Page:Scientific Papers of Josiah Willard Gibbs.djvu/176

140 the variations of which the body is capable. Such cases present no especial difficulty; there is indeed nothing in the physical and chemical properties of such bodies, so far as a certain range of experiments is concerned, which is different from what might be, if the proximate components were incapable of farther reduction or transformation. Yet among the various phases of the kinds of matter concerned, represented by the different sets of values of the variables which satisfy the fundamental equation, there is a certain class which merits especial attention. These are the phases for which the entropy has a maximum value for the same matter, as determined by the ultimate analysis of the body, with the same energy and volume. To fix our ideas let us call the proximate components $$S_{1},... S_{n}$$ and the ultimate components $$S_{a},... S_{h}$$ and let $$m_{1},... m_{n}$$ denote the quantities of the former, and $$m_{a},... m_{h}$$ the quantities of the latter. It is evident that $$m_{a},... m_{h}$$ are homogeneous functions of the first degree of $$m_{1},... m_{n};$$ and that the relations between the substances $$S_{1},... S_{n}$$ might be expressed by homogeneous equations of the first degree between the units of these substances, equal in number to the difference of the numbers of the proximate and of the ultimate components. The phases in question are those for which $$\eta$$ is a maximum for constant values of $$\epsilon, v, m_{a},... m_{h}$$; or, as they may also be described, those for which $$\epsilon$$ is a minimum for constant values of $$\eta, v, m_{a},... S_{h};$$ or for which $$\zeta$$ is a minimum for constant values of $$t, p, m_{a},... m_{h}.$$ The phases which satisfy this condition may be readily determined when the fundamental equation (which will contain the quantities $$m_{1},... m_{n}$$ or $$\mu_{1},... \mu_{n},$$) is known. Indeed it is easy to see that we may express the conditions which determine these phases by substituting $$\mu_{1},... \mu_{n}$$ for the letters denoting the units of the corresponding substances in the equations which express the equivalence in ultimate analysis between these units.

These phases may be called, with reference to the kind of change which we are considering, phases of dissipated energy. That we have used a similar term before, with reference to a different kind of changes, yet in a sense entirely analogous, need not create confusion.

It is characteristic of these phases that we cannot alter the values of $$m_{1},... m_{n}$$ in any real mass in such a phase, while the volume of the mass as well as its matter remain unchanged, without diminishing the energy or increasing the entropy of some other system. Hence, if the mass is large, its equilibrium can be but slightly disturbed