Page:Scientific Papers of Josiah Willard Gibbs.djvu/290

the single and double accents referring respectively to the interior and exterior masses. If we write $[\epsilon ],[\eta ],[m_{1}],[m_{2}]$, etc., for the excess of the total energy, entropy, etc., in and about the globular mass above what would be in the same space if it were uniformly filled with matter of the phase of the exterior mass, we shall have necessarily with reference to the whole dividing surface

$\epsilon ^{S}=[\epsilon ]-v'(\epsilon _{V'}-\epsilon _{V''}),$	$\eta ^{S}=[\eta ]-v'(\eta _{V'}-\eta _{V''}),$
$m_{1}^{S}=[m_{1}]-v'(\gamma _{1}'-\gamma _{1}''),$	$m_{2}^{S}=[m_{2}]-v'(\gamma _{2}'-\gamma _{2}''),$	etc.,

where $\epsilon _{V'},\epsilon _{V''},\eta _{V'},\eta _{V''},\gamma _{1}',\gamma _{1}''$, etc. denote, in accordance with our usage elsewhere, the volume-densities of energy, of entropy, and of the various components, in the two homogeneous masses. We may thus obtain from equation (502)

${\begin{aligned}\sigma s=[\epsilon ]&-v'(\epsilon _{V'}-\epsilon _{V''})-t[\eta ]+tv'(\eta _{V'}-\eta _{V''})\\&-\mu _{1}[m_{1}]+\mu _{1}v'(\gamma _{1}'-\gamma _{1}'')-\mu _{2}[m_{2}]+\mu _{2}v'(\gamma _{2}'-\gamma _{2}'')-{\text{etc.}}\\\end{aligned}}$

(551)

But by (93),

$p'=-\epsilon _{V'}+t\eta _{V'}+\mu _{1}\gamma _{1}'+\mu _{2}\gamma _{2}'+{\text{etc.}}$ $p''=-\epsilon _{V''}+t\eta _{V''}+\mu _{1}\gamma _{1}''+\mu _{2}\gamma _{2}''+{\text{etc.}}$

Let us also write for brevity

$W=[\epsilon ]-t[\eta ]-\mu _{1}[m_{1}]-\mu _{2}[m_{2}]-{\text{etc.}}$

(552)

(It will be observed that the value of $W$ is entirely determined by the nature of the physical system considered, and that the notion of the dividing surface does not in any way enter into its definition.) We shall then have

$\sigma s=W+v'(p'-p''),$

(553)

or, substituting for $s$ and $v'$ their values in terms of $r$,

$4\pi ^{2}\sigma =W+{\tfrac {4}{3}}\pi r^{3}(p'-p''),$

(554)

and eliminating $\sigma$ by (550),

${\tfrac {2}{3}}\pi r^{3}(p'-p'')=W,$

(555)

$r=\left({\frac {3W}{2\pi (p'-p'')}}\right)^{\tfrac {1}{3}}\cdot$

(556)

If we eliminate $r$ instead of $\sigma$, we have

${\frac {16\pi \sigma ^{3}}{3(p'-p'')^{2}}}=W,$

(557)

$\sigma =\left({\frac {3W(p'-p'')^{2}}{16\pi }}\right)^{\tfrac {1}{3}}\cdot$

(558)

Now, if we first suppose the difference of the pressures in the homogeneous masses to be very small, so that the surface of discontinuity is nearly plane, since without any important loss of generality we