Page:Scientific Papers of Josiah Willard Gibbs.djvu/281

The condition of stability derived from (522) may in this case be written

$r{\frac {d(p'-p'')}{dv'}}<2{\frac {d\sigma }{dv'}}-(p'-p''){\frac {dr}{dv'}},$

(535)

where the quantities relating to the concave side of the surface of tension are distinguished by a single accent.

If both the masses are infinitely large, or if one which contains all the components of the system is infinitely large, $p'-p''$ and $\sigma$ will be constant, and the condition reduces to

${\frac {dr}{dv'}}<0.$

The equilibrium will therefore be stable or unstable according as the surface of tension is less or greater than a hemisphere.

To return to the general problem:—if we denote by $x$ the part of the axis of the circular orifice intercepted between the center of the orifice and the surface of tension, by $R$ the radius of the orifice, and by $V'$ the value of $v'$ when the surface of tension is plane, we shall have the geometrical relations

$R^{2}=2rx-x^{2},$

and	${\begin{aligned}v'&=V'+{\tfrac {2}{3}}\pi r^{3}x-{\tfrac {1}{3}}\pi R^{2}(r-x)\\&=V'+\pi rx^{2}-{\tfrac {1}{3}}\pi x^{3}.\\\end{aligned}}$

By differentiation we obtain

$(r-x)dx+xdr=0,$

and	$dv'=\pi x^{2}dr+(2\pi rx-\pi x^{2})dx;$

whence

$(r-x)dv'=-\pi rx^{2}dr.$

By means of this relation, the condition of stability may be reduced to the form

${\frac {dp'}{dv'}}-{\frac {dp''}{dv'}}-{\frac {2}{r}}{\frac {d\sigma }{dv'}}<(p'-p''){\frac {r-x}{\pi r^{2}x^{2}}}\cdot$

(537)

Let us now suppose that the temperature and all the potentials except one, $\mu _{1}$ are to be regarded as constant. This will be the case when one of the homogeneous masses is very large and contains all the components of the system except one, or when both these masses are very large and there is a single substance at the surface of discontinuity which is not a component of either; also when the whole system contains but a single component, and is exposed to a constant temperature at its surface. Condition (537) will reduce by (98) and (508) to the form

$\left(\gamma _{1}'-\gamma _{1}''+{\frac {2\Gamma _{1}}{r}}\right){\frac {d\mu _{1}}{dv'}}<(p'-p''){\frac {r-x}{\pi r^{2}x^{2}}}\cdot$

(538)