Page:Scientific Papers of Josiah Willard Gibbs.djvu/328

in the second case, each pair of the triangles $\alpha \beta \delta ''',\beta \gamma \delta ',\gamma \alpha \delta ''$ will overlap, at least when the tensions $\sigma _{DA},\sigma _{DB},\sigma _{DC}$ are only a little too great to be represented as in figure 15, and the sum of the angles of each of the triangles adb, bdc, cda will be greater than two right angles.

Let us denote by $v_{A},v_{B},v_{C}$ the portions of $v_{D}$ which were originally occupied by the masses $A,B,C$, respectively, by $s_{DA},s_{DB},s_{DC}$, the areas of the surfaces specified per unit of length of the mass $D$, and by $s_{AB},s_{BC},s_{CA}$, the areas of the surfaces specified which were replaced by the mass $D$ per unit of its length. In numerical value, $v_{A},v_{B},v_{C}$ will be equal to the areas of the curvilinear triangles $bcd,cad,abd$; and $s_{DA},s_{DB},s_{DC},s_{AB},s_{BC},s_{CA}$ to the lengths of the lines $bc,ca,ab,cd,ad,bd$. Also let

$W_{S}=\sigma _{DA}s_{DA}+\sigma _{DB}s_{DB}+\sigma _{DC}s_{DC}-\sigma _{AB}s_{AB}-\sigma _{BC}s_{BC}-\sigma _{CA}s_{CA},$

(626)

The general condition of mechanical equilibrium for a system of homogeneous masses not influenced by gravity, when the exterior of the whole system is fixed, may be written

$\textstyle \sum \displaystyle (\sigma \delta s)-\textstyle \sum \displaystyle (p\delta v)=0.$

(628)

(See (606).) If we apply this both to the original system consisting of the masses $A,B$, and $C$, and to the system modified by the introduction of the mass $D$, and take the difference of the results, supposing the deformation of the system to be the same in each case, we shall have

$\sigma _{DA}\delta s_{DA}+\sigma _{DB}\delta s_{DB}+\sigma _{DC}\delta s_{DC}-\sigma _{AB}\delta s_{AB}-\sigma _{BC}\delta s_{BC}-\sigma _{CA}\delta s_{CA}-p_{D}\delta v_{D}+p_{A}\delta v_{A}+p_{B}\delta v_{B}+p_{C}\delta v_{C}=0.$

(629)

In view of this relation, if we differentiate (626) and (627) regarding all quantities except the pressures as variable, we obtain

$dW_{S}-dW_{V}=s_{DA}d\sigma _{DA}+s_{DB}d\sigma _{DB}+s_{DC}d\sigma _{DC}-s_{AB}d\sigma _{AB}-s_{BC}d\sigma _{BC}-s_{CA}d\sigma _{CA}.$

(630)

Let us now suppose the system to vary in size, remaining always similar to itself in form, and that the tensions diminish in the same ratio as lines, while the pressures remain constant. Such changes will evidently not impair the equilibrium. Since all the quantities $s_{DA},\sigma _{DA},s_{DB},\sigma _{DB}$, etc. vary in the same ratio,

$s_{DA}d\sigma _{DA}={\tfrac {1}{2}}d(\sigma _{DA}s_{DA}),$$s_{DB}d\sigma _{DB}={\tfrac {1}{2}}d(\sigma _{DB}s_{DB}),$etc.

(631)

We have therefore by integration of (630)

$W_{S}-W_{V}={\tfrac {1}{2}}(\sigma _{DA}s_{DA}+\sigma _{DB}s_{DB}+\sigma _{DC}s_{DC}-\sigma _{AB}s_{AB}-\sigma _{BC}s_{BC}-\sigma _{CA}s_{CA}),$

(632)

whence, by (626),

$W_{S}=2W_{V}.$

(633)