-/ W. Gibhs — EquiUhriiim of Heterogeneoiis Substcmces. 199 



vapor of water and free hydrogen and oxygen, which we have just 

 considered, to a point at which the numbers of the] different sorts of 

 molecules are entirely determined by the temperature and pressure 

 and the total quantities of hydrogen and of oxygen which are present, 

 the fundamental equation of such a mass would involve but four inde- 

 pendent variables, which might be the four quantities just mentioned. 

 The fact of a certain part of the matter j^resent existing in the 

 form of vapor of water would, of course, be one of the facts which 

 determine the nature of the relation between ? and the independent 

 variables, which is expressed by the fundamental equation. 



But in the case first considered, in which the quantities of the 

 different sorts of molecules are not determined by the temperature 

 and pressure and the quantities of the difierent kinds of matter in the 

 body as determined by its ultimate analysis, the components of which 

 the quantities or the potentials appear in the fimdamental equation 

 must be those which are detei-mined by the proximate analysis of the 

 body, so that the variations in their quantities, with two variations 

 relating to the thermodynamic state of the body, shall include all the 

 variations of which the body is capable.* Such cases present no 

 especial difficulty; there is indeed nothing in the physical and 

 chemical jiroperties of such bodies, so far as a certain range of experi- 

 ments is concerned, Avhich is different from what might be, if the 

 proximate components were incapable of farther reduction or trans- 

 formation. Yet among the 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 

 merit 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 S^, . . . S„^ and the 

 ultimate components S„^ . . . *S/, ; and let m^, . . . m„ denote the 

 quantities of the former, and m„, . , . m^, the quantities of the latter. 

 It is evident that m^ . . . m^ are homogeneous functions of the first 

 degree of m,, . . . J7^„; and that the relations between the substances 

 aSj, . . . /8„ 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 com- 



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

 lute 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. 



