392 ,/ W. Gibbs on a Representation hy Surfaces 



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 

 mistable. 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 heloni 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 remahis to consider the case in which the surface, although it 

 does not anywhere fall below the iixed 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 7^ and ^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 tlie thermodynamic 

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

 curvatures, but somewhere falls below the tangent plane drawn 

 through that point, the equilibrium although unstable in regard to 

 discontinuous changes of state is stable in regard to continuous 

 changes, as appears on restricting the test of stability to the vicinity 

 of the point in question ; that is, if we suppose a body to be in a state 

 represented by such a point, although the equilibrium would show 

 itself unstable if we should introduce into the body a small portion 

 of the same substance in one of the states represented by j^oints 

 below the tangent plane, yet if the conditions necessary for such a 

 discontinuous change are not present, the equilibrium would be sta- 

 ble. A familiar example of this is afforded by liquid water when 



