Page:Scientific Papers of Josiah Willard Gibbs.djvu/127

cooling the system, there must be a state of uniform temperature for which (regarded as a variation of the original state)

$\delta \epsilon <0\,\,{\text{and}}\,\,\delta \eta =0.$

From this we may conclude that for systems of initially uniform temperature condition (2) will not be altered if we limit the variations to such as do not disturb the uniformity of temperature.

Confining our attention, then, to states of uniform temperature, we have by differentiation of (105)

$\delta \epsilon -t\delta \eta =\delta \psi +\eta \delta t.$

(112)

Now there are evidently changes in the system (produced by heating or cooling) for which

$\delta \epsilon -t\delta \eta =0\,\,{\text{and therefore}}\,\,\delta \psi +\eta \delta t=0,$

(113)

neither $\delta \eta$ nor $\delta t$ having the value zero. This consideration is sufficient to show that the condition (2) is equivalent to

$\delta \epsilon -t\delta \eta \geqq 0,$

(114)

and that the condition (111) is equivalent to

$\delta \psi +\eta \delta t\geqq 0,$

(115)

and by (112) the two last conditions are equivalent.

In such cases as we have considered on pages 62-82, in which the form and position of the masses of which the system is composed are immaterial, uniformity of temperature and pressure are always necessary for equilibrium, and the remaining conditions, when these are satisfied, may be conveniently expressed by means of the function $\zeta$, which has been defined for a homogeneous mass on page 87, and which we will here define for any mass of uniform temperature and pressure by the same equation

$\zeta =\epsilon -t\eta +pv.$

(116)

For such a mass, the condition of (internal) equilibrium is

$(\delta \zeta )_{t,p}\geqq 0.$

(117)

That this condition is equivalent to (2) will easily appear from considerations like those used in respect to (111).

Hence, it is necessary for the equilibrium of two contiguous masses identical in composition that the values of $\zeta$ as determined for equal quantities of the two masses should be equal. Or, when one of three contiguous masses can be formed out of the other two, it is necessary for equilibrium that the value of $\zeta$ for any quantity of the first mass should be equal to the sum of the values of $\zeta$ for such quantities of the second and third masses as together contain the same matter. Thus, for the equilibrium of a solution composed of $a$ parts of water