Page:Scientific Papers of Josiah Willard Gibbs.djvu/357

where $\epsilon _{V}'$ and $\epsilon _{V}''$ denote the volume-densities of energy in the crystal and fluid respectively, $s$ the area of the side on which the crystal grows, $\epsilon _{S(1)}$ the surface-density of energy on that side, $\epsilon _{S(1)}'$ the surface-density of energy on an adjacent side, $\omega '$ the external angle of these two sides, $l'$ their common edge, and the symbol $\textstyle \sum '$ a summation with respect to the different sides adjacent to the first. The increments of entropy and of the quantities of the several components will be represented by analogous formulae, and if we deduce as on pages 316, 317 the expression for the increase of energy in the whole system due to the growth of the crystal without change of the total entropy or volume, and set this expression equal to zero, we shall obtain for the condition of equilibrium

$(\epsilon _{V}'-t\eta _{V}'-\mu _{1}''\gamma _{1}+p'')s\,\delta N+\textstyle \sum '\displaystyle (\sigma 'l'\csc \omega '-\sigma l'\cot \omega ')\delta N=0,$

(664)

where $\sigma$ and $\sigma '$ relate respectively to the same sides as $\epsilon _{S(1)}$ and $\epsilon _{S(1)}'$ in the preceding formula. This gives

$\mu _{1}''={\frac {\epsilon _{V}'-t\eta _{V}'+p''}{\gamma _{1}'}}+{\frac {\textstyle \sum '\displaystyle (\sigma 'l'\csc \omega '-\sigma l'\cot \omega ')}{s\gamma _{1}'}}\cdot$

(665)

It will be observed that unless the side especially considered is small or narrow, we may neglect the second fraction in this equation, which will then give the same value of $\mu _{1}''$ as equation (387), or as equation (661) applied to a plane surface.

Since a similar equation must hold true with respect to every other side of the crystal of which the equilibrium is not affected by meeting some other body, the condition of equilibrium for the crystalline form (when unaffected by gravity) is that the expression

${\frac {\textstyle \sum '\displaystyle (\sigma 'l'\csc \omega '-\sigma l'\cot \omega ')}{s}}$

(666)

shall have the same value for each side of the crystal. (By the value of this expression for any side of the crystal is meant its value when $\sigma$ and $s$ are determined by that side and the other quantities by the surrounding sides in succession in connection with the first side.) This condition will not be affected by a change in the size of a crystal while its proportions remain the same. But the tendencies of similar crystals toward the form required by this condition, as measured by the inequalities in the composition or the temperature of the surrounding fluid which would counterbalance them, will be inversely as the linear dimensions of the crystals, as appears from the preceding equation.

If we write $v$ for the volume of a crystal, and $\textstyle \sum \displaystyle (\sigma s)$ for the sum of the areas of all its sides multiplied each by the corresponding value of $\sigma$, the numerator and denominator of the fraction (666), multiplied each by $\delta N$, may be represented by $\delta \textstyle \sum \displaystyle (\sigma s)$ and $\delta v$