Page:Scientific Papers of Josiah Willard Gibbs.djvu/123

then by differentiation and comparision with (86) we obtain

$d\psi =-\eta dt-pdv+\mu _{1}dm_{1}+\mu _{2}dm_{2}...+\mu _{n}dm_{n}.$

(88)

If, then, $\psi$ is known as a function of $t,v,m_{1},m_{2},...m_{n}$ we can find $\eta ,p,\mu _{1},\mu _{2},...\mu _{n}$ in terms of the same variables. If we then substitute for $\psi$ in our original equation its value taken from eq. (87), we shall have again $n+3$ independent relations between the same $2n+5$ variables as before.

then by (86),

$d\chi =td\eta +vdp+\mu _{1}dm_{1}+\mu _{2}dm_{2}...+\mu _{n}dm_{n}.$

(90)

If, then, $\chi$ be known as a function of $\eta ,p,m_{1},m_{2},...m_{n}$, we can find $t,v,\mu _{1},\mu _{2},...\mu _{n}$ in terms of the same variables. By eliminating $\chi$, we may obtain again $n+3$ independent relations between the same $2n+5$ variables as at first.

then, by (86),

$d\zeta =-\eta dt+vdp+\mu _{1}dm_{1}+\mu _{2}dm_{2}...+\mu _{n}dm_{n}.$

(92)

If, then, $\zeta$ is known as a function of $t,p,m_{1},m_{2},...m_{n}$, we can find $\eta ,v,\mu _{1},\mu _{2},...\mu _{n}$ in terms of the same variables. By eliminating $\zeta$, we may obtain again $n+3$ independent relations between the same $2n+5$ variables as at first.

If we integrate (86), supposing the quantity of the compound substance considered to vary from zero to any finite value, its nature and state remaining unchanged, we obtain

$\epsilon =t\eta -pv+\mu _{1}m_{1}+\mu _{2}m_{2}...+\mu _{n}m_{n},$

(93)

and by (87), (89), (91)

$\psi =-pv+\mu _{1}m_{1}+\mu _{2}m_{2}...+\mu _{n}m_{n},$

(94)

$\chi =t\eta +\mu _{1}m_{1}+\mu _{2}m_{2}...+\mu _{n}m_{n},$

(95)

$\zeta =\mu _{1}m_{1}+\mu _{2}m_{2}...+\mu _{n}m_{n},$

(96)

The last three equations may also be obtained directly by integrating (88), (90), and (92).

from a single function, which he calls a characteristic function of the fluid considered. In the papers cited, he introduces two different functions of this kind, viz., a function of the temperature and volume, which he denotes by $\psi$, the value of which in our notation would be ${\frac {-\epsilon +t\eta }{t}}$ or ${\frac {-\psi }{t}}$; and a function of the temperature and pressure, which he denotes by $\psi '$, the value of which in our notation would be ${\frac {-\epsilon +t\eta -pv}{t}}$ or ${\frac {-\zeta }{t}}\cdot$ In both cases he considers a constant quantity (one kilogram) of the fluid, which is regarded as invariable in composition.