Page:Scientific Papers of Josiah Willard Gibbs.djvu/104

will remain unchanged. Now as all values of the variations which satisfy equations (24) will also satisfy equations (25), it is evident that all the particular conditions of equilibrium which we have already deduced, (19), (20), (22), are necessary in this case also. When these are satisfied, the general condition (23) reduces to

$M_{1}\sum \delta m_{1}+M_{2}\sum \delta m_{2}+M_{3}\sum \delta m_{3}\geqq 0.$

(27)

For, although it may be that $\mu _{1}'$, for example, is greater than $M_{1}$ yet it can only be so when the following $\delta m'_{1}$ is incapable of a negative value. Hence, if (27) is satisfied, (23) must also be. Again, if (23) is satisfied, (27) must also be satisfied, so long as the variation of the quantity of every substance has the value in all the parts of which it is not an actual component. But as this limitation does not affect the range of the possible values of $\sum \delta m_{1}$, $\sum \delta m_{2}$, and $\sum \delta m_{3}$, it may be disregarded. Therefore the conditions (23) and (27) are entirely equivalent, when (19), (20), (22) are satisfied. Now, by means of the equations of condition (25), we may eliminate $\sum \delta m_{2}$ and $\sum \delta m_{3}$ from (27), which becomes

$-aM_{1}\sum \delta m_{3}+(a+b)M_{3}\delta m_{3}\geqq 0,$

(28)

i.e., as the value of $\sum \delta m_{3}$ may be either positive or negative,

$aM_{1}+bM_{2}=(a+b)M_{3},$

(29)

which is the additional condition of equilibrium which is necessary in this case.

The relations between the component substances may be less simple than in this case, but in any case they will only affect the equations of condition, and these may always be found without difficulty, and will enable us to eliminate from the general condition of equilibrium as many variations as there are equations of condition, after which the coefficients of the remaining variations may be set equal to zero, except the coefficients of variations which are incapable of negative values, which coefficients must be equal to or greater than zero. It will be easy to perform these operations in each particular case, but it may be interesting to see the form of the resultant equations in general.

We will suppose that the various homogeneous parts are considered as having in all n components, $S_{1},S_{2},...S_{n}$, and that there is no restriction upon their freedom of motion and combination. But we will so far limit the generality of the problem as to suppose that each of these components is an actual component of some part of the given mass.^[1] If some of these components can be formed out

1. When we come to seek the conditions of equilibrium relating to the formation of masses unlike any previously existing, we shall take up de novo the whole problem of the equilibrium of heterogeneous masses enclosed in a non-conducting envelop, and give it a more general treatment, which will be free from this limitation.