Page:Scientific Papers of Josiah Willard Gibbs.djvu/184

so small that the force of gravity can be regarded as constant in direction and in intensity, we may use $\Upsilon$ to denote the potential of the force of gravity, and express the variation of the part of the energy which is due to gravity in the form

$-\int \Upsilon \delta Dm-\int \Upsilon \Delta Dm.$

(238)

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

$\int \delta D\epsilon +\int \Delta D\epsilon -\int \Upsilon \delta Dm-\int \Upsilon \Delta Dm\geqq 0;$

(239)

and the equations of condition will be

$\int \delta D\eta +\int \Delta D\eta =0,$

(240)

$\int \delta Dm_{1}+\int \Delta Dm_{1}=0,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ (241)
$......................................$
$\int \delta Dm_{n}+\int \Delta Dm_{n}=0,$

We may obtain a condition of equilibrium independent of these equations of condition, by subtracting these equations, multiplied each by an indeterminate constant, from condition (239). If we denote these indeterminate constants by $T,M_{1},...M_{n}$, we shall obtain after arranging the terms

$\int {\overline {\delta D\epsilon -\Upsilon \delta Dm-T\delta D\eta -M_{1}\delta Dm_{1}...-M_{n}\delta Dm_{n}}}+\int {\overline {\Delta D\epsilon -\Upsilon \Delta Dm-T\Delta D\eta -M_{1}\Delta Dm_{1}...-M_{n}\Delta Dm_{n}}}\geqq 0.$

(242)

The variations, both infinitesimal and finite, in this condition are independent of the equations of condition (240) and (241), and are only subject to the condition that the varied values of $D\epsilon ,D\eta ,Dm_{1},...Dm_{n}$ for each element are determined by a certain change of phase. But as we do not suppose the same element to experience both a finite and an infinitesimal change of phase, we must have

$\delta D\epsilon -\Upsilon \delta Dm-T\delta D\eta -M_{1}\delta Dm_{1}...-M_{n}\delta Dm_{n}=0,$

(243)

and $\Delta D\epsilon -\Upsilon \Delta Dm-T\Delta D\eta -M_{1}\Delta Dm_{1}...-M_{n}\Delta Dm_{n}=0.$

(244)

By equation (12), and in virtue of the necessary relation (222), the first of these conditions reduces to

$(t-T)\delta D\eta +(\mu _{1}-\Upsilon -M_{1})\delta Dm_{1}...+(\mu _{n}-\Upsilon -M_{n})\delta Dm_{n}\geqq 0;$

(245)

for which it is necessary and sufficient that

$t=T,$

(246)

$\mu _{1}-\Upsilon =M_{1},$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ (247)[1]
$...........................;$
$\mu _{n}-\Upsilon =M_{n}.$

1. The gravitation potential is here supposed to be defined in the usual way. But if it were defined so as to decrease when a body falls, we should have the sign $+$ instead of $-$ in these equations; i.e., for each component, the sum of the gravitation and intrinsic potentials would be constant throughout the whole mass.