Page:Scientific Papers of Josiah Willard Gibbs.djvu/269

Rh not the saine value has very nearly the same density in the two homogeneous masses, in which case, the condition under which the variations take place is nearly equivalent to the condition that the pressures shall remain equal.

Accordingly, we cannot in general expect to determine the superficial density $$\Gamma_{1}$$ from its value $$- \left(\frac{d\sigma}{d\mu_{1}} \right)_{t, \mu}$$ by measurements of superficial tensions. The case will be the same with $$\Gamma_{2}, \Gamma_{3}$$, etc., and also with $$\eta_{S}$$, the superficial density of entropy.

The quantities $$\epsilon_{S}, \eta_{S}, \Gamma_{1}, \Gamma_{2}$$, etc., are evidently too small in general to admit of direct measurement. When one of the components, however, is found only at the surface of discontinuity, it may be more easy to measure its superficial density than its potential. But except in this case, which is of secondary interest, it will generally be easy to determine $$\sigma$$ in terms of $$t, \mu_{1}, \mu_{2}$$, etc., with considerable accuracy for plane surfaces, and extremely difficult or impossible to determine the fundamental equation more completely.

An equation giving $$\sigma$$ in terms of $$t, \mu_{1}, \mu_{2}$$, etc., which will hold true only so long as the surface of discontinuity is plane, may be called a fundamental equation for a plane surface of discontinuity.

It will be interesting to see precisely what results can be obtained from such an equation, especially with respect to the energy and entropy and the quantities of the component substances in the vicinity of the surface of discontinuity.

These results can be exhibited in a more simple form, if we deviate to a certain extent from the method which we have been following. The particular position adopted for the dividing surface (which determines the superficial densities) was chosen in order to make the term $$\tfrac{1}{2}(C_{1} + C_{2}) \delta(c_{1} + c_{2})$$ in (494) vanish. But when the curvature of the surface is not supposed to vary, such a position of the dividing surface is not necessary for the simplification of the formula. It is evident that equation (501) will hold true for plane surfaces (supposed to remain such) without reference to the position of the dividing surface, except that it shall be parallel to the surface of discontinuity. We are therefore at liberty to choose such a position for the dividing surface as may for any purpose be convenient.

None of the equations (502)–(513), which are either derived from (501), or serve to define new symbols, will be affected by such a