Page:Scientific Papers of Josiah Willard Gibbs.djvu/107

any variation in the state of the system are arbitrary, we may so define the parts which we have called original, that we may consider them as initially homogeneous and remaining so, and as initially constituting the whole system.

The most general value of the variation of the energy of the whole system is evidently

$\sum \delta \epsilon +\sum D\epsilon ,$

(35)

the first summation relating to all the original parts, and the second to all the new parts. (Throughout the discussion of this problem, the letter $\delta$ or $D$ following $\sum$ will sufficiently indicate whether the summation relates to the original or to the new parts.) Therefore the general condition of equilibrium is

$\sum \delta \epsilon +\sum D\epsilon \geqq 0,$

(36)

or, if we substitute the value of $\delta \epsilon$ taken from equation (12),

$\sum D\epsilon +\sum (t\delta \eta )-\sum (p\delta v)+\sum (\mu _{1}\delta m_{1})+\sum (\mu _{2}\delta m_{2})...+\sum (\mu _{n}\delta m_{n})\geqq 0.$

(37)

If any of the substances $S_{1},S_{2},...S_{n}$ can be formed out of others, we will suppose, as before (see page 69), that such relations are expressed by equations between the units of the different substances.

Let these be

$a_{1}{\mathfrak {S}}_{1}+a_{1}{\mathfrak {S}}_{2}...+a_{n}{\mathfrak {S}}_{n}=0$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ $r$ equations.(38)
$b_{1}{\mathfrak {S}}_{1}+b_{2}{\mathfrak {S}}_{2}...+b_{n}{\mathfrak {S}}_{n}=0$
etc.

The equations of condition will be (if there is no restriction upon the freedom of motion and composition of the components)

$\sum \delta \eta +\sum D\eta =0,$

(39)

$\sum \delta v+\sum Dv=0,$

(40)

and $n-r$ equations of the form

$h_{1}(\sum \delta m_{1}+\sum Dm_{1})+h_{2}(\sum \delta m_{2}+\sum Dm_{2})...+h_{n}(\sum \delta m_{n}+\sum Dm_{n})=0$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ (41)[1]
$i_{1}(\sum \delta m_{1}+\sum Dm_{1})+i_{2}(\sum \delta m_{2}+\sum Dm_{2})...+i_{n}(\sum \delta m_{n}+\sum Dm_{n})=0$
etc.

Now, using Lagrange's "method of multipliers,"^[2] we will subtract

1. In regard to the relation between the coefficients in (41) and those in (38), the reader will easily convince himself that the coefficients of any one of equations (41) are such as would satisfy all the equations (38) if substituted for $S_{1},S_{2},...S_{n}$; and that this is the only condition which these coefficients must satisfy, except that the $n-r$ sets of coefficients shall be independent, i.e., shall be such as to form independent equations; and that this relation between the coefficients of the two sets of equations is a reciprocal one.
2. On account of the sign $\geqq$ in (37), and because some of the variations are incapable of negative values, the successive steps in the reasoning will be developed at greater length than would be otherwise necessary.