164 J. W. Gihhs — Equilibrium of Heterogeneous Substances. 



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 diffi^ring 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 



— 1/ ^t-\-v ^p — m^ J/4^ . . . —m„JjJ„>0, (148) 



or 



Comparing (98), we see that it is necessary and sufficient for the sta- 

 bility 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 sev- 

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

 necessary conditions witb i-espect to the first difierential coefficients 

 were fiilfilled. 



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

 the form 



->-?/■' ~ t' if —p' v' -\- /.ii' m^' . . . -\. ^(J m„'>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' =. d", m^' = iii ^\ . . . in„' = rnj\ 



f ~'/'+{t" -t'),/'>0; (151) 



and that when t' — t" 



f +P' '^" - /< 1 ' >/*i" . • . 4- /'„' mj' 

 - f ~ P' ''' -\- M i "> i' ■ ■ ■ +/'„'/>?„' >U. (152) 



These conditions may be written in the form 



