THEEMODYNAMIC PROPERTIES OF SUBSTANCES. 45 



Phil. Trans., vol. 159, p. 575, has obtained by his experiments with 

 carbonic acid is that, in the case of this substance at least, the derived 

 surface which represents a compound of liquid and vapor is terminated 

 as follows : as the tangent plane rolls upon the primitive surface, 

 the two points of contact approach one another and finally fall 

 together. The rolling of the double tangent plane necessarily comes 

 to an end. The point where the two points of contact fall together is 

 the critical point. Before considering farther the geometrical character- 

 istics of this point and their physical significance, it will be convenient 

 to investigate the nature of the primitive surface which lies between 

 the lines which form the limit of absolute stability. 



Between two points of the primitive surface which have a common 

 tangent plane, as those represented by L' and V in figure 2, if there 

 is no gap in the primitive surface, there must evidently be a region 

 where the surface is concave toward the tangent plane in one of its 

 principal curvatures at least, and therefore represents states of un- 

 stable equilibrium in respect to continuous as well as discontinuous 

 changes (see pages 42, 43).* If we draw a line upon the primitive 

 surface, dividing it into parts which represent respectively stable and 

 unstable equilibrium, in respect to continuous changes, i.e., dividing 

 the surface which is concave upward in both its principal curvatures 

 from that which is concave downward in one or both, this line, which 

 may be called the limit of essential instability, must have a form 

 somewhat like that represented by ll'Cvv'ss' in figure 2. It touches 

 the limit of absolute stability at the critical point C. For we may 

 take a pair of points in LC and VC having a common tangent plane 

 as near to C as we choose, and the line joining them upon the primi- 

 tive surface made by a plane section perpendicular to the tangent 

 plane, will pass through an area of instability. 



The geometrical properties of the critical point in our surface may 

 be made more clear by supposing the lines of curvature drawn upon 

 the surface for one of the principal curvatures, that one, namely, 

 which has different signs upon different sides of the limit of essential 

 instability. The lines of curvature which meet this line will in 

 general cross it. At any point where they do so, as the sign of their 

 curvature changes, they evidently cut a plane tangent to the surface, 

 and therefore the surface itself cuts the tangent plane. But where 

 one of these lines of curvature touches the limit of essential instability 

 without crossing it, so that its curvature remains always positive 

 (curvatures being considered positive when the concavity is on the 

 upper side of the surface), the surface evidently does not cut the 



* This is the same result as that obtained by Professor J. Thomson in connection with 

 the surface referred to in the note on page 34. 



