Page:Scientific Papers of Josiah Willard Gibbs.djvu/333

Rh equivalent expression $$\tfrac{1}{3}w_{S}$$. Hence the line of intersection of the surfaces of discontinuity $$\text{A-B, B-C, C-A}$$ is stable for values of greater than the other pressures (and therefore for all values of so long as our method is to be regarded as accurate, which will be so long as the mass $$D$$ which would be in equilibrium has a sensible size.

In certain cases in which the tensions of the new surfaces are much too large to be represented as in figure 15, the reasoning of the two last paragraphs will cease to be applicable. These are cases in which the six tensions cannot be represented by the sides of a tetrahedron. It is not necessary to discuss these cases, which are distinguished by the different shape which the mass $$D$$ would take if it should be formed, since it is evident that they can constitute no exception to the results which we have obtained. For an increase of the values of $$\sigma_{DA}, \sigma_{DB}, \sigma_{DC}$$ cannot favor the formation of $$D$$, and hence cannot impair the stability of the line considered, as deduced from our equations. Nor can an increase of these tensions essentially affect the fact that the stability thus demonstrated may fail to be realized when $$p_{D}$$ is considerably greater than the other pressures, since the a priori demonstration of the stability of any one of the surfaces $$\text{A-B, B-C, C-A}$$, taken singly, is subject to the limitation mentioned. (See pages 261, 262.)

Let four different fluid masses $$A, B, C, D$$ meet about a point, so as to form the six surfaces of discontinuity $$\text{A-B, B-C, C-A, D-A, D-B, D-C}$$, which meet in the four lines $$\text{A-B-C, B-C-D, C-D-A, D-A-B}$$, these lines meeting in the vertical point. Let us suppose the system stable in other respects, and consider the conditions of stability for the vertical point with respect to the possible formation of a different fluid mass $$E$$.

If the system can be in equilibrium when the vertical point has been replaced by a mass $$E$$ against which the four masses $$A, B, C, D$$ abut, being truncated at their vertices, it is evident that $$E$$ will have four vertices, at each of which six surfaces of discontinuity meet. (Thus at one vertex there will be the surfaces formed by $$A, B, C$$, and $$E$$.) The tensions of each set of six surfaces (like those of the six surfaces formed by $$A, B, C$$, and $$D$$) must therefore be such that they can be represented by the six edges of a tetrahedron. When the tensions do not satisfy these relations, there will be no particular condition of stability for the point about which $$A, B, C$$, and $$D$$ meet, since if a mass $$E$$ should be formed, it would distribute itself along some of the lines or surfaces which meet at the vertical point, and it