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

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

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

Now, using Lagrange's "method of multipliers," we will subtract