of the Thermody namic Properties of Substances. 399 



foi'm a solid figure unbounded in some directions, but bounded in 

 others by tliis surface.* 



The lines traced upon the primitive surface by the rolling double 

 tangent plane, which have been called the limit of absolute stability, 

 do not end at the vertices of the triangle which represents a mixture 

 of those states. For when the plane is tangent to the primitive sur- 

 face in these three })oints, it can commence to roll upon the surface as 

 a double tangent plane not only by leavuig the surface at one of 

 these points, but also by a rotation in the opposite direction. In the 

 latter case, however, the lines traced upon the primitive surface by 

 the points of contact, although a continuation of the lines previously 

 described, do not form any part of the limit of absolute stability. 

 And the parts of the envelops of the rolling plane between these Hues, 

 although a continuation of the developable surfaces which have been 

 described, and representing states of the body, of which some at least 

 may be realized, are of minor interest, as they form no part of the 



* This description of tlie surface of dissipated energy is intended to apply to a sub- 

 stance capaljle of existing as solid, liquid, and vapor, and vs^hich presents no anoma- 

 lies in its thermodynamic properties. But, whatever the form of the primitive sur- 

 face may be, if we take the parts of it for every point of which the tangent plane does 

 not cut the primitive surface, together with all the plane and developable derived sur- 

 face, which can be formed in a manner analogous to those described in the preceding 

 pages, by fixed and rolling tangent planes which do not cut the primitive surface, — 

 such surfaces taken together will form a continuous sheet, which, if we reject the 

 part, if any, for which p <^ 0, forms the surface of dissipated energy and has the geo- 

 metrJcal properties mentioned above. 



There will, however, be no such part in which p <[ 0, if there is any assignable tem- 

 perature t' at which the substance has the properties of a perfect gas except when its 

 volume is less than a certain quantity v'. For the equations of an isothermal line in 

 the thermodynamic surface of a perfect gas are (see equations (b) and (e) on pages 

 321-322 of this volume.) 



e= G 



7/ = a log w + C". 

 The isothermal of t' in the thermodynamic surface of the substance in question must 

 therefore have the same equations in the part in which v exceeds the constant v'. 

 Now if at any point in this surface /< •< and t > the equation of the tangent plane 

 for that point will be 



E = in 1] + M « + C", 

 where m denotes the temperature and —n the pressure for the point of contact, so that 

 m and n are both positive. Now it is evidently possible to give so large a value to v 

 in the equations of the isothermal that the point thus determined shall fall below the 

 tangent plane. Therefore, the tangent plane cuts the primitive surface, and the point 

 of the thermodynamic surface for which i^ < cannot belong to the surfaces men- 

 tioned in the last paragraph as forming a continuous sheet. 



