Page:Scientific Papers of Josiah Willard Gibbs.djvu/184

148 so small that the force of gravity can be regarded as constant in direction and in intensity, we may use $$\Upsilon$$ to denote the potential of the force of gravity, and express the variation of the part of the energy which is due to gravity in the form We shall then have, for the general condition of equilibrium,  and the equations of condition will be

We may obtain a condition of equilibrium independent of these equations of condition, by subtracting these equations, multiplied each by an indeterminate constant, from condition (239). If we denote these indeterminate constants by $$T, M_{1},...M_{n}$$, we shall obtain after arranging the terms The variations, both infinitesimal and finite, in this condition are independent of the equations of condition (240) and (241), and are only subject to the condition that the varied values of $$D\epsilon, D\eta, Dm_{1},...Dm_{n}$$ for each element are determined by a certain change of phase. But as we do not suppose the same element to experience both a finite and an infinitesimal change of phase, we must have  By equation (12), and in virtue of the necessary relation (222), the first of these conditions reduces to  for which it is necessary and sufficient that