Page:Scientific Papers of Josiah Willard Gibbs.djvu/334

298 is therefore sufficient to consider the stability of these lines and surfaces. We shall suppose that the relations mentioned are satisfied.

If we denote by $$W_{V}$$ the work gained in forming the mass $$E$$ (of such size and form as to be in equilibrium) in place of the portions of the other masses which are suppressed, and by $$W_{S}$$ the work expended in forming the new surfaces in place of the old, it may easily be shown by a method similar to that used on page 292 that whence also, that when the volume $$E$$ is small, the equilibrium of $$E$$ will be stable or unstable according as $$W_{S}$$ and $$W_{V}$$ are negative or positive.

A critical relation for the tensions is that which makes equilibrium possible for the system of the five masses $$A, B, C, D, E$$, when all the surfaces are plane. The ten tensions may then be represented in magnitude and direction by the ten distances of five points in space $$\alpha, \beta, \gamma, \delta, \epsilon$$, viz., the tension of $$\text{A-B}$$ and the direction of its normal by the line $$\alpha \beta$$, etc. The point $$\epsilon$$ will lie within the tetrahedron formed by the other points. If we write $$v_{E}$$ for the volume of $$E$$, and $$v_{A}, v_{B}, v_{C}, v_{D}$$ for the volumes of the parts of the other masses which are suppressed to make room for $$E$$, we have evidently Hence, when all the surfaces are plane, $$W_{V} = 0$$, and $$W_{S} = 0$$. Now equilibrium is always possible for a given small value of $$v_{E}$$ with any given values of the tensions and of $$p_{A}, p_{B}, p_{C}, p_{D}$$. When the tensions satisfy the critical relation, $$W_{S} = 0$$, if $$p_{A} = p_{B} = p_{C} = p_{D}$$. But when $$v_{E}$$ is small and constant, the value of $$W_{S}$$ must be independent of $$p_{A}, p_{B}, p_{C}, p_{D}$$, since the angles of the surfaces are determined by the tensions and their curvatures may be neglected. Hence, $$W_{S} = 0$$, and $$W_{V} = 0$$, when the critical relation is satisfied and $$v_{E}$$ small. This gives In calculating the ratios of $$v_{A}, v_{B}, v_{C}, v_{D}, v_{E}$$, we may suppose all the surfaces to be plane. Then $$E$$ will have the form of a tetrahedron, the vertices of which may be called $$\text{a, b, c, d}$$ (each vertex being named after the mass which is not found there), and $$v_{A}, v_{B}, v_{C}, v_{D}$$, will be the volumes of the tetrahedra into which it may be divided by planes passing through its edges and an interior point $$\text{e}$$. The volumes of these tetrahedra are proportional to those of the five tetrahedra of the figure $$\alpha\beta\gamma\delta\epsilon$$, as will easily appear if we recollect that the line $$\text{ab}$$ is common to the surfaces $$\text{C-D, D-E, E-C}$$, and therefore perpendicular to the surface common to the lines $$\gamma\delta, \delta\epsilon, \epsilon\gamma$$, i.e. to the surface $$\gamma\delta\epsilon$$, and so in other cases (it will be observed that $$\gamma, \delta$$, and $$\epsilon$$ are the letters which do not correspond to $$\text{a}$$ or $$\text{b}$$); also