Page:Scientific Papers of Josiah Willard Gibbs.djvu/187

appearing in any other form or combination, but solely as constituting the gas in question (in a state of purity), we may without loss of generality give to $E$ and $H$ the value zero, or any other arbitrary values. But when the scope of our investigations is not thus limited we may have determined the states of the substance of the gas for which $\epsilon =0$ and $\eta =0$ with reference to some other form in which the substance appears, or, if the substance is compound, the states of its components for which $\epsilon =0$ and $\eta =0$ may be already determined; so that the constants $E$ and $H$ cannot in general be treated as arbitrary.

We obtain from (255) by differentiation

${\frac {c}{\epsilon -Em}}d\epsilon ={\frac {1}{m}}d\eta -{\frac {a}{v}}dv+\left({\frac {cE}{\epsilon -Em}}+{\frac {c+a}{m}}-{\frac {\eta }{m^{2}}}\right)dm,$

(256)

whence, in virtue of the general relation expressed by (86),

$t={\frac {\epsilon -Em}{cm}},$

(257)

$p=a{\frac {\epsilon -Em}{cv}},$

(258)

$\mu =E+{\frac {\epsilon -Em}{cm^{2}}}(cm+am-\eta ).$

(259)

We may obtain the fundamental equation between $\psi ,t,v$, and $m$ from equations (87), (255), and (257). Eliminating $\epsilon$ we have

$\psi =Em+cmt-t\eta ,$

and $c\log t={\frac {\eta }{m}}-H+a\log {\frac {m}{v}};$

and eliminating $\eta$, we have the fundamental equation

$\psi =Em+mt\left({\frac {\eta }{m}}-H+a\log {\frac {m}{v}}\right).$

(260)

Differentiating this equation, we obtain

$d\psi =-m\left(H+c\log t+a\log {\frac {v}{m}}\right)dt-{\frac {amt}{v}}dv+\left(E+t\left(c+a-H-c\log t+a\log {\frac {m}{v}}\right)\right)dm;$

(261)

whence, by the general equation (88),

$\eta =m\left(H+c\log t+a\log {\frac {m}{v}}\right),$

(262)

$p={\frac {amt}{v}},$

(263)

$\mu =E+t\left(c+a-H-c\log t+a\log {\frac {m}{v}}\right).$

(264)