EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 109 



Again, by (91) and (96), the condition (142) may be brought to 

 the form 



-f -f^+pV+ftX' . + M.X'>0. (161) 



Therefore, for the stability of all phases within any given limits it is 

 necessary and sufficient that within the same limits 



[AM-*A*-vAp] w <0, (162) 



and [Af-ftAm,... -// n Am n ] fil> >0, (163) 



as may easily be proved by the method used with (153) and (154). 

 The first of these formulae expresses the thermal and mechanical 

 conditions of stability for a body considered as unchangeable in 

 composition, and the second the conditions of chemical stability for 

 a body considered as maintained at a constant temperature and 

 pressure. If n = l, the second condition falls away, and as in this 

 case f =mfJL, condition (162) becomes identical with (148). 



The foregoing discussion will serve to illustrate the relation of the 

 general condition of stability in regard to continuous changes to 

 some of the principal forms of fundamental equations. It is evident 

 that each of the conditions (146), (149), (154), (162), (163) involves 

 in general several particular conditions of stability. We will now 

 give our attention to the latter. Let 



$ = - t'l\ +p'v - yM^i ... - t* n ' m n> ( 164 ) 



the accented letters referring to one phase and the unaccented to 

 another. It is by (142) the necessary and sufficient condition of the 

 stability of the first phase that, for constant values of the quantities 

 relating to that phase and of v, the value of <3? shall be a minimum 

 when the second phase is identical with the first. Differentiating 

 (164), we have by (86) 



d3> = (t-t')dn-(p-p')dv + ( f ji l -H l ')dm 1 ... +(fjL n -fjL n ')dm n . (165) 



Therefore, the above condition requires that if we regard v, m 1? . . . m n 

 as having the constant values indicated by accenting these letters, 

 t shall be an increasing function of q, when the variable phase differs 

 sufficiently little from the fixed. But as the fixed phase may be any 

 one within the limits of stability, t must be an increasing function 

 of i\ (within these limits) for any constant values of v , m^ . . . m n . 

 This condition may be written 



(} >0. (166) 



\{\1]/ Vf mi} ... mn 



When this condition is satisfied, the value of 4>, for any given values 

 of v, m v . . . m n , will be a minimum when t = t'. And therefore, in 

 applying the general condition of stability relating to the value of 

 3>, we need only consider the phases for which t = tf. 



