340 



J. W, Gibbs on GrajjMcal Methods in the 



Fig. 13. 



transmission of lieat, the pressui-e should increase as the vokime di- 

 minishes, i. e., that [djy : (7y] ^ < 0. Tlirough the point in question, 

 A (fig. 13), let there be drawn the isometric vv' and the isentropic 

 rpf, and let the positive sides of these lines be indicated as in the 



figure. The conditions i£\ > and [dp : <^y] ^ < require that the 



pressure at v and at rj shall be greater than at A, and hence, that the 



isopiestic shall fall as pp' in the 

 figure, and have its positive side 

 turned as indicated. Again, the 

 conditions 



require that the temperature at 

 // and at p shall be greater than 

 at A, and hence, that the iso- 

 thermal shall fall as tt' and have 

 its positive side turned as indi- 

 cated. As it is not necessary that 



j-^j ^ 0, the lines pp' and tt' may be tangent to one another at A, 



provided that they cross one another, so as to have the same order 

 about the point A as is represented in the figure ; i. e., they may have 

 a contact of the second (or any even) order.* But the condition that 



\-j—] ^ 0, and hence \-j—\ <^ 0, does not allow pp' to be tangent 



vv', nor tt' to ?;//'. 



to 



If 



dp 



drf 



be still positive, but the equilibrium be neutral, it will be 



possible for the body to change its state without change either of 

 temperature or of pi-essiire ; i. e., the isothermal and isopiestic will be 



condition requires that the ratio of the differences of temperature and entropy between 

 the point in question and any other infinitely near to it and upon the same isopiestic 

 should be positive. It is not necessary that the limit of this ratio should be positive. 



* An example of this is doubtless to be found at the critical point of a fluid. See 

 Dr. Andrews " On the continuity of the gaseous and liquid states of matter." Phil. 

 Trans., vol. 159, p. 575. 



If the isothermal and isopiestic have a simple tangency at A, on one side of that 

 point they will have such directions as will express an unstable equilibrium. A line 

 drawn through all such points in the diagram will form a boundary to the possible part 

 of the diagram. It may be that the part of the diagram of a fluid, which represents 

 the superheated liquid state, is bounded on one side by such a line. 



