Page:Scientific Papers of Josiah Willard Gibbs.djvu/324

288 of the system we can construct two triangles having similar relations to the surfaces of discontinuity meeting at $$O'$$ and $$O''$$. But the directions of the normals to the surfaces $$\text{A-B}$$ and $$\text{B-C}$$ at $$O'$$ and to $$\text{C-D}$$ and $$\text{D-A}$$ at $$O''$$ in the varied state of the system differ infinitely little from the directions of the corresponding normals at $$O$$ in the initial state. We may therefore regard $$\alpha \beta, \beta \gamma$$ as two sides of the triangle representing the surfaces meeting at $$O'$$, and $$\gamma \delta, \delta \alpha$$ as two sides of the triangle representing the surfaces meeting at $$O''$$. Therefore, if we join $$\alpha \gamma$$, this line will represent the direction of the normal to the surface $$\text{A-C}$$, and the value of its tension. If the tension of a surface between such masses as $$A$$ and $$C$$ had been greater than that represented by $$\alpha \gamma$$, it is evident that the initial state of the system of surfaces (represented in figure 11) would have been stable with respect to the possible formation of any such surface. If the tension had been less, the state of the system would have been at least practically unstable. To determine whether it is unstable in the strict sense of the term, or whether or not it is properly to be regarded as in equilibrium, would require a more refined analysis than we have used. We may here remark that a nearer approximation in the theory of equilibrium and stability might be attained by taking special account, in our general equations, of the lines in which surfaces of discontinuity meet. These lines might be treated in a manner entirely analogous to that in which we have treated surfaces of discontinuity. We might recognize linear densities of energy, of entropy, and of the several substances which occur about the line, also a certain linear tension. With respect to these quantities and the temperature and potentials, relations would hold analogous to those which have been demonstrated for surfaces of discontinuity. (See pp. 229–231.) If the sum of the tensions of the lines $$L'$$ and $$L''$$, mentioned above, is greater than the tension of the line $$L$$, this line will be in strictness stable (although practically unstable) with respect to the formation of a surface between $$A$$ and $$C$$, when the tension of such a surface is a little less than that represented by the diagonal $$\alpha \gamma$$. The different use of the term practically unstable in different parts of this paper need not create confusion, since the general meaning of the term is in all cases the same. A system is called practically unstable when a very small (not necessarily indefinitely small) disturbance or variation in its condition will produce a considerable change. In the former part of this paper, in which the influence of surfaces of discontinuity was neglected, a system was regarded as practically unstable when such a result would be produced by a disturbance of the same order of magnitude as the quantities relating to surfaces of discontinuity which were neglected. But where surfaces of discontinuity are considered, a system is not regarded as practically unstable, unless the disturbance which will produce such a result is very small compared with the quantities relating to surfaces of discontinuity of any appreciable magnitude.

The result which we have obtained may be generalized as follows. When more than three surfaces of discontinuity in a fluid system meet in equilibrium along a line, with respect to the surfaces and masses immediately adjacent to any point of this line, we may form a polygon of which the angular points shall correspond in order to the different masses separated by the surfaces of discontinuity, and