Page:Scientific Papers of Josiah Willard Gibbs.djvu/394

then, by (5),

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

(8)

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.^[1]

If we integrate (5), (6) and (8), supposing the quantity of the compound substance considered to vary from zero to any finite value, its nature and state remaining unchanged, we obtain

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

(9)

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

(10)

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

(11)

If we differentiate (9) in the most general manner, and compare the result with (5), we obtain

$-vdp+\eta dt+m_{1}d\mu _{1}+m_{2}d\mu _{2}...+m_{n}d\mu _{n}=0,$

(12)

or

$dp={\frac {\eta }{v}}dt+{\frac {m_{1}}{v}}d\mu _{1}+{\frac {m_{2}}{v}}d\mu _{2}...+{\frac {m_{n}}{v}}d\mu _{n}=0.$

(13)

Hence, there is a relation between the $n+2$ quantities $t,p,\mu _{1},\mu _{2},...\mu _{n}$, which, if known, will enable us to find in terms of these quantities all the ratios of the $n+2$ quantities $\eta ,v,m_{1},m_{2},...m_{n}$. With (9), this will make $n+3$ independent relations between the same $2n+5$ variables as at first.

Any equation, therefore, between the quantities

	$\epsilon ,$	$\eta ,$	$v,$	$p,$	$m_{1},$	$m_{2},...m_{n}$
or	$\psi ,$	$t,$	$v,$	$p,$	$m_{1},$	$m_{2},...m_{n}$
or	$\zeta ,$	$t,$	$p,$	$p,$	$m_{1},$	$m_{2},...m_{n}$
or		$t,$	$p,$	$\mu _{1}$	$\mu _{2},...\mu _{n},$

is a fundamental equation, and any such is entirely equivalent to any other.

Coexistent phases.—In considering the different homogeneous bodies which can be formed out of any set of component substances, it is convenient to have a term which shall refer solely to the composition

1. The properties of the quantities $-\psi$ and $-\zeta$ regarded as functions of the temperature and volume, and temperature and pressure, respectively, the composition of the body being regarded as invariable, have been discussed by M. Massieu in a memoir entitled "Sur les fonctions caractéristiques des divers fluides et sur la théorie des vapours" (Mém. Savants Étrang., t. xxii). A brief sketch of his method in a form slightly different from that ultimately adopted is given in Comptes Rendus, t. lxix (1869), pp. 868 and 1057, and a report on his memoir by M. Bertrand in Comptes Rendus, t. lxxi, p. 257. M. Massieu appears to have been the first to solve the problem of representing all the properties of a body of invariable composition which are concerned in reversible processes by means of a single function.