Page:Scientific Papers of Josiah Willard Gibbs.djvu/286

consider those varied surfaces of tension which satisfy the same condition. We may therefore regard the surface of tension as determined by $v'$, the volume of one of the homogeneous masses. But the state of the system would evidently be completely determined by the position of the surface of tension and the temperature and potentials, if the entropy and the quantities of the components were variable; and therefore, since the entropy and the quantities of the components are constant, the state of the system must be completely determined by the position of the surface of tension. We may therefore regard all the quantities relating to the system as functions of $v'$, and the condition of stability may be written

${\frac {d\epsilon }{dv'}}dv'+{\frac {1}{2}}{\frac {d^{2}\epsilon }{dv^{'2}}}dv^{'2}+{\text{etc.}}>0,$

where $\epsilon$ denotes the total energy of the system. Now the conditions of equilibrium require that

${\frac {d\epsilon }{dv'}}=0.$

Hence, the general condition of stability is that

${\frac {d^{2}\epsilon }{dv^{'2}}}>0.$

(541)

Now if we write $\epsilon ',\epsilon '',\epsilon ^{S}$ for the energies of the two masses and of the surface, we have by (86) and (501), since the total entropy and the total quantities of the several components are constant,

$d\epsilon =d\epsilon '+d\epsilon ''+d\epsilon ^{S}=-p'dv'-p''dv''+\sigma ds,$

or, since $dv''=-dv'$,

${\frac {d\epsilon }{dv'}}=-p'+p''+\sigma {\frac {ds}{dv'}}\cdot$

(542)

Hence,

${\frac {d^{2}\epsilon }{dv^{'2}}}=-{\frac {dp'}{dv'}}+{\frac {dp''}{dv'}}+{\frac {d\sigma }{dv'}}{\frac {ds}{dv'}}+\sigma {\frac {d^{2}s}{dv^{'2}}},$

(543)

and the condition of stability may be written

$\sigma {\frac {d^{2}s}{dv^{'2}}}>{\frac {dp'}{dv'}}-{\frac {dp''}{dv'}}-{\frac {d\sigma }{dv'}}{\frac {ds}{dv'}}\cdot$

(544)

If we now simplify the problem by supposing, as in the similar case on page 245, that we may disregard the variations of the temperature and of all the potentials except one, the condition will reduce to

$\sigma {\frac {d^{2}s}{dv^{'2}}}>\left(\gamma _{1}'-\gamma _{1}''+\Gamma _{1}{\frac {ds}{dv'}}\right){\frac {d\mu _{1}}{dv'}}\cdot$

(545)