EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 107 



we see that the stability of any phase in regard to continuous changes 

 depends upon the same conditions in regard to the second and higher 

 differential coefficients of the density of energy regarded as a function 

 of the density of entropy and the densities of the several components^ 

 which would make the density of energy a minimum,, if the necessary 

 conditions in regard to the first differential coefficients were fulfilled. 

 When n = l, it may be more convenient to regard m as constant 

 in (145) than v. Regarding m a constant, it appears that the stability 

 of a phase depends upon the same conditions in regard to the second 

 and higher differential coefficients of the energy of a unit of mass 

 regarded as a function of its entropy and volume, which would make 

 the energy a minimum, if the necessary conditions in regard to the 

 first differential coefficients were fulfilled. 



The formula (144) expresses the condition of stability for the phase 

 to which t', p, etc., relate. But it is evidently the necessary and 

 sufficient condition of the stability of all phases of certain kinds of 

 matter, or of all phases within given limits, that (144) shall hold true 

 of any two infinitesimally differing phases within the same limits, or, 

 as the case may be, in general. For the purpose, therefore, of such 

 collective determinations of stability, we may neglect the distinction 

 between the two states compared, and write the condition in the form 



... -m n A/z n >0, (148) 



or Ap> -A^ + T i A/u 1 ... -\ -A/* n . (149) 



Comparing (98), 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 limits the same conditions should be 

 fulfilled in respect to the second and higher differential coefficients 

 of the pressure regarded as a function of the temperature and the 

 several potentials, which would make the pressure a minimum, if 

 the necessary conditions with respect to the first differential co- 

 efficients were fulfilled. 



By equations (87) and (94), the condition (142) may be brought to 

 the form 



m n ' > 0. (150) 



For the stability of all phases within any given limits it is necessary 

 and sufficient that within the same limits this condition shall hold 

 true of any two phases which differ infinitely little. This evidently 

 requires that when v' = v", m^ = m^ f , ... m n ' = m 



n , 



(151) 



