Page:Scientific Papers of Josiah Willard Gibbs.djvu/263

we distinguish the quantities determined for ${\mathsf {s}}'$ and for ${\mathsf {s}}''$ by the marks $'$ and $''$, we may therefore write

$\sigma '\delta s'+{\tfrac {1}{2}}(C_{1}'+C_{2}')\delta (c_{1}'+c_{2}')=\sigma ''+\delta s''{\tfrac {1}{2}}(C_{1}''+C_{2}'')\delta (c_{1}''+c_{2}'').$

Now if we make$\delta s''=0,$

we shall have by geometrical necessity

$\delta s'=s\lambda \delta (c_{1}''+c_{2}'').$

Hence

$\sigma 's'\lambda \delta (c_{1}''+c_{2}'')+{\tfrac {1}{2}}(C_{1}'+C_{2}')\delta (c_{1}'+c_{2}')={\tfrac {1}{2}}(C_{1}''+C_{2}'')\delta (c_{1}''+c_{2}'').$

But$\delta (c_{1}'+c_{2}')=\delta (c_{1}''+c_{2}'').$

Therefore,$C_{1}'+C_{2}'+2\sigma 's\lambda =C_{1}''+C_{2}''.$

This equation shows that we may give a positive or negative value to $C_{1}''+C_{2}''$ by placing ${\mathsf {s}}''$ a sufficient distance on one or on the other side of ${\mathsf {s}}'$. Since this is true when the (unvaried) surface is plane, it must also be true when the surface is nearly plane. And for this purpose a surface may be regarded as nearly plane, when the radii of curvature are very large in proportion to the thickness of the non-homogeneous film. This is the case when the radii of curvature have any sensible size. In general, therefore, whether the surface of discontinuity is plane or curved it is possible to place the surface ${\mathsf {s}}$ so that $C_{1}+C_{2}$ in equation (494) shall vanish.

Now we may easily convince ourselves by equation (493) that if ${\mathsf {s}}$ is placed within the non-homogeneous film, and $s=1$, the quantity $\sigma$ is of the same order of magnitude as the values of $\epsilon ^{S},\eta ^{S},m_{1}^{S},m_{2}^{S}$, etc., while the values of $C_{1}$ and $C_{2}$ are of the same order of magnitude as the changes in the values of the former quantities caused by increasing the curvature of ${\mathsf {s}}$ by unity. Hence, on account of the thinness of the non-homogeneous film, since it can be very little affected by such a change of curvature in ${\mathsf {s}}$, the values of $C_{1}$ and $C_{2}$ must in general be very small relatively to $\sigma$. And hence, if ${\mathsf {s}}'$ be placed within the non-homogeneous film, the value of $\lambda$ which will make $C_{1}''+C_{2}''$ vanish must be very small (of the same order of magnitude as the thickness of the non-homogeneous film). The position of ${\mathsf {s}}$, therefore, which will make $C_{1}+C_{2}$ in (494) vanish, will in general be sensibly coincident with the physical surface of discontinuity.

We shall hereafter suppose, when the contrary is not distinctly indicated, that the surface ${\mathsf {s}}$, in the unvaried state of the system, has such a position as to make $C_{1}+C_{2}=0$. It will be remembered that the surface s is a part of a larger surface $S$, which we have called the dividing surface, and which is coextensive with the physical surface of discontinuity. We may suppose that the position of the dividing surface is everywhere determined by similar considerations. This