108 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



and that when t' t" 



Tj,"+p'v"-v l 'm l "...+ f i n 'm n " 



\l/ p'v' -j- ^-iffii ... -f-// w 'm n '> 0. (152) 



These conditions may be written in the form 



(153) 

 <>0, (154) 



in which the subscript letters indicate the quantities which are to 

 be regarded as constant, m standing for all the quantities m x . . . m n . 

 If these conditions hold true within any given limits, (150) will also 

 hold true of any two infinitesimally differing phases within the same 

 limits. To prove this, we will consider a third phase, determined by 

 the equations 



t" = tf, (155) 



and v'" = v", m/" = m/', . . . m n '" = m/. (156) 



Now by (153), \!s'"-\ls"+(t'"-t")ri" <0; (157) 



and by (154), \/r"'+ p'v'" jJ-{m-{" ... Hn m n" 



\l/ p'v' +/* 1 / m 1 / ... -\-ju. n 'm n '>Q. (158) 



Hence, \fr" + 1" q" +p'v f " - /ij'm/" ... - fJL n 'm n '" 



which by v (1^5) and (156) is equivalent to (150). Therefore, the 

 conditions (153) and (154) in respect to the phases within any given 

 limits are necessary and sufficient for the stability of all the phases 

 within those limits. It will be observed that in (153) we have the 

 condition of thermal stability of a body considered as unchange- 

 able in composition and in volume, and in (154), the condition of 

 mechanical and chemical stability of the body considered as main- 

 tained at a constant temperature. Comparing equation (88), we see 



that the condition (153) will be satisfied, if -rr<0, i.e., if -^ or t-A 



(the specific heat for constant volume) is positive. When n = l, i.e., 

 when the composition of the body is invariable, the condition (154) 

 will evidently not be altered, if we regard m as constant, by which 

 the condition will be reduced to 



(160) 



This condition will evidently be satisfied if -r^r>0, i.e., if 



* dv 2 dv 



Off) 



or v-f- (the elasticity for constant temperature) is positive. But 



when n > 1, (154) may be abbreviated more symmetrically by making 

 v constant. 



