204 ./. W. Gibbs — Equilihrmm of Heterogeneous Substances. 



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

 its position in space. Let Dtn denote an element of the mass, Ds the 

 intrinsic energy of this element, h its height aboA'e 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 



fI)e-\-fghDm., (219) 



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

 the general condition of equilibrium will be 



dfBe + 6fg h 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 quanti- 

 ties of each of the component substances is constant. We shall sup- 

 pose that there are no otlier 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 Z>//, Dv., JJm^, . . . J)m„ denote the entropy of the element 

 J)ni, its volume, and the quantities which it contains of the various 

 components. Then 



Dm = Dm^ . . . + Dm„, (221) 



and 



dJ)m= dBm^ • • • + ^-Z>m„. (222) 



Also, by equation (12), 



6D€ = t SDrj — ^ dUv + ju^ SJJm^ . . . -f yM„ 6J)m„. (223) 

 By these equations the general condition of equilibrium may be 

 reduced to the form 



ft SDi] ~fp 6Dv +f/i, SBm^ . . . -f ///„ dDm„ 



+ fg 6h Biti -\-fg h 6 Dm ^ . . . -\- fgh dDm„ ^0. (224 ) 



Now it will be observed that the different equations of condition 

 affect different parts of this condition, so that we must have, sepa- 

 rately, 



ft 6Dt] i 0, if fSDt] = ; (225) 



