156 BUTLER ART. D 



31. Condition with Respect to the Variation of the Pressure. 

 Substituting the value 



e = t ri — p V +/xiOTi .,.-t-/x„ ntn 



in (209), we obtain 



- v"{t' - t") + v"{v' - V") - m,"{y.,' - Ml") 



- W(m2' - M2") ... > 0. (213) [144] 



This formula expresses the condition of stability for the phase to 

 which t', p', etc. relate. But if all phases (within any given 

 hmits) are stable, (213) will hold for any two infinitesimally 

 differing phases (within the same hmits) and the phase (") 

 may be regarded as the phase of which the stabiUty is in ques- 

 tion, and (') as the infinitestimal variant of it. Then (213) can 

 be written 



- r]At + vAp - miA/ii ... - m,Apin > 0, (214) [148] 



or 



Ap > ^ Ai + -^ Ami . . . + - AMn. (215) [149] 



V V V 



But by (56) 



dp= ^dt-\- '-^ dfJi,... + "^ d^n, (216) 



V V V 



so that "we see that it is necessary and sufficient for the stability 

 in regard to continuous changes of all the phases within any 

 given limits, that within those hmits the same conditions should 

 be fulfilled in respect to the second and higher differential 

 coefficients of the pressure regarded as a function of the tem- 

 perature and the several potentials, which would make the 

 pressure a minimum, if the necessary conditions with regard 

 to the first differential coefficients were fulfilled." 



32. Conditions oj Stability in Terms of the Functions \p and T- 

 Writing 



e" = lA" + t'W. 



