Page:Scientific Papers of Josiah Willard Gibbs.djvu/150

constant the quantities occurring in the numerators of the others together with $v$, will have the value zero. But if one such has the value zero, all such will in general have the same value. For if

$\left({\frac {d\mu _{n}}{dm_{n}}}\right)_{t,v,\mu _{1},...\mu _{n-1}},$

(182)

for example, has the value zero, we may change the density of the component $S_{n}$ without altering (if we disregard infinitesimals of higher orders than the first) the temperature or the potentials, and therefore, by (98), without altering the pressure. That is, we may change the phase without altering any of the quantities $t,p,\mu _{1},...\mu _{n}$. (In other words, the phases adjacent to the limits of stability exhibit

approximately the relations characteristic of neutral equilibrium.) Now this change of phase, which changes the density of one of the components, will in general change the density of the others and the density of entropy. Therefore, all the other differential coefficients formed after the analogy of (182), i.e., formed from the fractions in (181) by taking as constants for each the quantities in the numerators of the others together with v, will in general have the value zero at the limit of stability. And the relation which characterizes the limit of stability may be expressed, in general, by setting any one of these differential coefficients equal to zero. Such an equation, when the fundamental equation is known, may be reduced to the form of an equation between the independent variables of the fundamental equation.

Again, as the determinant (173) is equal to the product of the differential coefficients obtained by writing $d$ for $\Delta$ in the first members of (166)-(169), the equation of the limit of stability may be expressed by setting this determinant equal to zero. The form of the differential equation as thus expressed will not be altered by the interchange of the expressions $\eta ,m_{1},...m_{n},$ but it will be altered by the substitution of $v$ for any one of these expressions, which will be allowable whenever the quantity for which it is substituted has not the value zero in any of the phases to which the formula is to be applied.

The condition formed by setting the expression (182) equal to zero is evidently equivalent to this, that

$\left[{\frac {d\mu _{n}}{d{\tfrac {m_{n}}{v}}}}\right]_{t,\mu _{1},...\mu _{n-1}}=0,$

(183)

that is, that

$\left[{\frac {d{\tfrac {m_{n}}{v}}}{d\mu _{n}}}\right]_{t,\mu _{1},...\mu _{n-1}}=\infty ,$

(183)