Page:Scientific Papers of Josiah Willard Gibbs.djvu/235

It will also be observed, that if we regard the forces acting upon the sides of the solid parallelepiped as composed of the hydrostatic pressure $p$ together with additional forces, the work done in any infinitesimal variation of the state of strain of the solid by these additional forces will be represented by the second member of the equation.

We will first consider the case in which the fluid is identical in substance with the solid. We have then, by equation (97), for a mass of the fluid equal to that of the solid,

$\eta _{F}dt-v_{F}dp+md\mu _{1}=0,$

(405)

$\eta _{F}$ and $v_{F}$ denoting the entropy and volume of the fluid. By subtraction we obtain

$-(\eta _{F}-\eta )dt+(v_{F}-v)dp=\left(X_{X'}+p{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)d{\frac {dx}{dx'}}+X_{Y'}d{\frac {dx}{dy'}}+\left(Y_{Y'}+p{\frac {dz}{dz'}}{\frac {dx}{dx'}}\right)d{\frac {dy}{dy'}}\cdot$

(406)

Now if the quantities ${\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dy}{dy'}}$ remain constant, we shall have for the relation between the variations of temperature and pressure which is necessary for the preservation of equilibrium

${\frac {dy}{dp}}={\frac {v_{F}-v}{\eta _{F}-\eta }}=t{\frac {v_{F}-v}{Q}},$

(407)

where $Q$ denotes the heat which would be absorbed if the solid body should pass into the fluid state without change of temperature or pressure. This equation is similar to (131), which applies to bodies subject to hydrostatic pressure. But the value of ${\frac {dt}{dp}}$ will not generally be the same as if the solid were subject on all sides to the uniform normal pressure $p$; for the quantities $v$ and $\eta$ (and therefore $Q$) will in general have different values. But when the pressures on all sides are normal and equal, the value of ${\frac {dt}{dp}}$ will be the same, whether we consider the pressure when varied as still normal and equal on all sides, or consider the quantities ${\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dy}{dy'}}$ as constant.

But if we wish to know how the temperature is affected if the pressure between the solid and fluid remains constant, but the strain of the solid is varied in any way consistent with this supposition, the differential coefficients of $t$ with respect to the quantities which express the strain are indicated by equation (406). These differential coefficients all vanish, when the pressures on all sides are normal and equal, but the differential coefficient ${\frac {dt}{dp}}$, when ${\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dy}{dy'}}$ are