Page:Scientific Papers of Josiah Willard Gibbs.djvu/181

By these equations the general condition of equilibrium may be reduced to the form

$\int \delta D\eta -\int p\delta Dv+\int \mu _{1}\delta Dm_{1}...+\int \mu _{1}\delta Dm_{n}+\int g\delta hDm+\int gh\delta Dm_{1}...+\int gh\delta Dm_{n}\geqq 0.$

(224)

Now it will be observed that the different equations of condition affect different parts of this condition, so that we must have, separately,

$\int t\delta D\eta \geqq 0,\,\,{\text{if}}\,\,\int \delta D\eta =0;$

(225)

$-\int p\delta Dv+\int ghDm\geqq 0,$

(226)

if the bounding surface is unvaried;

$\int \mu _{1}\delta Dm_{1}+\int gh\delta Dm_{1}\geqq 0,\,\,{\text{ if }}\,\,\int Dm_{1}=0;$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ (227)
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$\int \mu _{n}\delta Dm_{n}+\int gh\delta Dm_{n}\geqq 0,\,\,{\text{ if }}\,\,\int Dm_{n}=0.$

From (225) we may derive the condition of thermal equilibrium,

$t={\text{const.}}$

(228)

Condition (226) is evidently the ordinary mechanical condition of equilibrium, and may be transformed by any of the usual methods. We may, for example, apply the formula to such motions as might take place longitudinally within an infinitely narrow tube, terminated at both ends by the external surface of the mass, but otherwise of indeterminate form. If we denote by $m$ the mass, and by $v$ the volume, included in the part of the tube between one end and a transverse section of variable position, the condition will take the form

$-\int p\delta dv+\int ghdm\geqq 0,$

(229)

in which the integrations include the whole contents of the tube. Since no motion is possible at the ends of the tube,

$\int p\delta dv+\int \delta vdp=\int d(p\delta v)=0.$

(230)

Again, if we denote by $\gamma$ the density of the fluid,

$\int g\delta hdm=\int g{\frac {dh}{dv}}\delta v\gamma dv=\int g\gamma \delta vdh.$

(231)

By these equations condition (229) may be reduced to the form

$\int \delta v(dp+g\gamma dh)\geqq 0.$

(232)

Therefore, since $\delta v$ is arbitrary in value,

$dp=-g\gamma dh,$

(233)

which will hold true at any point in the tube, the differentials being taken with respect to the direction of the tube at that point. Therefore, as the form of the tube is indeterminate, this equation must hold true, without restriction, throughout the whole mass. It evidently