J. W. Gibbs — JEquilihrium of Heterogeneous Substances. 119 



(18) 



drn^' -\- dm/' + dm/" + etc. = 0, ] 

 dm./ + dm/' + dm/" + etc. =: 0, 



and dm/ -\- dm/' -j- dm/" -\- etc. = 0. 

 For this it is evidently necessary and sufficient that 



t' = t" =zt"'z:i etc. (19) 



y =y =y' — etc. (20) 



/Yj' = //,"=///"= etc.^ 



f.i/ — H/' z= ^i/" = etc. [ ^21) 



lA,! z= pi/' = fx/" =. etc. J 



Equations (19) and (20) express the conditions of thermal and 

 mechanical equilibrium, viz., that the temperature and the pressure 

 must be constant throughout the whole mass. In equations (21) we 

 have the conditions characteristic of chemical equilibrium. If we 

 call a quantity //„ as defined by such an equation as (12), the potential 

 for the substance >S, in the homogeneous mass considered, these con- 

 ditions may be expressed as follows : 



The potential for each cotnponent substance must be constant 

 throughout the lohole mass. 



It will be remembered that we have supposed that there is no 

 restriction upon the freedom of motion or combination of the com- 

 ponent substances, and that each is an actual component of all parts 

 of the given mass. 



The state of the whole mass will be completely determined (if we 

 regard as immaterial the position and form of the various homoge- 

 neous parts of which it is composed), when the values are determined 

 of the quautities of whicli the variations occur in (15). The number 

 of these quantities, which we may call the independent variables, is 

 evidently {n -\- 2) k, k denoting the number of homogeneous parts 

 into which the whole mass is divided. All the quantities which 

 occur in (19), (20), (21), are functions of these variables, and may be 

 regarded as known functions, if the energy of each part is known as 

 a function of its entropy, volume, and the quantities of its com- 

 ponents. (See eq. (12).) Therefore, equations (19), (20), (21), may 

 be regarded as {v - 1) {n -\- 2) independent equations between the 

 independent variables. The volume of the whole mass and the total 

 quantities of the various substances being known afford n-\- \ addi- 

 tional equations. If we also know the total energy of the given 

 mass, or its total entropy, we will have as many equations as there 

 are independent variables. 



