144 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



We shall hereafter, in the discussion of the fundamental equations 

 of gases, have an example of the derivation of the fundamental 

 equation for phases of dissipated energy (with respect to the mole- 

 cular changes on which the proximate composition of the body 

 depends) from the more general form of the fundamental equation. 



The Conditions of Equilibrium for Heterogeneous Masses under 



the Influence of Gravity. 



Let us now seek the conditions of equilibrium for a mass of various 

 kinds of matter subject to the influence of gravity. It will be con- 

 venient to suppose the mass enclosed in an immovable envelop which 

 is impermeable to matter and to heat, and in other respects, except 

 in regard to gravity, to make the same suppositions as on page 62. 

 The energy of the mass will now consist of two parts, one of which 

 depends upon its intrinsic nature and state, and the other upon its 

 position in space. Let Dm denote an element of the mass, De the 

 intrinsic energy of this element, h its height above a fixed horizontal 

 plane, and g the force of gravity ; then the total energy of the mass 

 (when without sensible motions) will be expressed by the formula 



fDe+fghDm, (219) 



in which the integrations include all the elements of the mass ; and 

 the general condition of equilibrium will be 



SfDe + Sfgh Dm ^ 0, (220) 



the variations being subject to certain equations of condition. < These 

 must express that the entropy of the whole mass is constant, that 

 the surface bounding the whole mass is fixed, and that the total 

 quantity of each of the component substances is constant. We shall 

 suppose that there are no other equations of condition, and that 

 the independently variable components are the same throughout the 

 whole mass ; and we shall at first limit ourselves to the consideration 

 of the conditions of equilibrium with respect to the changes which 

 may be expressed by infinitesimal variations of the quantities which 

 define the initial state of the mass, without regarding the possibility 

 of the formation at any place of infinitesimal masses entirely different 

 from any initially existing in the same vicinity. 



Let Dq, Dv, Dm l ,...Dm n denote the entropy of the element Dm, 

 its volume, and the quantities which it contains of the various com- 

 ponents. Then 



Dm = Dm l ... +Dm n , (221) 



and SDm = SDm l ... +SDm n . (222) 



Also, by equation (12), 



(223) 



