Page:Scientific Papers of Josiah Willard Gibbs.djvu/224

the fluid (within the envelop), and $\textstyle \sum _{1}$ denotes a summation with regard to those independently variable components of the fluid of which the solid is composed. Where the solid does not consist of substances which are components, actual or possible (see page 64), of the fluid, this term is of course to be cancelled.

If we wish to take account of gravity, we may suppose that it acts in the negative direction of the axis of $Z$. It is evident that the variation of the energy due to gravity for the whole mass considered is simply

$\iiint g\Gamma '\delta z\delta x'\delta y'\delta z',$

(359)

where $g$ denotes the force of gravity, and $\Gamma '$ the density of the element in the state of reference, and the triple integration, as before, extends throughout the solid.

We have, then, for the general condition of equilibrium,

$\iiint t\delta \eta _{V'}dx'dy'dz'+\iiint \textstyle \sum \sum \,'\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right)dx'dy'dz'+\iiint g\Gamma '\delta z\delta x'\delta y'\delta z'+\int \epsilon _{V'}\delta N'Ds'+\int ^{F}t\delta D\eta -\int ^{F}p\delta Dv+\textstyle \sum _{1}\displaystyle \int ^{F}\mu _{1}\delta Dm_{1}\geqq 0.$

(360)

The equations of condition to which these variations are subject are:

(1) that which expresses the constancy of the total entropy,

$\iiint \delta \eta _{V'}\delta x'\delta y'\delta z'+\int \eta _{V'}\delta N'Ds'+\int ^{F}\delta D\eta =0;$

(361)

(2) that which expresses how the value of $\delta Dv$ for any element of the fluid is determined by changes in the solid,

$\delta Dv=-(\alpha \delta x+\beta \delta y+\gamma \delta z)Ds-v_{V'}\delta N'Ds',$

(362)

where $\alpha ,\beta ,\gamma$ denote the direction cosines of the normal to the surface of the body in the state to which $x,y,z$ relate, $Ds$ the element of the surface in this state corresponding to $Ds'$ in the state of reference, and $v_{V'}$ the volume of an element of the solid divided by its volume in the state of reference;

(3) those which express how the values of $\delta Dm_{1},\delta Dm_{2}$, etc. for any element in the fluid are determined by the changes in the solid,

$\delta Dm_{1}=-\Gamma _{1}'\delta N'Ds',$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ (363)
$\delta Dm_{2}=-\Gamma _{2}'\delta N'Ds',$
etc.,

where $\Gamma _{1}',\Gamma _{2}'$, etc. denote the separate densities of the several components in the solid in the state of reference.

Now, since the variations of entropy are independent of all the other variations, the condition of equilibrium (360), considered with regard to the equation of condition (361), evidently requires that throughout the whole system

$t=const.$

(364)