Page:Scientific Papers of Josiah Willard Gibbs.djvu/314

by the fundamental equation of the mass or surface concerned, or may be immediately derived from it. (See pp. 85–89 and 229–231.)

The variations in (600) are subject to the conditions which arise from the nature of the system and from the supposition that the changes in the system are not such as to affect external bodies. This supposition is necessary, unless we are to consider the variations in the state of the external bodies, and is evidently allowable in seeking the conditions of equilibrium which relate to the interior of the system.^[1] But before we consider the equations of condition in detail, we may divide the condition of equilibrium (600) into the three conditions

$\int t\delta D\eta ^{V}+\int t\delta D\eta ^{S}=0,$

(601)

$-\int p\delta Dv+\int \sigma \delta Ds+\int g\delta zDm^{V}+\int g\delta zDm^{S}=0,$

(602)

$\int \mu _{1}\delta Dm_{1}^{V}+\int \mu _{1}\delta Dm_{1}^{S}+\int gz\delta Dm_{1}^{V}+\int gz\delta Dm_{1}^{S}$ $+\int \mu _{2}\delta Dm_{2}^{V}+\int \mu _{2}\delta Dm_{2}^{S}+\int gz\delta Dm_{2}^{V}+\int gz\delta Dm_{2}^{S}$ $+{\text{etc.}}=0.$

(603)

For the variations which occur in any one of the three are evidently independent of those which occur in the other two, and the equations of condition will relate to one or another of these conditions separately.

The variations in condition (601) are subject to the condition that the entropy of the whole system shall remain constant. This may be expressed by the equation

$\int \delta D\eta ^{V}+\int \delta D\eta ^{S}=0.$

(604)

To satisfy the condition thus limited it is necessary and sufficient that

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

(605)

throughout the whole system, which is the condition of thermal equilibrium.

The conditions of mechanical equilibrium, or those that relate to the possible deformation of the system, are contained in (602), which may also be written

$-\int p\delta Dv+\int \sigma \delta Ds+\int g\gamma \delta zDv+\int g\Gamma \delta zDs=0.$

(606)

1. We have sometimes given a physical expression to a supposition of this kind, problems in which the peculiar condition of matter in the vicinity of surfaces discontinuity was to be neglected, by regarding the system as surrounded by a rigid and impermeable envelop. But the more exact treatment which we are now to give the problem of equilibrium would require us to take account of the influence of the envelop on the immediately adjacent matter. Since this involves the consideration of surfaces of discontinuity between solids and fluids, and we wish to limit ourselves at present to the consideration of the equilibrium of fluid masses, we shall give up the conception of an impermeable envelop, and regard the system as bounded simply by an imaginary surface, which is not a surface of discontinuity. The variations of the system must be such as do not deform the surface, nor affect the matter external to it.