J. W. Gihhs — Equilihrium of Heterogeneous Substances. 165 



[JV'4-//Z/«]„,^<0, (153) 



{Aip+pAv~i.i^Jm^ . . . -/<„Jw„],>0, (154) 



in which the subscript letters indicate the quantities which are to be 

 resjarded as constant, m standing for all the quantities m, . . . m„. 

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

 hold true of any two iniinitesimally differing phases within the same 

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



by the equations 



t"' = t', (155) 



and 



v"' = v", m/" = m,", . . . m„"' = m„". (156) 



Now by (153), 



r'-'/'"+(«"'-«") v"<o; (157) 



and by (154), 



- //'' —p'v' H-/<,'//^i' . . . 4- yU„' //<„'> 0. (158) 



Hghcg 



' r + t" rf+p'v"'-fx,'m,"' . . . -yu„'m„"' 



_^/ ^t'" if -p'v' +j.{^'m,' . . . +jj„'m„'>0, (159) 



which by (155) and (156) is equivalent to (150). Therefore, the con- 

 ditions (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 unchangeable 

 in composition and in volume, and in (154), the condition of mechan- 

 ical and chemical stability of the body considered as maintained at a 

 constant temperature. Comparing equation (88), we see that the 



condition (153) will be satisfied, if -^ <0, i. e., if --^ or #-^ (the spe- 

 cific heat for constant volume) is positive. When n=. 1, 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 



[z/z/'-fjo J4,,„>0. (160) 



d^ lb dp 



This condition will evidently be satisfied if 3-^ ^^^ i- *?-, if -7- or 



_ rf^JL (the elasticity for constant temperature) is positive. But 



dv 

 when 7i> 1, (154) may be abbreviated more symmetrically by making 

 v constant. 



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

 the form 



