140 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



the variations of which the body is capable.* Such cases present 

 no especial difficulty; there is indeed nothing in the physical and 

 chemical properties of such bodies, so far as a certain range of 

 experiments is concerned, which is different from what might be, 

 if the proximate components were incapable of farther reduction or 

 transformation. Yet among the various phases of the kinds of matter 

 concerned, represented by the different sets of values of the variables 

 which satisfy the fundamental equation, there is a certain class which 

 merits especial attention. These are the phases for which the entropy 

 has a maximum value for the same matter, as determined by the 

 ultimate analysis of the body, with the same energy and volume. 

 To fix our ideas let us call the proximate components 8 lt ... S n , and 

 the ultimate components S at ...S h ; and let m v ...m n denote the 

 quantities of the former, and m a , ... m h) the quantities of the latter. 

 It is evident that m a ,...m h are homogeneous functions of the first 

 degree of m lt . . . m n ; and that the relations between the substances 

 S v ... S n might be expressed by homogeneous equations of the first 

 degree between the units of these substances, equal in number to 

 the difference of the numbers of the proximate and of the ultimate 

 components. The phases in question are those for which r\ is a 

 maximum for constant values of e, v, m a , ... tn h ; or, as they may also 

 be described, those for which e is a minimum for constant values 

 of 77, v, m a , ...?%; or for which f is a minimum for constant values 

 of t, p, m a ,...m fe . The phases which satisfy this condition may be 

 readily determined when the fundamental equation (which will 

 contain the quantities m v ...m n or fjL v ... fji n ,) is known. Indeed it 

 is easy to see that we may express the conditions which determine 

 these phases by substituting yu p . . . // for the letters denoting the 

 units of the corresponding substances in the equations which express 

 the equivalence in ultimate analysis between these units. 



These phases may be called, with reference to the kind of change 

 which we are considering, phases of dissipated energy. That we 

 have used a similar term before, with reference to a different kind 

 of changes, yet in a sense entirely analogous, need not create 

 confusion. 



It is characteristic of these phases that we cannot alter the values 

 of m lt . . . m n in any real mass in such a phase, while the volume of 

 the mass as well as its matter remain unchanged, without diminishing 

 the energy or increasing the entropy of some other system. Hence, 

 if the mass is large, its equilibrium can be but slightly disturbed 



*The terms proximate or ultimate are not necessarily to be understood in an 

 absolute sense. All that is said here and in the following paragraphs will apply 

 to many cases in which components may conveniently be regarded as proximate or 

 ultimate, which are such only in a relative sense. 



