42 REPEESENTATION BY SURFACES OF THE 



seen (page 37), have a common tangent plane, which is identical with 

 the tangent plane for the point in the derived surface. 



Now, if the form of the surface be such that it falls above the tan- 

 gent plane except at the single point of contact, the equilibrium is 

 necessarily stable ; for if the condition of the body be slightly altered, 

 either by imparting sensible motion to any part of the body, or by 

 slightly changing the state of any part, or by bringing any small 

 part into any other thermodynamic state whatever, or in all of these 

 ways, the point representing the volume, entropy, and energy of the 

 whole body will then occupy a position above the original tangent 

 plane, and the proposition above enunciated shows that processes 

 will ensue which will diminish the distance of this point from that 

 plane, and that such processes cannot cease until the body is brought 

 back into its original condition, when they will necessarily cease on 

 account of the form supposed of the surface. 



On the other hand, if the surface have such a form that any part 

 of it falls below the fixed tangent plane, the equilibrium will be 

 unstable. For it will evidently be possible by a slight change in the 

 original condition of the body (that of equilibrium with the surround- 

 ing medium and represented by the point or points of contact) to 

 bring the point representing the volume, entropy, and energy of the 

 body into a position below the fixed tangent plane, in which case we 

 see by the above proposition that processes will occur which will 

 carry the point still farther from the plane, and that such processes 

 cannot cease until all the body has passed into some state entirely 

 different from its original state. 



It remains to consider the case in which the surface, although it 

 does not anywhere fall below the fixed tangent plane, nevertheless 

 meets the plane in more than one point. The equilibrium in this 

 case, as we might anticipate from its intermediate character between 

 the cases already considered, is neutral. For if any part of the 

 body be changed from its original state into that represented by 

 another point in the thermodynamic surface lying in the same tan- 

 gent plane, equilibrium will still subsist. For the supposition in 

 regard to the form of the surface implies that uniformity in tempera- 

 ture and pressure still subsists, nor can the body have any necessary 

 tendency to pass entirely into the second state or to return into the 

 original state, for a change of the values of T and P less than any 

 assignable quantity would evidently be sufficient to reverse such a 

 tendency if any such existed, as either point at will could by such an 

 infinitesimal variation of T and P be made the nearer to the plane 

 representing T and P. 



It must be observed that in the case where the thermodynamic 

 surface at a certain point is concave upward in both its principal 



