THERMODYNAMICAL SYSTEM OF GIBBS 157 



and 



_ ^' _ p'v' + (jiMi ... + fj^n'mn' = 0, 

 (209) becomes 



(rP" - ^') + it" - t')v" + {v" - v')v' - (mi" - m/W 



... - (m„" - mn')nn' > 0. (217) [150] 



As in (213), when all phases within any given limits are stable, 

 this condition holds for any two phases which differ infinitely 

 little. When 



v' = v", mi = nil", . . . lUn = Mn", 



ir - ^') + it" - t'W > 0, (218) [151] 



or 



(^' - r) + {f - t")ri" < 0, (219) 



which may be written 



[^^P + -nM],, ^ < 0. (220) [153] 



Note that the phase, the stability of which is in question here 

 is that to which t]" refers; hence Axp = 4/' — \p". Similarly, 

 when t' = t", 



ir - ^') + V\v" - y') - m/(wi" - m/) 



... - /xn'(w„" - w„') > 0, (221) [152] 



or 



[A^P + pAv - HiAmi ... - HnAmn]t > 0. (222) [154] 



The phase of which the stability is in question is now that 

 distinguished by single accents. 



We may first observe that since, by (45), {d^/dt\rn = — »7> 

 (220) requires that d^rp/dP < 0, i.e., d-q/dt or td-q/dt is positive, 

 tdr}/dt being the specific heat of the phase in question at constant 



