Page:Scientific Papers of Josiah Willard Gibbs.djvu/270

change in the position of the dividing surface. But the expressions $\epsilon ^{S},\eta ^{S},m_{1}^{S},m_{2}^{S}$, etc., as also $\epsilon _{S},\eta _{S},\Gamma _{1},\Gamma _{2}$, etc., and $\psi ^{S}$, will of course have different values when the position of that surface is changed. The quantity $\sigma$, however, which we may regard as defined by equations (501), or, if we choose, by (502) or (507), will not be affected in value by such a change. For if the dividing surface be moved a distance $\lambda$ measured normally and toward the side to which $v''$ relates, the quantities

$\epsilon _{S},$⁠$\eta _{S},$⁠$\Gamma _{1},$⁠$\Gamma _{2},$⁠etc.,

will evidently receive the respective increments

$\lambda (\epsilon _{V}''-\epsilon _{V}'),$⁠$\lambda (\eta _{V}''-\eta _{V}'),$⁠$\lambda (\gamma _{1}''-\gamma _{1}'),$⁠$\lambda (\gamma _{2}''-\gamma _{2}'),$⁠etc.,

$\epsilon _{V}',\epsilon _{V}'',\eta _{V}',\eta _{V}''$ denoting the densities of energy and entropy in the two homogeneous masses. Hence, by equation (507), $\sigma$ will receive the increment

$\lambda (\epsilon _{V}''-\epsilon _{V}')-t\lambda (\eta _{1}''-\eta _{1}')-\mu _{1}\lambda (\gamma _{1}''-\gamma _{1}')-\mu _{2}\lambda (\gamma _{2}''-\gamma _{2}')-{\text{etc.}}$

But by (93)

$-p''=\epsilon _{V}''-t\eta _{V}''-\mu _{1}\gamma _{1}''-\mu _{2}\gamma _{2}''-{\text{etc.,}}$ $-p'=\epsilon _{V}'-t\eta _{V}'-\mu _{1}\gamma _{1}'-\mu _{2}\gamma _{2}'-{\text{etc.}}$

Therefore, since $p'=p''$, the increment in the value of $\sigma$ is zero. The value of $\sigma$ is therefore independent of the position of the dividing surface, when this surface is plane. But when we call this quantity the superficial tension, we must remember that it will not have its characteristic properties as a tension with reference to any arbitrary surface. Considered as a tension, its position is in the surface which we have called the surface of tension, and, strictly speaking, nowhere else. The positions of the dividing surface, however, which we shall consider, will not vary from the surface of tension sufficiently to make this distinction of any practical importance.

It is generally possible to place the dividing surface so that the total quantity of any desired component in the vicinity of the surface of discontinuity shall be the same as if the density of that component were uniform on each side quite up to the dividing surface. In other words, we may place the dividing surface so as to make any one of the quantities $\Gamma _{1},\Gamma _{2}$, etc., vanish. The only exception is with regard to a component which has the same density in the two homogeneous masses. With regard to a component which has very nearly the same density in the two masses such a location of the dividing surface might be objectionable, as the dividing surface might fail to coincide sensibly with the physical surface of discontinuity. Let us suppose that $\gamma _{1}'$ is not equal (nor very nearly equal) to $\gamma _{1}''$, and that the dividing surface is so placed as to make $\Gamma _{1}=0$. Then equation (508) reduces to

$d\sigma =-\eta _{S(1)}dt-\Gamma _{2(1)}d\mu _{2}-\Gamma _{3(1)}d\mu _{3}-{\text{etc.,}}$

(514)