of the T/iermodyncunic Projjertieti of /Substances. 393 



heated at any pressure above the tem{)erature of boiling water at 

 that pressure.* 



Leading Features of the Tliermodynaynic Surface for Substances 

 which take the forms of Solid, JAquid and Va^yor. 



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 whicli takes the forms of solid, licpiid, and va}»or. The prim- 

 itive 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 jaoints of contact, is the derived sur- 

 face which represents a compound of the three states of the sub- 

 stance. 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 per- 

 manentlj^ in contact. In this way six lines are traced upon the sur- 

 face. 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 i>i general, for, as we shall see hereafter, this 

 statement does not liold good for the critical point. A tangent i)lane 

 at any point of the surface outside of these lines has the surface en- 

 tirely al)ove it, except the single point of contact. A tangent plane 

 at any point of the })riraitive surface within these lines will cut the 

 surface. These lines, therefore, taken together may be called the 

 limit of absolute stability, and the surface outside of them, the sur- 

 face of cd)solute stability. That })art of the envelop of the rolling 

 plane, which lies between the pair of lines which the plane traces on 

 the surfoce, is a part of the derived surface, and represents a mixture 

 of two states of the substance. 



* If we wish to express in a single equation the necessary and sufl5cient condition 

 of thermod}''namic equilibrium for a substance when surrounded by a medium of con- 

 stant pressure P and temperature T, this equation may be written 



6{e - Ti] + Pv) = 0, 

 when 6 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 propor- 

 tion 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. 



