Page:Scientific Papers of Josiah Willard Gibbs.djvu/232

(see (381)), which will in general be different for the different pairs of opposite sides, and may be denoted by $p',p'',p'''$. (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 $\mu _{1}',\mu _{1}'',\mu _{1}''',$ determined by the equations

$\epsilon -t\eta +p'v=\mu _{1}'m,$

(393)

$\epsilon -t\eta +p''v=\mu _{1}''m,$

(394)

$\epsilon -t\eta +p'''v=\mu _{1}'''m.$

(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 $\mu _{1}'$, 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 $[\epsilon ]_{p'},[\eta ]_{p'},[v]_{p'},$ and $[\mu _{1}]_{p'}$ for the values which $\epsilon ,\eta ,v,$ and $\mu _{1}$ would receive on this supposition, we shall have

$[\epsilon ]_{p'}-t[\eta ]_{p'}+p'[v]_{p'}=[\mu _{1}]_{p'}m.$

(396)

Subtracting this from (393), we obtain

$\epsilon -[\epsilon ]_{p'}-t\eta +t[\eta ]_{p'}+p'v-p'[v]_{p'}=\mu _{1}m-[\mu _{1}]_{p'}m.$

(397)

Now it follows immediately from the definitions of energy and entropy that the first four terms of this equation represent the work spent upon the solid in bringing it from the state of hydrostatic stress to the other state without change of temperature, and $p'v-p'[v]_{p'}$ evidently denotes the work done in displacing a fluid of pressure $p'$ surrounding the solid during the operation. Therefore, the first number of the equation represents the total work done in bringing the solid when surrounded by a fluid of pressure $p'$ from the state of hydrostatic stress $p'$ to the state of stress $p',p'',p'''$. This quantity is necessarily positive, except of course in the limiting case when $p'=p''=p'''$. If the quantity of matter of the solid body be unity, the increase of the potential in the fluid on the side of the solid on which the pressure remains constant, which will be necessary to maintain equilibrium, is equal to the work done as above described. Hence, $\mu _{1}'$ is greater than $[\mu _{1}]_{p'}$, and for similar reasons $\mu _{1}''$ is greater than the value of the potential which would be necessary for equilibrium if the solid were subjected to the uniform pressure $p''$, and $\mu _{1}'''$ greater than that which would be necessary for equilibrium if the solid were subjected to the uniform pressure $p'''$. That is (if we