipal axes of stress are partially or wholly indeterminate, (so 

 that these cases may be regarded as included in the first,) but the 

 values of the principal tractions are always determinate, although not 

 always ditl^erent. 



If!, therefore, a solid which is homogeneous in nature and in state of 



strain is bounded by six surfaces perpendicular to the principal axes 



of strain, the mechanical conditions of equilibrium for these surfaces 



may be satisfied by the contact of fluids having the proper pressures, 



[see (381),] which will in general be different for the different pairs of 



opposite sides, and may be denoted by jk>', p'\ p'". (These pressures 



are equal to the principal tractions of the solid taken negatively.) 



It will then be necessary for equilibrium with respect to the tendency 



of the solid to dissolve that the potential for the substance of the 



solid in the fluids shall have values yw/, /i/', yu/" detemiined by the 



equations 



e — t)f -|-/)' V = // , ' »i, (393) 



€ — t r^ -\- p" V z:^ fx^" m, (394) 



s—tf^ + p"'v = /iii"'m. (395) 



These values, it will be observed, are entirely determined by the 

 nature and state of the solid, and their differences are equal to the 

 differences of the corresponding pressures divided by the density of 

 the solid. 



It may be interesting to compare one of these potentials, as ///, 

 with the potential (for the same substance) in a fluid of the same 

 temperature t and pressure p' which would be in equilibrium with the 

 same solid subjected on all sides to the uniform pressure p'. If we 

 write [f];,, , ['/]/,», [^]/,»5 and [/'i]^,, for the values which e, ?], i\ and 

 /<, would receive on this supposition, we shall have 



V^l' - t bf\r> +P' Vvl.. = [//J,, m. (396) 



Subtracting this from (393), we obtain 



^ - l^]p' —tV-^t Vfl' +P V - p' [y],,, = /<, m — [/',],, rn. (397) 



Now it follows immediately from the definitions of energy and entropy 



