THERMODYNAMICAL SYSTEM OF GIBBS 161 



occiirring tn each of the other terms, but not such as to make the 

 phases identical." 



It is evident that when v is taken as constant, there are a 

 number of ways in which one of the quantities in each of n of the 

 remaining n -\- 1 terms can be made zero. We can thus obtain 

 different sets of n + 1 conditions, Hke (229) and (230). Gibbs 

 points out that it is not always possible to substitute the con- 

 dition that the pressure shall be constant for the condition that 

 the volume shall be constant, without imposing a restriction on 

 the variations of the phase. 



It may be pointed out with regard to the equations (229), 

 (230), that if the sign A is replaced by d we obtain conditions 

 which are sufficient for stability. 

 It is evident that if 



the condition 



\dmn/i. V. ^„ 



/A/xA 



\AmnJt, V, w. . . 



> 0, (232) 



Mn — 1 



> (233) 



Mn— 1 



must also hold true, i.e., the condition of stabihty is satisfied. 

 But (233) may also hold true if 



= (234) 



' Mn — 1 



(when one or more of the higher differential coefficients are 

 positive). The expression (233) cannot hold true when the 

 differential coefficient term (232) is negative, so that it is 

 necessary for stability that 



^ 0. (235) 



lin—i. 



34. Limits of Stability. At the limits of stability (i.e., the 

 limits which divide stable from unstable phases) with respect to 

 continuous changes, one of the conditions (229), (230) must 



