Page:Scientific Papers of Josiah Willard Gibbs.djvu/147

true of any two infinitesimally differing phases within the limits of stability. Combining these two conditions we have

$(t''-t')(\eta ''-\eta ')-(p''-p')(v''-v')+(\mu _{1}''-\mu _{1}')(m_{1}''-m_{1}')...(\mu _{n}''-\mu _{n}')(m_{n}''-m_{n}')>0,$

(170)

which may be written more briefly

$\Delta t\Delta \eta -\Delta p\Delta v+\Delta \mu _{1}\Delta m_{1}...\Delta \mu _{n}\Delta m_{n}>0.)$

(171)

This must hold true of any two infinitesimally differing phases within the limits of stability. If, then, we give the value zero to one of the differences in every term except one, but not so as to make the phases completely identical, the values of the two differences in the remaining term will have the same sign, except in the case of $\Delta p$ and $\Delta v$, which will have opposite signs. (If both states are stable this will hold true even on the limits of stability.) Therefore, within the limits of stability, either of the two quantities occurring (after the sign $\Delta$) in any term of (171) is an increasing function of the other,—except $p$ and $v$, of which the opposite is true,—when we regard as constant one of the quantities occurring in each of the other terms, but not such as to make the phases identical.

If we write $d$ for $\Delta$ in (166)–(169), we obtain conditions which are always sufficient for stability. If we also substitute $\geqq$ for $>$, we obtain conditions which are necessary for stability. Let us consider the form which these conditions will take when $\eta ,v,m_{1},...m_{n}$ are regarded as independent variables. When $dv=0$, we shall have

$dt={\frac {dt}{d\eta }}d\eta +{\frac {dt}{dm_{1}}}dm_{1}...+{\frac {dt}{dm_{n}}}dm_{n}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ (172)
$d\mu _{1}={\frac {d\mu _{1}}{d\eta }}d\eta +{\frac {d\mu _{1}}{dm_{1}}}dm_{1}...+{\frac {d\mu _{1}}{dm_{n}}}dm_{n}$
. . . . . .
$d\mu _{2}={\frac {d\mu _{2}}{d\eta }}d\eta +{\frac {d\mu _{2}}{dm_{1}}}dm_{1}...+{\frac {d\mu _{2}}{dm_{n}}}dm_{n}$

Let us write $R_{n+1}$ for the determinant of the order $n+1:$

${\begin{vmatrix}{\frac {d^{2}\epsilon }{d\eta ^{2}}}&{\frac {d^{2}\epsilon }{dm_{1}d\eta }}&...&{\frac {d^{2}\epsilon }{dm_{n}d\eta }}\\{\frac {d^{2}\epsilon }{d\eta dm_{1}}}&{\frac {d^{2}\epsilon }{dm_{1}^{2}}}&...&{\frac {d^{2}\epsilon }{dm_{n}dm_{1}}}\\.&.&...&.\\{\frac {d^{2}\epsilon }{d\eta dm_{n}}}&{\frac {d^{2}\epsilon }{dm_{1}dm_{n}}}&...&{\frac {d^{2}\epsilon }{dm_{n}^{2}}}\end{vmatrix}},$

(173)

of which the constituents are by (86) the same as the coefficients in equations (172), and $R_{n},R_{n-1},$ etc., for the minors obtained by erasing the last column and row in the original determinant and in the