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Physics. — ''Some remarks on the direction of the hinodal curve.^ 

 in the v — x-diagram in a three-phase-equilihrium.'' By Prof. 



.]. P. KUENEN. 



Professor Schreinemakers lias privately communicated to me the 

 following rule which he has derived from the general theory of 

 plaits and which occurs in a joint paper about to be published by 

 him and Professor D. J. Korteweg: 



In a three-phase-equilibrium the two hinodal curves wldcli pass 

 through an. angle of the three-phase-triangle lie either both inside or 

 both outside the triangle. 



For the special case of the «f>surface for binary mixtures this rule 

 may be proved as follows. Choose one of the angles, say 1, and 

 start from the condition where the phases 2 and 3 coincide, a con- 

 dition which we may always imagine, even though it may not be 

 physically realisable: at this moment the two binodal curves through 

 1 form one continuous curve. If now 2 and 3 separate the two 

 curves meet at an angle and obviously in the beginning they both 

 lie outside, the triangle. As the surface continues to change one of 

 the binodal curves (say the one of the equilibrium 1,2) may pass 

 into the triangle by its direction at a definite moment coinciding 

 with the side 1 — 3 of the triangle. But it may be shown that at 

 the same moment the second binodal curve coincides with the side 

 1 — 2, in consequence of which this curve also passes into the triangle. 



The equation which expresses the special position referred to of 

 the binodal curve 1,2 is as follows ') : 



V — V 



Solving for — we find : 



Or ÖxQv 



in other words the curve 1,3 coincides with the side 1 — 2, q. e. d. 



Looking through the literature concerning the if?-surface one finds 



that in the diagrams the above rule has been often sinned against. 



1) All the differential coefficients without an index refer to the point 1. 



