Page:Scientific Papers of Josiah Willard Gibbs.djvu/252

We have, therefore, for a state of hydrostatic stress,

$d\epsilon _{V'}=td\eta _{V'}-pdv_{V'}+L_{a}d\Gamma _{a}'+L_{b}d\Gamma _{b}'+{\text{etc.,}}$

(465)

and multiplying by the volume of the element in the state of reference, which we may regard as constant,

$d\epsilon =td\eta -pdv+L_{a}dm_{a}+L_{b}dm_{b}+{\text{etc.,}}$

(466)

where $\epsilon ,\eta ,v,m_{a},m_{b}$, etc., denote the energy, entropy, and volume of the element, and the quantities of its several fluid components. It is evident that the equation will also hold true, if these symbols are understood as relating to a homogeneous body of finite size. The only limitation with respect to the variations is that the element or body to which the symbols relate shall always contain the same solid matter. The varied state may be one of hydrostatic stress or otherwise.

But when the body is in a state of hydrostatic stress, and the solid matter is considered invariable, we have by equation (12)

$d\epsilon =td\eta -pdv+\mu _{a}dm_{a}+\mu _{b}dm_{b}+{\text{etc.}}$

(467)

It should be remembered that the equation cited occurs in a discussion which relates only to bodies of hydrostatic stress, so that the varied state as well as the initial is there regarded as one of hydrostatic stress. But a comparison of the two last equations shows that the last will hold true without any such limitation, and moreover, that the quantities $L_{a},L_{b}$, etc., when determined for a state of hydrostatic stress, are equal to the potentials $\mu _{a},\mu _{b}$, etc.

Since we have hitherto used the term potential solely with reference to bodies of hydrostatic stress, we may apply this term as we choose with regard to other bodies. We may therefore call the quantities $L_{a},L_{b}$, etc., the potentials for the several fluid components in the body considered, whether the state of the body is one of hydrostatic stress or not, since this use of the term involves only an extension of its former definition. It will also be convenient to use our ordinary symbol for a potential to represent these quantities. Equation (462) may then be written

$d\epsilon _{V'}=td\eta _{V'}+\textstyle \sum \sum '\displaystyle \left(X_{X'}d{\frac {dx}{dx'}}\right)+\mu _{a}d\Gamma _{a}'+\mu _{b}d\Gamma _{b}'+{\text{etc.}}$

(468)

This equation holds true of solids having fluid components without any limitation with respect to the initial state or to the variations, except that the solid matter to which the symbols relate shall remain the same.

In regard to the conditions of equilibrium for a body of this kind, it is evident in the first place that if we make $\Gamma _{a}',\Gamma _{b}'$, etc., constant, we shall obtain from the general criterion of equilibrium