168 J. W. Gihbs — Equilibrium of Heterogeneous Substances. 



Then the condition that that quantity shall be constant would create 

 a restriction upon the variations of the phase, and cannot be substi- 

 tuted for the condition that the volume shall be constant in the state- 

 ment of the general condition of stability relative to the minimum 

 value of 0. 



To indicate more distinctly all these particular conditions at once, 

 we observe that the condition (144), and therefore also the condition 

 obtained by interchanging the single and double accents, must hold 

 true of any two infinitesimally difiering phases within the limits of 

 stability. Combining these two conditions we have 



i^t" - t') [rf - rf) - {p" -p') W - ^') 



+ (/^i" - /^i') (^i" - '-'^x) • • ' (/^"" - Z^"') «'-O>0, (170) 



which may be written more briefly 



AtAr] — ApAv-{- Jf^^Am^ . . . +J//„Jm„>0. (IVI) 



This must hold true of any two infinitesimally differing phases within 

 the limits of stability. If, then, we give the value zero to one of the 

 differences in every term except one, but not so as to make the phases 

 completely identical, the values of the two differences in the remain- 

 ing term will have the same sign, except in the case of Ap and Av, 

 which will have opposite signs. (If both states are stable this will 

 hold true even on the limits of stability.) Therefore, within the 

 limits of stability, either of the two quantities occurring (after the 

 sign A) in any term of (IVI) in an increasing function of the other, 

 — except p and v, of which the opposite is true, — when we regard as 

 constant one of the quantities occurring in each of the other terms, 

 but not such as to make the phases identical. 



If we write <^? for A in (166)-(169), we obtain conditions which are 

 always sufficient for stability. If we also substitute ^ for >, we 

 obtain conditions which are necessary for stability. Let us consider 

 the form which these conditions will take when ?/, v, m,, . . . m.„ are 

 regarded as independent variables. When dv = 0, we shall have 



dt dt , dt ^ 



at=i— drj -\- - — dm , . . . + -^ — dm. 

 dt] dm^ dm^ 



d\x.—^-^di]-\-^^dm. . . . + 4^dm„ [ .,^„, 



^* d}] dm^ ^ dm„ \ (172) 



_ d^„ , , dfA^ , c?w_ , 



du„=z -^-dn -{--z — dm, . . . -\--~^dm„ 

 dt] dm^^ ^ dm„ 



