Page:Scientific Papers of Josiah Willard Gibbs.djvu/258

222 If we substitute $$\geqq$$ for $$=$$ in this equation, the formula will hold true of all variations whether reversible or not; for if the variation of energy could have a value less than that of the second member of the equation, there must be variation in the condition of $$M$$ in which its energy is diminished without change of its entropy or of the quantities of its various components.

It is important, however, to observe that for any given values of $$\delta \eta, \delta m_{1}, \delta m_{2}$$, etc., while there may be possible variations of the nature and state of $$M$$ for which the value of $$\delta \epsilon$$ is greater than that of the second member of (477), there must always be possible variations for which the value of $$\delta \epsilon$$ is equal to that of the second member. It will be convenient to have a notation which will enable us to express this by an equation. Let $$\mathfrak{d} \epsilon$$ denote the smallest value (i.e., the value nearest to $$\infty$$) of $$\delta \epsilon$$ consistent with given values of the other variations, then For the internal equilibrium of the whole mass which consists of the parts $$M, M', M''$$, it is necessary that  for all variations which do not affect the enclosing surface or the total entropy or the total quantity of any of the various components. If we also regard the surfaces separating $$M, M'$$, and $$M$$ as invariable, we may derive from this condition, by equations (478) and (12), the following as a necessary'' condition of equilibrium:—