Page:Scientific Papers of Josiah Willard Gibbs.djvu/108

$T(\sum \delta \eta +\sum D\eta )-P(\sum \delta v+\sum Dv)$ from the first member of the general condition of equilibrium (37), $T$ and $P$ being constants of which the value is as yet arbitrary. We might proceed in the same way with the remaining equations of condition, but we may obtain the same result more simply in another way. We will first observe that

$(\sum \delta m_{1}+\sum Dm_{2}){\mathfrak {S}}_{1}+(\sum \delta m_{2}+\sum Dm_{2}){\mathfrak {S}}_{2}...+(\sum \delta m_{n}+\sum Dm_{n}){\mathfrak {S}}_{n}$

(42)

which equation would hold identically for any possible values of the quantities in the parentheses, if for $r$ of the letters ${\mathfrak {S}}_{1},{\mathfrak {S}}_{2},...{\mathfrak {S}}_{n}$ were substituted their values in terms of the others as derived from equations (38). (Although ${\mathfrak {S}}_{1},{\mathfrak {S}}_{2},...{\mathfrak {S}}_{n}$ do not represent abstract quantities, yet the operations necessary for the reduction of linear equations are evidently applicable to equations (38).) Therefore, equation (42) will hold true if for ${\mathfrak {S}}_{1},{\mathfrak {S}}_{2},...{\mathfrak {S}}_{n}$ we substitute $n$ numbers which satisfy equations (38). Let $M_{1},M_{2},...M_{n}$ be such numbers, i.e., let

$a_{1}M_{1}+a_{2}M_{2}+...+a_{n}M_{n}=0,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ $r$ equations. (43)
$b_{1}M_{1}+b_{2}M_{2}+...+b_{n}M_{n}=0,$
etc.

then

$M_{1}(\sum \delta m{1}+\sum Dm_{1})+M_{2}(\sum \delta m{2}+\sum Dm_{2})...+M_{n}(\sum \delta m{n}+\sum Dm_{n})=0.$

(44)

This expression, in which the values of $n-r$ of the constants $M_{1},M_{2},...M_{n}$ are still arbitrary, we will also subtract from the first member of the general condition of equilibrium (37), which will then become

$\sum D\epsilon +\sum (t\delta \eta )-\sum (p\delta v)+\sum (\mu _{1}\delta m_{1})..+\sum (\mu _{n}\delta m_{n})-T\sum \delta \eta +P\sum \delta v-M_{1}\sum \mu _{1}\delta m_{1}...-M_{n}\sum \mu _{n}\delta m_{n}-T\sum D\eta +PDv-M_{1}\sum Dm_{1}...-M_{n}\sum Dm_{n}\geqq 0.$

(45)

That is, having assigned to $T,P,M_{1},M_{2},...M_{n}$ any values consistent with (43), we may assert that it is necessary and sufficient for equilibrium that (45) shall hold true for any variations in the state of the system consistent with the equations of condition (39), (40), (41). But it will always be possible, in case of equilibrium, to assign such values to $T,P,M_{1},M_{2},...M_{n}$, without violating equations (43), that (45) shall hold true for all variations in the state of the system and in the quantities of the various substances composing it, even though these variations are not consistent with the equations of condition (39), (40), (41). For, when it is not possible to do this, it must be possible by applying (45) to variations in the system not necessarily restricted by the equations of condition (39), (40), (41) to