286 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



in which, moreover, the conditions of equilibrium relating to tem- 

 perature and the potentials are satisfied, and the relations expressed 

 by the fundamental equations of the masses and surfaces are to be 

 regarded as satisfied, although the state of the system is not one 

 of complete equilibrium. Let us imagine the state of the system 

 to vary continuously in the course of time in accordance with these 

 conditions and use the symbol d to denote the simultaneous changes 

 which take place at any instant. If we denote the total energy of 

 the system by E, the value of dE may be expanded like that 

 of SE in (599) and (600), and then reduced (since the values of 

 ^> l ui i+9 z > Pz+yZ) e tc., are uniform throughout the system, and the 

 total entropy and total quantities of the several components are 

 constant) to the form 



dE = -fp dDv +fg dz Dm v +fo- dDs +fg dz Dm* 



= -fp dDv+fg y dz Dv+fo- dDs+fg T dz Ds, (621) 



where the integrations relate to the elements expressed by the 

 symbol D. The value of p at any point in any of the various 

 masses, and that of a- at any point in any of the various surfaces 

 of discontinuity are entirely determined by the temperature and 

 potentials at the point considered. If the variations of t and M v 

 M 2 , etc. are to be neglected, the variations of p and or will be 

 determined solely by the change in position of the point considered. 

 Therefore, by (612) and (614), 



dp=gydz, dar=gTdz', 



and ,- . 



dE = -fp dDv -fdp Dv +f<r dDs +fd<r Ds 



= - dfp Dv + dfa- Ds. (622) 



If we now integrate with respect to d, commencing at the given state 

 of the system, we obtain 



AE = - &fp Dv + A/<r Ds, (623) 



where A denotes the value of a quantity in a varied state of the 

 system diminished by its value in the given state. This is true for 

 finite variations, and is therefore true for infinitesimal variations 

 without neglect of the infinitesimals of the higher orders. The con- 

 dition of stability is therefore that 



A/p Dv - A/o- Ds < 0, (624) 



or that the quantity 



fpDv-fcrDs (625) 



has a maximum value, the values of p and cr, for each different mass 

 or surface, being regarded as determined functions of z. (In ordinary 

 cases cr may be regarded as constant in each surface of discontinuity, 

 and p as a linear function of z in each different mass.) It may easily 



