EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 143 



quantities of these proximate components are not capable of inde- 

 pendent variation, and the phase may be completely defined by the 

 quantities of the ultimate components with the temperature and 

 pressure. There may be, at certain intermediate temperatures and 

 pressures, a condition with respect to the independence of the 

 proximate components intermediate in character, in which the 

 quantities of the proximate components are independently variable 

 when we consider all phases, the essentially transitory as well as the 

 permanent, but in which these quantities are not independently 

 variable when we consider the permanent phases alone. Now we 

 have no reason to believe that the passing of a body in a state of 

 dissipated energy from one to another of the three conditions men- 

 tioned has any necessary connection with any discontinuous change 

 of state. Passing the limit which separates one of these states from 

 another will not therefore involve any discontinuous change in the 

 values of any of the quantities enumerated in (99)-(103) on page 88, 

 if m lt ra 2 , ... ra n , // 1? yu 2 ,.../z n are understood as always relating to 

 the ultimate components of the body. Therefore, if we regard masses 

 in the different conditions mentioned above as having different 

 fundamental equations (which we may suppose to be of any one 

 of the five kinds described on page 88), these equations will agree 

 at the limits dividing these conditions not only in the values of 

 all the variables which appear in the equations, but also in all the 

 differential coefficients of the first order involving these variables. 

 We may illustrate these relations by supposing the values of t, p, 

 and f for a mass in which the quantities of the ultimate components 

 are constant to be represented by rectilinear coordinates. Where the 

 proximate composition of such a mass is not determined by t and p, 

 the value of f will not be determined by these variables, and the 

 points representing connected values of t, p, and f will form a solid. 

 This solid will be bounded in the direction opposite to that in which 

 f is measured, by a surface which represents the phases of dissipated 

 energy. In a part of the figure, all the phases thus represented may 

 be permanent, in another part only the phases in the bounding surface, 

 and in a third part there may be no such solid figure (for any phases 

 of which the existence is experimentally demonstrable), but only a 

 surface. This surface together with the bounding surfaces representing 

 phases of dissipated energy in the parts of the figure mentioned above 

 forms a continuous sheet, without discontinuity in regard to the 

 direction of its normal at the limits dividing the different parts of 

 the figure which have been mentioned. (There may, indeed, be 

 different sheets representing liquid and gaseous states, etc., but if we 

 limit our consideration to states of one of these sorts, the case will 

 be as has been stated.) 



