452 J. Wl Gibbs — Equilibrium of Heterogeneous Siibstances. 



whole system or to considerable parts of it. To determine the ques- 

 tion of the stability of a given fluid system under the influence of 

 gravity, when all the (!onditions of equilibrium are satisfied as well 

 as those conditions of stability which relate to small parts of the sys- 

 tem taken separately, we may use the method described on page 

 413, the demonstration of which (pages 411, 412) will not require 

 any essential modification on account of gravity. 



When the variations of temperature and of the quantities J/j, J/o, 

 etc. [see (617)] involved in the changes considered are so small that 

 they may be neglected, the condition of stability takes a very simple 

 form, as we have already seen to be the case with respect to a sys- 

 tem uninfluenced by gravity. (See page 415.) 



We have to consider a varied state of the system in which the 

 total entropy and the total quantities of the various components are 

 unchanged, and all variations vanish at the exterior of the system, — 

 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 condi- 

 tions 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 t, /.i^+gz, 

 l^2~\'9 "^1 ^^^- '"^^'^ uniform throughout the system, and the total entropy 

 and total quantities of the several components are constant) to the 

 form 



dE——j'p dBv -\-J'g dz Ihn'''+fo- dDs -\-fg dz Dm^ 



= —fP <ll^'^ + yV/ ydzDv+fa dDs -\-fg V dz Ds, (621) 



where the integrations relate to the elements expressed by the symbol 

 D. The value of ^; at any point in any of the various masses, and 

 that of o' 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 ^, il/g, etc. are to be 

 neglected, the variations of p and G will be determined solely by the 

 change in position of the point considered. Therefore, by (612) and 

 (614), 



dp-=. — g y/ dz, da =: g I dz ; 

 and 



