36 REPRESENTATION BY SURFACES OF THE 



the thermodynamic surface, for many substances at least, can be 

 divided into two parts, of which one represents the homogeneous 

 states, the other those which are not so. We shall see that, when 

 the former part of the surface is given, the latter can readily be 

 formed, as indeed we might expect. We may therefore call the 

 former part the primitive surface, and the latter the derived surface. 



To ascertain the nature of the derived surface and its relations to 

 the primitive surface sufficiently to construct it when the latter is 

 given, it is only necessary to use the principle that the volume, 

 entropy, and energy of the whole body are equal to the sums of the 

 volumes, entropies, and energies respectively of the parts, while the 

 pressure and temperature of the whole are the same as those of each 

 of the parts. Let us commence with the case in which the body is 

 in part solid, in part liquid, and in part vapor. The position of the 

 point determined by the volume, entropy, and energy of such a com- 

 pound will be that of the center of gravity of masses proportioned 

 to the masses of solid, liquid, and vapor placed at the three points of 

 the primitive surface which represent respectively the states of com- 

 plete solidity, complete liquidity, and complete vaporization, each at 

 the temperature and pressure of the compound. Hence, the part of 

 the surface which represents a compound of solid, liquid, and vapor is 

 a plane triangle, having its vertices at the points mentioned. The 

 fact that the surface is here plane indicates that the pressure and 

 temperature are here constant, the inclination of the plane indicating 

 the value of these quantities. Moreover, as these values are the same 

 for the compound as for the three different homogeneous states cor- 

 responding to its different portions, the plane of the triangle is 

 tangent at each of its vertices to the primitive surface, viz: at one 

 vertex to that part of the primitive surface which represents solid, at 

 another to the part representing liquid, and at the third to the part 

 representing vapor. 



When the body consists of a compound of two different homo- 

 geneous states, the point which represents the compound state will be 

 at the center of gravity of masses proportioned to the masses of the 

 parts of the body in the two different states and placed at the points 

 of the primitive surface which represent these two states (i.e., which 

 represent the volume, entropy, and energy of the body, if its whole 

 mass were supposed successively in the two homogeneous states which 

 occur in its parts). It will therefore be found upon the straight line 



so that the results are in general strictly valid only in cases in which the influence 

 of these particulars may be neglected. When, therefore, two states of the substance 

 are spoken of as in contact, it must be understood that the surface dividing them 

 is plane. To consider the subject in a more general form, it would be necessary to 

 introduce considerations which belong to the theories of capillarity and crystallization. 



