Page:Scientific Papers of Josiah Willard Gibbs.djvu/318

with respect to the equations of condition, which are of the form

$\int \delta Dm_{1}^{V}+\int \delta Dm_{2}^{V}=0,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ (616)
$\int \delta Dm_{1}^{V}+\int \delta Dm_{2}^{V}=0,$
etc.,

(It is here supposed that the various components are independent, i.e., that none can be formed out of others, and that the parts of the system in which any component actually occurs are not entirely separated by parts in which it does not occur.) To satisfy the condition (603), subject to these equations of condition, it is necessary and sufficient that the conditions

$\mu _{1}+gz=M_{1},$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ (617)
$\mu _{2}+gz=M_{2},$
etc.,

($M_{1},M_{2}$, etc. denoting constants,) shall each hold true in those parts of the system in which the substance specified is an actual component. We may here add the condition of equilibrium relative to the possible absorption of any substance (to be specified by the suffix $_{a}$) by parts of the system of which it is not an actual component, viz., that the expression $\mu _{a}+gz$ must not have a less value in such parts of the system than in a contiguous part in which the substance is an actual component.

From equation (613) with (605) and (617) we may easily obtain the differential equation of a surface of tension (in the geometrical sense of the term), when $p',p''$, and $\sigma$ are known in terms of the temperature and potentials. For $c_{1}+c_{2}$ and $\theta$ may be expressed in terms of the first and second differential coefficients of $z$ with respect to the horizontal co-ordinates, and $p',p'',\sigma$, and $\Gamma$ in terms of the temperature and potentials. But the temperature is constant, and for each of the potentials we may substitute—$gz$ increased by a constant. We thus obtain an equation in which the only variables are $z$ and its first and second differential coefficients with respect to the horizontal co-ordinates. But it will rarely be necessary to use so exact a method. Within moderate differences of level, we may regard $\gamma ',\gamma ''$, and $\sigma$ as constant. We may then integrate the equation {derived from (612)}

$d(p'-p'')=g(\gamma ''-\gamma ')dz,$

which will give

$p'-p''=g(\gamma ''-\gamma ')z,$

(618)

where $z$ is to be measured from the horizontal plane for which $p'=p''$. Substituting this value in (613), and neglecting the term containing $\Gamma$, we have

$c_{1}+c_{2}={\frac {g(\gamma ''-\gamma ')}{\sigma }}z,$

(619)