46 REPRESENTATION BY SURFACES OF THE 



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

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

 where the line of that principal curvature which changes its sign 

 is tangent to the line which separates positive from negative 

 curvatures. 



From the last paragraphs we may derive the following physical 

 property of the critical state : Although this is a limiting state 

 between those of stability and those of instability in respect to con- 

 tinuous changes, and although such limiting states are in general 

 unstable in respect to such changes, yet the critical state is stable in 

 regard to them. A similar proposition is true in regard to absolute 

 stability, i.e., if we disregard the distinction between continuous and 

 discontinuous changes, viz : that although the critical state is a limit- 

 ing state between those of stability and instability, and although the 

 equilibrium of such limiting states is in general neutral (when we 

 suppose the substance surrounded by a medium of constant pressure 

 and temperature), yet the critical point is stable. 



From what has been said of the curvature of the primitive surface 

 at the critical point, it is evident, that if we take a point in this 

 surface infinitely near to the critical point, and such that the tangent 

 planes for these two points shall intersect in a line perpendicular to 

 the section of least curvature at the critical point, the angle made by 

 the two tangent planes will be an infinitesimal of the same order as 

 the cube of the distance of these points. Hence, at the critical point 



//7 2 TA //7 2 r>\ //7 2 /\ /<7 2 /\ 



(^) = (^)=0 ( 1=0 ( ]=0 



1 7 9 / v -' V 7 <>i / **J \ 7 O I V J I 7 O I V J 



\dv i Jt \dr}*/t \dv*/p \drj 2 /p 



and if we imagine the isothermal and isopiestic (line of constant 

 pressure) drawn for the critical point upon the primitive surface, 

 these lines will have a contact of the second order. 



Now the elasticity of the substance at constant temperature and 

 its specific heat at constant pressure may be defined by the equations r 



_ (dp\ _j.(dt)\ 



therefore at the critical point 



e=0, 1 = 0, 



g\ 0> gf) =0> gi) =0> gh =a 

 \dv/t \dr]/t \dv/p \driJp 



The last four equations would also hold good if p were substituted 

 for t t and vice versa. 



