30 



GRAPHICAL METHODS IN THE 



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

 r\i\ ', and let the positive sides of these lines be indicated as in the 



figure. The conditions (-) > and [dp : dv]^ < require that the 



pressure at v and at r\ 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 (-T-) <0 and [dt : dt]] p >0 



require that the temperature at ?/ and at p shall be greater than at A, 

 and hence, that the isothermal shall fall as tt' and have its positive 



side turned as indicated. As it is not necessary that f-y-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 (-^-) >0, and 



\dr]/ v 



hence ( -7- ) < 0, does not allow pp' to be tangent to vv', nor tt' to r\r\ ' . 



If f -^- J be still positive, but the equilibrium be neutral, it will be 



possible for the body to change its 

 state without change either of tem- 

 perature or of pressure ; i.e., the 

 t' isothermal and isopiestic will be 

 identical. The lines will fall as in 

 figure 13, except that the isothermal 

 and isopiestic will be superposed. 



I?) < > it may 



t - 



Fig. 13. 



In like manner, if 



be proved that the lines will fall as 

 in figure 14 for stable equilibrium, 

 and in the same way for neutral 



equilibrium, except that pp' and tt' will be superposed.! 



*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. 



i When it is said that the arrangement of the lines in the diagram must be like that 

 in figure 13 or in figure 14, it is not meant to exclude the case in which the figure 

 (13 or 14) must be turned over, in order to correspond with the diagram. In the case, 

 however, of diagrams formed by any of the methods mentioned in this article, if the 



