1:^2 J. TF. Gibbs—Eqailibviiim, of Heterogeneoiis Svbstances. 



formation of the body, the joint entropy of the medium and the 

 hody, their joint volumes and joint quantities of matter, were the 

 same as the entropy, etc., of the medium before the formation of the 

 body. Tliis consideration may convince us that for any given finite 

 values of v and of T, P, 31^ , etc. this expression cannot be infinite 

 when f, //, m,, etc. are determined by any real body, whether homo- 

 geneous or not, (but of the given volume), even when T, P, 3/j, etc. 

 do not represent the values of the temperature, pressure, and poten- 

 tials of any real substance. (If the substances *S',, /Sg, . . . S„ are 

 all actual components of any homogeneous part of the system of 

 which the equilibrium is discussed, that part will aiford an example 

 of a body having the temperature, pressure, and potentials of the 

 medium supposed.) 



Now by integrating equation (12) on the supposition that the 

 nature and state of the mass considered i-emain unchanged, we obtain 

 the equation 



which will hold true of any homogeneous mass whatever. Therefore 

 for any one of the original parts, by (50) and (51), 



f - T)]-\-Pv-M^ m J - J/2 »«2 • • • — ^^n ^''„ = 0. (56) 

 If the condition (52) is not satisfied in regard to all possible new 

 parts, let JVhe a new part occurring in an original part O, for which 

 the condition is not satisfied. It is evident that the value of the 

 expression 



s—Ti] + Pv - M^ m^ — 31^ m^ . . . —3f„m„ (57) 



applied to a mass like including some very small masses like JV, 

 will be negative, and will decrease if the number of these masses like 

 JV is increased, until there remains within the whole mass no portion 

 of any sensible size without these masses like iV, which, it will be 

 remembered, have no sensible size. But it cannot decrease without 

 limit, as the value of (54) cannot become infinite. Now we need not 

 inquire whether the least value of (57) (for constant values of T, P, 

 M^, J/g* • • • -^^") would be obtained by excluding entirely the 

 mass like 0, and filling the whole space considered with masses like 

 iV, or whether a certain mixture would give a smaller value, — it is 

 certain that the least possible value of (57) per unit of volume, and 

 that a negative value, will be realized by a mass having a certain 

 homogeneity. If the new part iVfor which the condition (52) is not 

 satisfied occurs between two diflferent original parts 0' and 6>", the 

 aigument need not be essentially varied. We may consider the 



