Page:Scientific Papers of Josiah Willard Gibbs.djvu/323

Rh be shown (compare page 252) that this condition is always sufficient for stability with reference to motion of surfaces of discontinuity, even when the variations of $$t, M_{1}, M_{2}$$, etc. cannot be neglected in the determination of the necessary condition of stability with respect to such changes.

When more than three surfaces of discontinuity between homogeneous masses meet along a line, we may conceive of a new surface being formed between any two of the masses which do not meet in a surface in the original state of the system. The condition of stability with respect to the formation of such a surface may be easily obtained by the consideration of the limit between stability and instability, as exemplified by a system which is in equilibrium when a very small surface of the kind is formed.

To fix our ideas, let us suppose that there are four homogeneous masses $$A, B, C$$, and $$D$$, which meet one another in four surfaces, which we may call $$\text{A-B, B-C, C-D}$$, and $$\text{D-A}$$, these surfaces all meeting along a line $$L$$. This is indicated in figure 11 by a section of the surfaces cutting the line $$L$$ at right angles at a point $$O$$. In an infinitesimal variation of the state of the system, we may conceive of a small surface being formed between $$A$$ and $$C$$ (to be called $$\text{A-C}$$), so that the section of the surfaces of discontinuity by the same plane takes the form indicated in figure 12. Let us suppose that the condition of equilibrium (615) is satisfied both for the line $$L$$ in which the surfaces of discontinuity meet in the original state of the system, and for the two such lines (which we may call $$L'$$ and $$L$$) in the varied state of the system, at least at the points $$O'$$ and $$O$$ where they are cut by the plane of section. We may therefore form a quadrilateral of which the sides $$\alpha \beta, \beta \gamma, \gamma \delta, \delta \alpha$$ are equal in numerical value to the tensions of the several surfaces $$\text{A-B, B-C, C-D, D-A}$$, and are parallel to the normals to these surfaces at the point $$O$$ in the original state of the system. In like manner, for the varied state