Page:Scientific Papers of Josiah Willard Gibbs.djvu/322

in which, moreover, the conditions of equilibrium relating to temperature and the potentials are satisfied, and the relations expressed by the fundamental equations of the masses and surfaces are to be regarded as satisfied, although the state of the system is not one of complete equilibrium. Let us imagine the state of the system to vary continuously in the course of time in accordance with these conditions and use the symbol d to denote the simultaneous changes which take place at any instant. If we denote the total energy of the system by $E$, the value of $dE$ may be expanded like that of $\delta E$ in (599) and (600), and then reduced (since the values of $t,\mu _{1}+gz,\mu _{2}+gz$, etc., are uniform throughout the system, and the total entropy and total quantities of the several components are constant) to the form

${\begin{aligned}dE&=-\int pdDv+\int gdzDm^{V}+\int \sigma dDs+\int gdzDm^{S}\\&=-\int pdDv+\int g\gamma dzDv+\int \sigma dDs+\int g\Gamma dzDs,\\\end{aligned}}$

(621)

where the integrations relate to the elements expressed by the symbol $D$. The value of $p$ at any point in any of the various masses, and that of $\sigma$ at any point in any of the various surfaces of discontinuity are entirely determined by the temperature and potentials at the point considered. If the variations of $t$ and $M_{1},M_{2}$, etc. are to be neglected, the variations of $p$ and $\sigma$ will be determined solely by the change in position of the point considered. Therefore, by (612) and (614),

$dp=-g\gamma dz,$$d\sigma =g\Gamma dz;$

and

${\begin{aligned}dE&=-\int pdDv-\int dpDv+\int \sigma dDs+\int d\sigma Ds\\&=-d\int pDv+d\int \sigma Ds.\\\end{aligned}}$

(622)

If we now integrate with respect to $d$, commencing at the given state of the system, we obtain

$\Delta E=-\Delta \int pDv+\Delta \int \sigma Ds,$

(623)

where $\Delta$ denotes the value of a quantity in a varied state of the system diminished by its value in the given state. This is true for finite variations, and is therefore true for infinitesimal variations without neglect of the infinitesimals of the higher orders. The condition of stability is therefore that

$\Delta \int pDv-\Delta \int \sigma Ds<0,$

(624)

or that the quantity

$\int pDv-\int \sigma Ds$

(625)

has a maximum value, the values of $p$ and $\sigma$, for each different mass or surface, being regarded as determined functions of $z$. (In ordinary cases $\sigma$ may be regarded as constant in each surface of discontinuity, and $p$ as a linear function of $z$ in each different mass.) It may easily