Page:Scientific Papers of Josiah Willard Gibbs.djvu/236

constant, or when the pressures on all sides are normal and equal, vanishes only when the density of the fluid is equal to that of the solid.

The case is nearly the same when the fluid is not identical in substance with the solid, if we suppose the composition of the fluid to remain unchanged. We have necessarily with respect to the fluid

$d\mu _{1}=\left({\frac {d\mu _{1}}{dt}}\right)_{p,m}^{(F)}dt+\left({\frac {d\mu _{1}}{dp}}\right)_{t,m}^{(F)}dp,$[1]

(408)

where the index (f) is used to indicate that the expression to which it is affixed relates to the fluid. But by equation (92)

$\left({\frac {d\mu _{1}}{dt}}\right)_{p,m}^{(F)}=-\left({\frac {d\eta }{dm_{1}}}\right)_{t,p,m}^{(F)}$,and$\left({\frac {d\mu _{1}}{dp}}\right)_{t,m}^{(F)}=\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(F)}\cdot$

(409)

Substituting these values in the preceding equation, transposing terms, and multiplying by $m$, we obtain

$m\left({\frac {d\eta }{dm_{1}}}\right)_{t,p,m}^{(F)}-m\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(F)}dp+md\mu _{1}=0.$

(410)

By subtracting this equation from (404) we may obtain an equation similar to (406), except that in place of $\eta _{F}$ and $v_{F}$ we shall have the expressions

$m\left({\frac {d\eta }{dm_{1}}}\right)_{t,p,m}^{(F)}$and$m\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}^{(F)}\cdot$

The discussion of equation (406) will therefore apply mutatis mutandis to this case.

We may also wish to find the variations in the composition of the fluid which will be necessary for equilibrium when the pressure $p$ or the quantities ${\frac {dx}{dx'}},{\frac {dx}{dy'}},{\frac {dy}{dy'}}$ are varied, the temperature remaining constant. If we know the value for the fluid of the quantity represented by $\xi$ on page 87 in terms of $t,p$, and the quantities of the several components $m_{1},m_{2},m_{3}$, etc., the first of which relates to the substance of which the solid is formed, we can easily find the value of $\mu _{1}$ in terms of the same variables. Now in considering variations in the composition of the fluid, it will be sufficient if we make all but one of the components variable. We may therefore give to $m_{1}$ a constant value, and making $t$ also constant, we shall have

$d\mu _{1}=\left({\frac {d\mu _{1}}{dp}}\right)_{t,m}^{(F)}dp+\left({\frac {d\mu _{1}}{dm_{2}}}\right)_{t,p,m}^{(F)}dm_{2}+\left({\frac {d\mu _{1}}{dm_{3}}}\right)_{t,p,m}^{(F)}dm_{3}+{\text{etc.}}$

1. A suffixed $m$ stands here, as elsewhere in this paper, for all the symbols $m_{1},m_{2}$, etc., except such as may occur in the differential coefficient.