Page:Scientific Papers of Josiah Willard Gibbs.djvu/259

the variations being subject to the equations of condition

$\delta \eta +\delta \eta '+\delta \eta ''=0,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ (481)
$\delta m_{1}+\delta m_{1}'+\delta m_{1}''=0,$
$\delta m_{2}+\delta m_{2}'+\delta m_{2}''=0,$
etc.

It may also be the case that some of the quantities | $\delta m_{1}',\delta m_{1}'',\delta m_{2}',\delta m_{2}''$, etc., are incapable of negative values or can only have the value zero. This will be the case when the substances to which these quantities relate are not actual or possible components of $M'$ or $M''$. (See page 64.) To satisfy the above condition it is necessary and sufficient that

$t=t'=t'',$

(482)

$\mu _{1}'\delta m_{1}'\geqq \mu _{1}\delta m_{1}',$⁠$\mu _{2}'\delta m_{2}'\geqq \mu _{2}\delta m_{2}',$⁠etc.,

(483)

$\mu _{1}''\delta m_{1}''\geqq \mu _{1}\delta m_{1}'',$⁠$\mu _{2}''\delta m_{2}''\geqq \mu _{2}\delta m_{2}'',$⁠etc.

(484)

It will be observed that, if the substance to which $\mu _{1}$ for instance, relates is an actual component of each of the homogeneous masses, we shall have $\mu _{1}=\mu _{1}'=\mu _{1}''.$ If it is an actual component of the first only of these masses, we shall have $\mu _{1}=\mu _{1}'.$ If it is also a possible component of the second homogeneous mass, we shall also have $\mu _{1}\geqq \mu _{1}''.$ If this substance occurs only at the surface of discontinuity, the value of the potential $\mu _{1}$ will not be determined by any equation, but cannot be greater than the potential for the same substance in either of the homogeneous masses in which it may be a possible component.

It appears, therefore, that the particular conditions of equilibrium relating to temperature and the potentials which we have before obtained by neglecting the influence of the surfaces of discontinuity (pp. 65, 66, 74) are not invalidated by the influence of such discontinuity in their application to homogeneous parts of the system bounded like $M'$ and $M''$ by imaginary surfaces lying within the limits of homogeneity,—a condition which may be fulfilled by surfaces very near to the surfaces of discontinuity. It appears also that similar conditions will apply to the non-homogeneous films like $M$, which separate such homogeneous masses. The properties of such films, which are of course different from those of homogeneous masses, require our farther attention.

The volume occupied by the mass $M$ is divided by the surface ${\mathsf {s}}$ into two parts which we will call $v'''$ and $v'''',v'''$ lying next to $M'$, and $v''''$ to $M''$. Let us imagine these volumes filled by masses having throughout the same temperature, pressure and potentials, and the same densities of energy and entropy, and of the various components,