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

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

$W_{S}={\tfrac {3}{2}}W_{V},$

(637)

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

$W_{V}=p_{E}v_{E}-p_{A}v_{A}-p_{B}v_{B}-p_{C}v_{C}-p_{D}v_{D}.$

(639)

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

$p_{E}={\frac {v_{A}p_{A}+v_{B}p_{B}+v_{C}p_{C}+v_{D}p_{D}}{v_{E}}}\cdot$

(640)

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