832 



The tiaiisition case which causes fig-. 4rt to pass into fig. 5a 

 now also often occurs for the practicall}' occurring heterogeneous 

 equilibria. (An instance of this is discussed in ^ 6). When naaiely 

 the point S lies on the prolongation of the line AB, the nodal lines 

 ?ig and iif coincide (n, is the nodal line drawn from a fluid to the 

 solid phase, rif to the other fluid phase). The theorem of ^ 2a then 

 requires that also bg and />; coincide, in other words that the two 

 binodal lines are in contact. 



When a .solid phase lies on a nodal line of the fiuids, the binodal 

 lines jiuid-jiuid and fluid-solid touch each other in the nodes. 



In a perfectl}' analogous way it also follows that: 



//' the nodal line solid-jiuid touches the binodal line jiuid-jluid, the 

 binodal line solid-Jiuid touches the nodal line of the fluid phases. 



All the cases that can occur for a stable plait, have now been 

 discussed. It, however, repeatedly occurs that part of the plait 

 indicates unstable states; in the points of the binodal lines which 

 are situated within the spinodal line the surface is namely convex- 

 concave, and the points themselves are hyperbolical. In analogy 

 with figs. 4(2 and ba we can now again construct two figures, which 

 are applicable when one of the binodal lines consists of hyperbolical 

 points. (The case, thai Itoth binodal lines consist of hyperbolical 

 points is not considered ; the discussion is self-evidently just as 

 simple). In these cases we get figs. 6 and 7. In both point A is 



Fig. 6. Fig. 7. 



given as hyperbolical, point B as elliptical point. The corresponding 

 P-.r-diagrams are easy to construct; they have, therefore, been 

 omitted ; besides the coexistences are not realizable, and are accord- 

 ingly devoid of practical importance. 



5. The Four- Phase Equilibria. 



When on the if'-surface simultaneous coexistence with solid occurs 

 for a three-phase equilibrium of fluid phases, the number of nodal 



