THERMODYNAMIC PROPERTIES OF SUBSTANCES. 49 



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 surface 

 in these three points, it can commence to roll upon the surface as 

 a double tangent plane not only by leaving 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 lines, 

 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 

 surface of dissipated energy on the one hand, nor have the theoretical 

 interest of the primitive surface on the other. 



Problems relating to the Surface of Dissipated Energy. 



The surface of dissipated energy has an important application to a 

 certain class of problems which refer to the results which are theo- 

 retically possible with a given body or system of bodies in a given 

 initial condition. 



For example, let it be required to find the greatest amount of 

 mechanical work which can be obtained from a given quantity of a 

 certain substance in a given initial state, without increasing its total 

 volume or allowing heat to pass to or from external bodies, except 



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- 

 metrical properties mentioned above. 



There will, however, be no such part in which ^><0, if there is any assignable 

 temperature 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 12-13) 



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 p < and t> the equation of the tangent plane for 

 that point will be 



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 />-<0 cannot belong to the surfaces mentioned 

 in the last paragraph as forming a continuous sheet. 

 G. I. D 



