J. W. Gi.bbs — Equilibrium of Heterogeneous Substances. 203 



understood as always relating to the ultimate components of the body. 

 Therefore, if we regard masses in the diiferent conditions mentioned 

 above as having different fundamental equations, (which we may sup- 

 pose to be of any one of the five kinds described on page 143,) 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 difi'erential coefficients of the first order involving these 

 variables. We may illustrate these relations by supposing the values 

 of t, />, and 'Q for a mass in which the quantities of the ultimate com- 

 ponents are constant to be represented by rectilinear coordinates. 

 Where the proximate composition of such a mass is not determined 

 by t and jo, the value of I will not be determined by these variables, 

 and the points representing connected values of t, ^>, and ^ will form 

 a solid. This solid will be bounded in the direction opposite to that 

 in which l is measured, by a surface which represents the phases of 

 dissipated energy. In a part of the figure, all the phases thus repre- 

 sented 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.) 



We shall hereafter, in the discussion of the fundamental equations 

 of gases, have an example of the derivation of the fundamental equa- 

 tion for phases of dissipated energy (with respect to the molecular 

 changes on which the proximate composition of the body depends) 

 from the more general form of the fundamental equation. 



THE CONDITIONS OF EQUILIBRIUM FOR HETEROGENEOUS MASSES UNDER 

 THE INFLUENCE OF GRAVITY. 



Let US now seek the conditions of equilibrium for a mass of various 

 kinds of matter subject to the influence of gravity. It will be con- 

 venient to suppose the mass enclosed in an immovable envelop which 

 is impermeable to matter and to heat, and in other respects, except 

 in regard to gravity, to make the same suppositions as on pages 115, 

 116. The energy of the mass will now consist of two parts, one of 



