Page:Scientific Papers of Josiah Willard Gibbs.djvu/105

of others, all such relations can be expressed by equations such as

$\alpha {\mathfrak {S}}_{a}+\beta {\mathfrak {S}}_{b}+{\text{etc.}}=\kappa {\mathfrak {S}}_{k}+\gamma {\mathfrak {S}}_{l}+{\text{etc.}}$

(30)

where ${\mathfrak {S}}_{a},{\mathfrak {S}}_{b},{\mathfrak {S}}_{k}$ etc. denote the units of the substances $S_{a},S_{b},S_{k}$, etc., (that is, of certain of the substances $S_{1},S_{2},S_{3}$,) and $\alpha ,\beta ,\kappa$, etc. denote numbers. These are not, it will be observed, equations between abstract quantities, but the sign $=$ denotes qualitative as well as quantitative equivalence. We will suppose that there are $r$ independent equations of this character. The equations of condition relating to the component substances may easily be derived from these equations, but it will not be necessary to consider them particularly. It is evident that they will be satisfied by any values of the variations which satisfy equations (18); hence, the particular conditions of equilibrium (19), (20), (22) must be necessary in this case, and, if these are satisfied, the general equation of equilibrium (15) or (23) will reduce to

$M_{1}\sum \delta m_{1}+M_{2}\sum \delta m_{2}...+M_{n}\sum \delta m_{n}\geqq 0.$

(31)

This will appear from the same considerations which were used in regard to equations (23) and (27). Now it is evidently possible to give to $\sum \delta m_{a}$, $\sum \delta m_{b}$, $\sum \delta m_{k}$, etc. values proportional to $\alpha$, $\beta$, $\kappa$, etc. in equation (30), and also the same values taken negatively, making $\sum \delta m=0$ in each of the other terms; therefore

$\alpha M_{a}+\beta M_{b}+etc...-\kappa M_{k}-\lambda M_{l}-etc.=0,$

(32)

${\text{or}}\,\,\alpha M_{a}+\beta M_{b}+etc=\kappa M_{k}+\lambda M_{l}+etc.=0.$

(33)

It will be observed that this equation has the same form and coefficients as equation (30), $M$ taking the place of ${\mathfrak {S}}$. It is evident that there must be a similar condition of equilibrium for every one of the $r$ equations of which (30) is an example, which may be obtained simply by changing in these equations into $M$. When these conditions are satisfied, (31) will be satisfied with any possible values of $\sum \delta m_{1},\sum \delta m_{2},...\sum \delta m_{n}$. For no values of these quantities are possible, except such that the equation

$\left(\sum \delta m_{1}\right){\mathfrak {S}}_{1}+\left(\sum \delta m_{2}\right){\mathfrak {S}}_{2}...+\left(\sum \delta m_{n}\right){\mathfrak {S}}_{n}=0$

(34)

after the substitution of these values, can be derived from the $r$ equations like (30), by the ordinary processes of the reduction of linear equations. Therefore, on account of the correspondence between (31) and (34), and between the r equations like (33) and the $r$ equations like (30), the conditions obtained by giving any possible values to the variations in (31) may also be derived from the r equations like (33); that is, the condition (31) is satisfied if the r equations like (33) are satisfied. Therefore the $r$ equations like (33) are with (19), (20), and (22) the equivalent of the general condition (15) or (23).