Page:Scientific Papers of Josiah Willard Gibbs.djvu/279

equilibrium will be stable in respect to change in one direction and unstable in respect to change in the opposite direction, and is therefore to be considered unstable. In general, therefore, we may call (523) the condition of stability.

When the interior mass and the surface of discontinuity are formed entirely of substances which are components of the external mass, $p'$ and $\sigma$ cannot vary, and condition (524) being satisfied the equilibrium is unstable.

But if either the interior homogeneous mass or the surface of discontinuity contains substances which are not components of the enveloping mass, the equilibrium may be stable. If there is but one such substance, and we denote its densities and potential by $\gamma _{1}',\Gamma _{1}$, and $\mu _{1}$, the condition of stability (523) will reduce to the form

$\left(r{\frac {dp'}{d\mu _{1}}}-2{\frac {d\sigma }{d\mu _{1}}}\right){\frac {d\mu _{1}}{dr}}<p''-p',$

or, by (98) and (508),

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

(526)

In these equations and in all which follow in the discussion of this case, the temperature and the potentials $\mu _{2},\mu _{3}$, etc. are to be regarded as constant. But

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

which represents the total quantity of the component specified by the suffix, must be constant. It is evidently equal to

${\tfrac {4}{3}}\pi r^{3}\gamma _{1}'+4\pi r^{2}\Gamma _{1}.$

Dividing by $4\pi$ and differentiating, we obtain

$(r^{2}\gamma _{1}'+2r\Gamma _{1})dr+{\tfrac {1}{3}}r^{3}d\gamma _{1}'+r^{2}d\Gamma _{1}=0,$

or, since $\gamma _{1}'$ and $\Gamma _{1}$ are functions of $\mu _{1}$,

$(r\gamma _{1}'+2\Gamma _{1})dr+\left({\frac {r^{2}}{3}}{\frac {d\gamma _{1}'}{d\mu _{1}}}+r{\frac {d\Gamma _{1}}{d\mu _{1}}}\right)d\mu _{1}=0.$

(527)

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

${\frac {(r\gamma _{1}'+2\Gamma _{1})^{2}}{{\frac {r^{2}}{3}}{\frac {d\gamma _{1}'}{d\mu _{1}}}+r{\frac {d\Gamma _{1}}{d\mu _{1}}}}}>p'-p''.$

(528)

If we eliminate $r$ by equation (522), we have

${\frac {\left({\frac {\gamma _{1}'}{p'-p''}}+{\frac {\Gamma _{1}}{\sigma }}\right)^{2}}{{\frac {1}{3(p'-p'')}}{\frac {d\gamma _{1}'}{d\mu _{1}}}+{\frac {1}{2\sigma }}{\frac {d\Gamma _{1}}{d\mu _{1}}}}}<1.$

(529)

If $p'$ and $\sigma$ are known in terms of $t,\mu _{1},\mu _{2}$, etc., we may express the first member of this condition in terms of the same variables and $p''$.