( 518 ) 



In fig. 7 .y.v' represents the spinodal line; rah and r'ed are two 

 branches of the curve of contact having in a and e a tangent passing- 

 through point P. In these points a and /Mve have the case considered 

 siib Ib; n and e lie therefore both on the hyperbolicallv curved 

 part of the surface. 



If we pursue the branches ah and ed, these can of course pass 

 into each other ^); in fig. 7 this continuation is represented by the 

 dotted curve bed. 



In fig. 9 the curve of contact consists of the two branches rr' 

 and abed, sepanited from each other bv the spinodal line ss' "). 



The equation (I9j can, however, also represent an isolated point; 

 the curve of contact then consists of a single isolated point, lying 

 on the spinodal line. For a small change in parameter this point 

 then vanishes or a closed curve of contact is generated. 



Inversely the closed curve of contact abed of fig. 9 can thus 

 contract so as to disappear in a point of the s[)inodal line. 



To investigate whether the curve can possess other nodes or isolated 

 points (in ordinary not conical points), we cause the }"-axis to 

 coincide with OP. This is of course always possible and then we 

 have p = 0. 



From (2) follows now as condition for a node c./_/:=0 and 2t'32 = 0. 



So we find : 



1 



Cj r= Cj = and therefore also ''i ''3 c% =r 0. 



4 



This is just the condition for the generation of case IIb • So we 

 find the nodes and isolated points only in case 7/j5 , except of course 

 in the points of osculation which can be regarded as a special case 

 of it, where c^ = 0. 

 ' So we can say : 



"Nodes and isolated points of the curve of contact always lie on 

 the spinodal line. " 



There would be an exception only if point P were on the surface 

 itself; then of course there would alwiiys be in that place an isolated 

 point or node; this howe\er we do not discuss. 



IIb.u- Point is a plaitpomt. 



We assume (fig. 10) OP as 1^-axis so that besides p = 0, c, = 

 and Cg = 0, we find also d^ = ^). 



1) Gomp. F. A. H. Schreixemakers. Z. f. Phys. Cliem. 22. 532 (1897). 

 ~) Comp. F. A. H. ScHREiNEMAKERs. Z. f. Plivs. Gliem. 22 581 (1897). 

 3) D. J. KoRTEWEG. Ai'di. Néeil. (I). 24 61. (1891). 



I 



