THERMODYNAMIC PROPERTIES OF SUBSTANCES. 43 



curvatures, but somewhere falls below the tangent plane drawn 

 through that point, the equilibrium although unstable in regard to 

 discontinuous changes of state is stable in regard to continuous 

 changes, as appears on restricting the test of stability to the vicinity 

 of the point in question ; that is, if we suppose a body to be in a state 

 represented by such a point, although the equilibrium would show 

 itself unstable if we should introduce into the body a small portion 

 of the same substance in one of the states represented by points 

 below the tangent plane, yet if the conditions necessary for such a 

 discontinuous change are not present, the equilibrium would be 

 stable. A familiar example of this is afforded by liquid water when 

 heated at any pressure above the temperature of boiling water at 

 that pressure.* 



Leading Features of the Thermodynamic Surface for Substances 

 which take the forms of Solid, Liquid, and Vapor. 



We are now prepared to form an idea of the general character of 

 the primitive and derived surfaces and their mutual relations for a 

 substance which takes the forms of solid, liquid, and vapor. The 

 primitive surface will have a triple tangent plane touching it at the 

 three points which represent the three states which can exist in 

 contact. Except at these three points, the primitive surface falls 

 entirely above the tangent plane. That part of the plane which forms 

 a triangle having its vertices at the three points of contact, is the 

 derived surface which represents a compound of the three states of the 

 substance. We may now suppose the plane to roll on the under side 

 of the surface, continuing to touch it in two points without cutting it. 

 This it may do in three ways, viz : it may commence by turning about 

 any one of the sides of the triangle aforesaid. Any pair of points 

 which the plane touches at once represent states which can exist 

 permanently in contact. In this way six lines are traced upon the 

 surface. These lines have in general a common property, that a 

 tangent plane at any point in them will also touch the surface in 

 another point. We must say in general, for, as we shall see hereafter, 

 this statement does not hold good for the critical point. A tangent 

 plane at any point of the surface outside of these lines has the surface 



*If we wish to express in a single equation the necessary and sufficient condition 

 of thermodynamic equilibrium for a substance when surrounded by a medium of constant 

 pressure P and temperature T, this equation may be written 



when 5 refers to the variation produced by any variations in the state of the parts of 

 the body, and (when different parts of the body are in different states) in the proportion 

 in which the body is divided between the different states. The condition of stable 

 equilibrium is that the value of the expression in the parenthesis shall be a minimum. 



