Page:Scientific Papers of Josiah Willard Gibbs.djvu/262

surface of discontinuity. For let us imagine the material system to remain unchanged, while the plane surface ${\mathsf {s}}$ without change of area or of form moves in the direction of its normal. As this does not affect the boundaries of the mass $M$,

$\delta \epsilon -t\delta \eta -\mu _{1}\delta m_{1}-\mu _{2}\delta m_{2}-{\text{etc.}}=0.$

Also $\delta s=0,\delta (c_{1}+c_{2})=0,\delta (c_{1}-c_{2})=0,$, and $\delta v'''=-\delta v''''.$. Hence $p'=p''$, when the surface of discontinuity is plane.

Let us now examine the effect of different positions of the surface ${\mathsf {s}}$ in the same material system upon the value of $C_{1}+C_{2}$, supposing at first that in the initial state of the system the surface of discontinuity is plane. Let us give the surface ${\mathsf {s}}$ some particular position. In the initial state of the system this surface will of course be plane like the physical surface of discontinuity, to which it is parallel. In the varied state of the system, let it become a portion of a spherical surface having positive curvature; and at sensible distances from this surface let the matter be homogeneous and with the same phases as in the initial state of the system; also at and about the surface let the state of the matter so far as possible be the same as at and about the plane surface in the initial state of the system. (Such a variation in the system may evidently take place negatively as well as positively, as the surface may be curved toward either side. But whether such a variation is consistent with the maintenance of equilibrium is of no consequence, since in the preceding equations only the initial state is supposed to be one of equilibrium.) Let the surface ${\mathsf {s}}$, placed as supposed, whether in the initial or the varied state of the surface, be distinguished by the symbol ${\mathsf {s}}'$. Without changing either the initial or the varied state of the material system, let us make another supposition with respect to the imaginary surface ${\mathsf {s}}$. In the unvaried system let it be parallel to its former position but removed from it a distance $\lambda$ on the side on which lie the centers of positive curvature. In the varied state of the system, let it be spherical and concentric with ${\mathsf {s}}'$, and separated from it by the same distance $\lambda$. It will of course lie on the same side of ${\mathsf {s}}'$ as in the unvaried system. Let the surface ${\mathsf {s}}$, placed in accordance with this second supposition, be distinguished by the symbol ${\mathsf {s}}''$. Both in the initial and the varied state, let the perimeters of ${\mathsf {s}}'$ and ${\mathsf {s}}''$ be traced by a common normal. Now the value of

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

in equation (496) is not affected by the position of ${\mathsf {s}}$, being determined simply by the body $M$. The same is true of $p'\delta v''',p''\delta v''''$ or $p'\delta (v'''+v''''),v'''+v''''$ being the volume of $M$. Therefore the second member of (496) will have the same value whether the expressions relate to ${\mathsf {s}}'$ or ${\mathsf {s}}''$. Moreover, $\delta (c_{1}-c_{2})=0$ both for ${\mathsf {s}}'$ and ${\mathsf {s}}''$. If