Page:Scientific Papers of Josiah Willard Gibbs.djvu/282

(the total quantity of the component specified by the suffix) must be constant; therefore, since

$dv''=-dv',$⁠and⁠$ds={\frac {2}{r}}dv',$

$\left(v'{\frac {d\gamma _{1}'}{d\mu _{1}}}+v''{\frac {d\gamma _{1}''}{d\mu _{1}}}+s{\frac {d\Gamma _{1}}{d\mu _{1}}}\right)+\left(\gamma _{1}'-\gamma _{1}''+{\frac {2\Gamma _{1}}{r}}\right)dv'=0.$

(538)

By this equation, the condition of stability is brought to the form

${\frac {\left(\gamma _{1}'-\gamma _{1}''+{\frac {2\Gamma _{1}}{r}}\right)^{2}}{v'{\frac {d\gamma _{1}'}{d\mu _{1}}}+v''{\frac {d\gamma _{1}''}{d\mu _{1}}}+s{\frac {d\Gamma _{1}}{d\mu _{1}}}}}>(p'-p''){\frac {x-r}{\pi x^{2}r^{2}}}\cdot$

(540)

When the substance specified by the suffix is a component of either of the homogeneous masses, the terms ${\frac {2\Gamma _{1}}{r}}$ and $s{\frac {d\Gamma _{1}}{d\mu _{1}}}$ may generally be neglected. When it is not a component of either, the terms $\gamma _{1}',\gamma _{1}'',v'{\frac {d\gamma _{1}'}{d\mu _{1}}},{\frac {d\gamma _{1}''}{d\mu _{1}}}$ may of course be cancelled, but we must not apply the formula to cases in which the substance spreads over the diaphragm separating the homogeneous masses.

In the cases just discussed, the problem of the stability of certain surfaces of tension has been solved by considering the case of neutral equilibrium,—a condition of neutral equilibrium affording the equation of the limit of stability. This method probably leads as directly as any to the result, when that consists in the determination of the value of a certain quantity at the limit of stability, or of the relation which exists at that limit between certain quantities specifying the state of the system. But problems of a more general character may require a more general treatment.

Let it be required to ascertain the stability or instability of a fluid system in a given state of equilibrium with respect to motion of the surfaces of tension and accompanying changes. It is supposed that the conditions of internal stability for the separate homogeneous masses are satisfied, as well as those conditions of stability for the surfaces of discontinuity which relate to small portions of these surfaces with the adjacent masses. (The conditions of stability which are here supposed to be satisfied have been already discussed in part and will be farther discussed hereafter.) The fundamental equations for all the masses and surfaces occurring in the system are supposed to be known. In applying the general criteria of stability which are given on page 57, we encounter the following difficulty.

The question of the stability of the system is to be determined by the consideration of states of the system which are slightly varied