120 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



regard to adjacent phases, as otherwise the case would be devoid of 

 interest. The point which represents the state of the composite 

 body will evidently be at the center of gravity of masses equal to 

 the parts of the body placed at the points representing the phases 

 of these parts. Hence from the surface representing the properties 

 of homogeneous bodies, which may be called the primitive surface, we 

 may easily construct the surface representing the properties of bodies 

 which are 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 combination 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 contact 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 im- 

 portant which gives the least value of f for any given values of 

 m v m 2 , m 3 . 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 rigid envelop impermeable 

 both to matter and to heat; and the state of any mass composed 

 of S v S 2 , S B in any proportions, in which the dissipation of energy 

 has been completed, 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 m-f 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 

 P 1 P 2 P 3 , (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-f lines for two component 

 substances 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 1, in which f is 



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