of the Therniodyna niu' Properties of /Snbstcmces. ;!95 



together. The rolling of the double t.'uigent 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 char- 

 acteristics of this point and their physical significance, it will he con- 

 venient to investigate the nature of the primitive sui-face which lies 

 between the Ihies 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 

 princi})al curvatui-es at least, and therefore represents states of iinsta- 

 ble equilibrium in respect to continuous as well as discontinuous 

 changes (see page :!92).* If we draw a line upon the primitive sur- 

 face, dividing it into parts which represent respectively stable and 

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

 the surface which is concave uj)ward in both its principal curvatures 

 from that which is concave downward in one or both, tliis 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 ujjon the primi- 

 tive surface made by a plane section perpendicular to the tangent 

 plane, will pass through an area of instability. 



The geometrical projjerties 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, 

 Avhich has diiferent signs upon diflferent sides of the limit of essential 

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

 eral 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 instabil- 

 ity 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 tan- 

 gent plane, but has a contact of the third order with it in tlie section 

 of least curvature. The critical point, therefore, must be a point 



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

 with the surface referred to in the note on page ;!82. 



Trans. Connecticut Academy, Vol. II. 34 Dec, 1873. 



