Page:Scientific Papers of Josiah Willard Gibbs.djvu/261

Rh therefore only to regard variations in the position and form of the limited surface $$\mathsf{s}$$, as this determines all of the surfaces in question lying within the region of non-homogeneity. Let us first suppose the form of $$\mathsf{s}$$ to remain unvaried and only its position in space to vary, either by translation or rotation. No change in (492) will be necessary to make it valid in this case. For the equation is valid if $$\mathsf{s}$$ remains fixed and the material system is varied in position; also, if the material system and $$\mathsf{s}$$ are both varied in position, while their relative position remains unchanged. Therefore, it will be valid if the surface alone varies its position.

But if the form of $$\mathsf{s}$$ be varied, we must add to the second member of (492) terms which shall represent the value of due to such variation in the form of $$\mathsf{s}$$. If we suppose $$\mathsf{s}$$ to be sufficiently small to be considered uniform throughout in its curvatures and in respect to the state of the surrounding matter, the value of the above expression will be determined by the variation of its area $$\delta s$$ and the variations of its principal curvatures $$\delta c_{1}$$and $$\delta c_{2}$$, and we may write or  $$\sigma, C_{1},$$ and $$C_{2}$$ denoting quantities which are determined by the initial state of the system and the position and form of $$\mathsf{s}$$. The above is the complete value of the variation of $$\epsilon^S$$ for reversible variations of the system. But it is always possible to give such a position to the surface $$\mathsf{s}$$ that $$C_{1} + C_{2}$$ shall vanish.

To show this, it will be convenient to write the equation in the longer form {see (490), (491)}

i.e., by (482)–(484) and (12), From this equation it appears in the first place that the pressure is the same in the two homogeneous masses separated by a plane