Page:Scientific Papers of Josiah Willard Gibbs.djvu/359

To take account of the influence of gravity, we must give to $\mu _{1}''$ and $p''$ in (665) their average values in the side considered. These coincide (when the fluid is in a state of internal equilibrium) with their values at the center of gravity of the side. The values of $\gamma _{1}',\epsilon _{V}',\eta _{V}'$, may be regarded as constant, so far as the influence of gravity is concerned. Now since by (612) and (617)

$dp''=-g\gamma ''dz,$

and

$d\mu _{1}''=-gdz,$

we have

$d(\gamma _{1}'\mu _{1}''-p'')=g(\gamma ''-\gamma _{1}')dz.$

Comparing (664), we see that the upper or the lower faces of the crystal will have the greater tendency to grow (other things being equal), according as the crystal is lighter or heavier than the fluid. When the densities of the two masses are equal, the effect of gravity on the form of the crystal may be neglected.

In the preceding paragraph the fluid is regarded as in a state of internal equilibrium. If we suppose the composition and temperature of the fluid to be uniform, the condition which will make the effect of gravity vanish will be that

${\frac {d(\gamma _{1}'\mu _{1}''-p'')}{dz}}=0,$

when the value of the differential coefficient is determined in accordance with this supposition. This condition reduces to

$\left({\frac {d\mu _{1}''}{dp}}\right)_{t,m}''={\frac {1}{\gamma _{1}'}},$[1]

which, by equation (92), is equivalent to

$\left({\frac {dv}{dm_{1}}}\right)_{t,p,m}''={\frac {1}{\gamma _{1}'}}\cdot$

(669)

The tendency of a crystal to grow will be greater in the upper or lower parts of the fluid, according as the growth of a crystal at constant temperature and pressure will produce expansion or contraction.

Again, we may suppose the composition of the fluid and its entropy per unit of mass to be uniform. The temperature will then vary with the pressure, that is, with $z$. We may also suppose the temperature of different crystals or different parts of the same crystal to be determined by the fluid in contact with them. These conditions express a state which may perhaps be realized when the fluid is gently stirred. Owing to the differences of temperature we cannot regard $\epsilon _{V}'$ and $\eta _{V}'$

1. A suffixed $m$ is used to represent all the symbols $m_{1},m_{2}$, etc., except such as may occur in the differential coefficient.