J. W. Gibbs — Equilibrium of Heterogeneous Srdistances. 10] 



tage ill the use of ;;, y, ^jj/n,, . . . m„ as independent variables. The 

 condition that the phase may be vai'ied without altering any of the 

 quantities t, //,, //.,, ...//„ will then be expressed by the equation 



i?„+i=0, (203) 



in which /i„^^ denotes the same determinant as on page 169. To 

 obtain the second equation characteristic of critical phases, we observe 

 that as a phase which is critical cannot become unstable when \aried 

 so that n of the quantities ^, jt), /<j, //g? • • • /'« remain constant, the 

 differentia] of ^n^., for constant volume, viz., 



^-^»+l^„-i- ^^-^"+l dm . -U ^J^ dm„ 



dtn„ 



+ 



(204) 



d// dm I 



cannot become negative when n of the equations (1V2) are satisfied. 

 Neither can it have a positive value, for then its value might become 

 negative by a change of sign of d?/, dm^, etc. Therefore the expres- 

 sion (204) has the value zero, if w of the equations (172) are satisfied. 

 This may be expressed by an equation 



aS=0, (205) 



in which S denotes a determinant in which the constituents are the 

 same as in ^„+i, except in a single horizontal line, in which the 

 differential coefficients in (204) are to be substituted. In whatever 

 line this substitution is made, the equation (205), as well as (203), 

 will hold true of every critical phase without exception. 



If we choose t, p, m^, m^, . . . m„ ?is independent variables, and 

 write V for the determinant 



d^i dH dn 



(206) 



and V for the determinant formed from this by substituting for the 

 constituents in any horizontal line the expressions 



IE, i^, . . . i^ (20V) 



the equations of critical phases will be 



Z7= 0, V— 0. (208) 



It results immediately from the definition of a critical phase, that 



an infinitesimal change in the condition of a mass in such a phase 



