J. W. Gibbs — EqulUbrituti of Heterogeneous tmbstances. 1G3 



the other phase by double accents. This condition is by (93) equiva- 

 lent to 



5" _ t' if +p' v" -II,' )>i," ... — //„' m„" 



— f' + «'?/—;/«' + /<, '/>i/ . . . -!-//„' w„'>0, (143) 



and to 



^t'ff+pv"-,i,'m," . . . -//:»?„" 

 4. t" if - if v" + 1.1 ,"m^" . . . + Mn" mj' > 0. (144) 



The condition (143) may be expressed more briefly in the form 



z/f> ^ J/; — ^>z/ti + /<, z/?Hj . . . -\-/.4„Jm„, (145) 



if we use the character J to signify that the condition, although 

 relating to infinitesimal differences, is not to be interpreted in accord- 

 ance with the usual convention in respect to differential equations 

 with neglect of infinitesimals of higher orders than the first, but is 

 to be interpreted strictly, like an equation between finite differences. 

 In fact, when a condition like (145) (interpreted strictly) is satisfied 

 for infinitesimal diffei'ences, it must be possible to assign limits within 

 which it shall hold true of finite differences. But it is to be remem- 

 bered that the condition is not to be applied to any arbitrary values 

 of Jyj, z/u, Zlm,, . . . Jnin, but only to such as are determined by a 

 change of phase. (If only the quantity of the body which determines 

 the value of the variables should vary and not its phase, the value of 

 the first member of (145) would evidently be zero.) We may free 

 ourselves from this limitation by making v constant, which will 

 cause the term — p Av to disappear. If we then divide by the con- 

 stant V, the condition will become 



in which form it will not be necessary to regard v as constant. As 

 we may obtain from (86) 



V V V V 



we see that the stability of any phase in regard to continuous changes 

 depends ujion the same conditions in regard to the second and higher 

 differential coefficients of the density of energy regarded as a function 

 of the density of entropy and the densities of the several components^ 

 which would make the density of energy a minimum, if the necessary 

 conditions in regard to the first differential coefficients were fulfilled. 

 When //= 1, it may be more convenient to regard m as constant 



