178 tT. W. Gibhs — Equilibrium of Heterogeneous Substances. 



may easily construct the surface representing the properties of bodies 

 which ai-e in equilibrium but not homogeneous. This may be called 

 the secondary or derived surface. It will consist, in general, of 

 various portions or sheets. The sheets which represent a combina- 

 tion of two phases may be formed by rolling a double tangent plane 

 upon the primitive surface : the part of the envelop of its successive 

 positions which lies between the curves traced by the points of con- 

 tact will belong to the derived surface. When the primitive surface 

 has a triple tangent plane or one of higher order, the triangle in the 

 tangent plane formed by joining the points of contact, or the smallest 

 polygon without re-entrant angles which includes all the points of 

 contact, will belong to the derived surface, and will represent masses 

 consisting in general of three or more phases. 



Of the whole thermodynamic surface as thus constructed for any 

 temperature and any positive pressure, that part is especially impor- 

 tant which gives the least value of !: for any given values of Wj, ?«2? 

 m^. The state of a mass represented by a point in this part of the 

 surface is one in which no dissipation of energy would be possible if 

 the mass were enclosed in a i-igid envelop impermeable both to 

 matter and to heat ; and the state of any mass composed of aS^, aSj, S^ 

 in any proportions, in which the dissipation of energy has been com- 

 pleted, so far as internal processes are concerned, (i. e., under the 

 limitations imposed by such an envelop as above supposed,) would be 

 represented by a point in the part which we are considering of the 

 in-'Q surface for the temperature and pressure of the mass. We may 

 therefore briefly distinguish this part of the surface as the surface of 

 dissipated energy. It is evident that it forms a continuous sheet, the 

 projection of which upon the X- Y plane coincides with the triangle 

 Pj P2 P3, (except when the pressure for which the m-? surface is 

 constructed is negative, in which case there is no surface of dissipated 

 energy,) that it nowhere has any convexity upward, and that the 

 states which it represents are in no case unstable. 



The general properties of the m-t, lines for two component sub- 

 stances are so similar as not to require separate consideration. We 

 now proceed to illustrate the use of both the surfaces and the lines 

 by the discussion of several particular cases. 



Three coexistent phases of two component substances may be 

 represented by the points A, B, and C, in figure ], in which I is 

 measured toward the top of the page from PjPg, '" , toward the left 

 from P2Q2, and m^ toward the right from P,Qi. It is supposed 

 that P1P2 = 1- Portions of the curves to which these points belong 



