Page:Scientific Papers of Josiah Willard Gibbs.djvu/280

This will enable us to determine, for any given state of the external mass, the values of $\mu _{1}$ which will make the equilibrium stable or unstable.

If the component to which $\gamma _{1}'$ and $\Gamma _{1}$ relate is found only at the surface of discontinuity, the condition of stability reduces to

${\frac {\Gamma _{1}^{2}}{\sigma }}{\frac {d\mu _{1}}{d\Gamma _{1}}}<{\frac {1}{2}}\cdot$

(530)

we may also write

${\frac {\Gamma _{1}}{\sigma }}{\frac {d\sigma }{d\Gamma _{1}}}<-{\frac {1}{2}},$⁠or⁠${\frac {d\log \sigma }{d\log \Gamma _{1}}}<-{\frac {1}{2}}$

(531)

Again, if $\Gamma _{1}=0$ and ${\frac {d\Gamma _{1}}{d\mu _{1}}}=0$, the condition of stability reduces to

${\frac {3\gamma _{1}^{'2}}{p'-p''}}{\frac {d\mu _{1}}{d\gamma _{1}'}}>1.$

(532)

we may also write

${\frac {\gamma _{1}'}{p'-p''}}{\frac {dp'}{d\gamma _{1}'}}>{\frac {1}{3}}$⁠or⁠${\frac {d\log(p'-p'')}{d\log \gamma _{1}'}}>{\frac {1}{3}}\cdot$

(533)

When $r$ is large, this will be a close approximation for any values of $\Gamma _{1}$, unless $\gamma _{1}'$ is very small. The two special conditions (531) and (533) might be derived from very elementary considerations.

Similar conditions of stability may be found when there are more substances than one in the inner mass or the surface of discontinuity, which are not components of the enveloping mass. In this case, we have instead of (526) a condition of the form

$(r\gamma _{1}'+2\Gamma _{1}){\frac {d\mu _{1}}{dr}}+(r\gamma _{1}'+2\Gamma _{1}){\frac {2\mu _{2}}{dr}}+{\text{etc.}}<p''-p',$

(534)

from which ${\frac {d\mu _{1}}{dr}},{\frac {d\mu _{2}}{dr}}$, etc. may be eliminated by means of equations derived from the conditions that

$\gamma _{1}'v'+\Gamma _{1}s,$⁠$\gamma _{2}'v'+\Gamma _{2}s,$⁠etc.

must be constant.

Nearly the same method may be applied to the following problem. Two different homogeneous fluids are separated by a diaphragm having a circular orifice, their volumes being invariable except by the motion of the surface of discontinuity, which adheres to the edge of the orifice;—to determine the stability or instability of this surface when in equilibrium.