Page:Scientific Papers of Josiah Willard Gibbs.djvu/156

120 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 important which gives the least value of $$\zeta$$ for any given values of $$m_{1}, m_{2}, m_{3}$$. 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 $$S_{1}, S_{2}, S_{3}$$ 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 $$m\!-\!\zeta$$ 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 $$m\! -\!\zeta$$ 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 $$m\! -\!\zeta$$ 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 $$\zeta$$ is