Page:Scientific Papers of Josiah Willard Gibbs.djvu/272

 The same physical relations may of course be deduced without giving up the use of the surface of tension as a dividing surface, but the formulæ which express them will be less simple. If we make $$t, \mu_{3}, \mu_{4}$$, etc., constant, we have by (98) and (508) where we may suppose $$\Gamma_{1}$$ and $$\Gamma_{2}$$ to be determined with reference to the surface of tension. Then, if $$dp' = dp$$, and  That is,  The reader will observe that $$\frac{\Gamma_{1}}{\gamma_{1}' - \gamma_{1}}$$ represents the distance between the surface of tension and that dividing surface which would make $$\Gamma_{1} = 0$$; the second number of the last equation is therefore equivalent to $$- \Gamma_{2(1)}$$.

If any component substance has the same density in the two homogeneous masses separated by a plane surface of discontinuity, the value of the superficial density for that component is independent of the position of the dividing surface. In this case alone we may derive the value of the superficial density of a component with reference to the surface of tension from the fundamental equation for plane surfaces alone. Thus in the last equation, when $$\gamma_{2}' = \gamma_{2}''$$, the second member will reduce to $$- \Gamma_{2}$$. It will be observed that to