Page:Scientific Papers of Josiah Willard Gibbs.djvu/257

any other (suitable) rule, the energy of the mass on the other side must be less by the same amount when determined by the first rule than when determined by the second, since the discontinuity relative to the second mass is equal but opposite in character to the discontinuity relative to the first.

If the entropy of the mass which occupies any one of the spaces considered is not in the nature of things determined without reference to the surrounding masses, we may suppose a similar method to be applied to the estimation of entropy.

With this understanding, let us return to the consideration of the equilibrium of the three masses $M,M'$, and $M''$. We shall suppose that there are no limitations to the possible variations of the system due to any want of perfect mobility of the components by means of which we express the composition of the masses, and that these components are independent, i.e., that no one of them can be formed out of the others.

With regard to the mass $M$, which includes the surface of discontinuity, it is necessary for its internal equilibrium that when its boundaries are considered constant, and when we consider only reversible variations (i.e., those of which the opposite are also possible), the variation of its energy should vanish with the variations of its entropy and of the quantities of its various components. For changes within this mass will not affect the energy or the entropy of the surrounding masses (when these quantities are estimated on the principle which we have adopted), and it may therefore be treated as an isolated system. For fixed boundaries of the mass $M$, and for reversible variations, we may therefore write

$\delta \epsilon =A_{0}\delta \eta +A_{1}\delta m_{1}+A_{2}\delta m_{2}+{\text{etc.,}}$

(476)

where $A_{0},A_{1},A_{2}$, etc., are quantities determined by the initial (unvaried) condition of the system. It is evident that $A_{0}$ is the temperature of the lamelliform mass to which the equation relates, or the temperature at the surface of discontinuity. By comparison of this equation with (12) it will be seen that the definition of $A_{1},A_{2}$, etc., is entirely analogous to that of the potentials in homogeneous masses, although the mass to which the former quantities relate is not homogeneous, while in our previous definition of potentials, only homogeneous masses were considered. By a natural extension of the term potential, we may call the quantities $A_{1},A_{2}$, etc., the potentials at the surface of discontinuity. This designation will be farther justified by the fact, which will appear hereafter, that the value of these quantities is independent of the thickness of the lamina (M) to which they relate. If we employ our ordinary symbols for temperature and potentials, we may write

$\delta \epsilon =t\delta \eta +\mu _{1}\delta m_{1}+\mu _{2}\delta m_{2}+{\text{etc.}}$

(477)