( 184 ) 



plaitpoint mentioned is then an homogeneous double plaitpoint. If' 

 then in the v, «-projection the plaitpoint line is drawn, it proceeds 

 again continuously from the left to the right side — and this con- 

 tinues to be the case also when the plaitpoint line mentioned lias 

 more intricate properties, e. g. when there are two heterogeneous 

 double plaitpoints, as discussed in "Contribution etc." and also treated 

 in These Proc. March 25, 1905, p. 621 and These Proc. June 24, 

 1905, p. 184. However, besides this plaitpoint line, another is possible. 

 The latter does not cross from the left side to the opposite side. So 

 only the possibility is left either that it is a closed curve in the 

 v, .«-projection, or that it begins and terminates on the borders v = b. 

 We shall proceed to discuss some properties of the special points 

 of this line, particularly of the double points of this line. Korteweg 

 has demonstrated that these double points are of two kinds. Either 

 it is a double point in which two homogeneous plaitpoints originate 

 or coincide — or it is a double point in which this is the case for 

 two heterogeneous plaitpoints. Though physically such plaitpoints 

 bear such different characters - mathematically they satisfy the 

 same criteria,, and on the plaitpoint line such an heterogeneous double 

 point is the transition point for a series of plaitpoints which might 

 be realized, and for a series of unrealisable plaitpoints. 



Minimum or maximum temperature for the plaitpoint line. 



If we suppose a double plaitpoint to originate or to disappear on 



the ^-surface at a certain value of T, two plaitpoints are found at 



somewhat higher or lower value of T. This holds both for the case 



that the double point is an homogeneous and an heterogeneous double 



dT 

 point, as we shall briefly call them. For the plaitpoint line — and 



dx 



dT dT 



— = in this case. But for an homogenous double point — is also 

 dv dp 



dp 

 = 0. This property follows from the shape of — , which has been 



derived in Verslag Kon. Ak. v. Wet. Deel IV p. 20 and p. 82 because 



éPv „ 



= in an homogeneous double point (Contribution etc. These Proc. 



dx^p 



dT 

 March 30, 1907, p. 745). For an heterogeneous double point — is 



dp 



dp 

 not equal to 0, as also appears from the value given for — , as for such 



d*v d'v d'v m 



a double point = 0, but =-— — . That in an heterogeneous 



r dx\, dx% dx' 



