829 



When one of the phases participating^ in the three-phase equilibrium 

 corresponds with an elliptic point on the \\)-surface, the prolongations 

 of the {stable) binodal lines lie either both inside or both outside the 

 th ree phase triang le . 



d> d> / d> \' „ ,. , . 



h. — — . = I 1 . Parabolical point. 



dv* da?' \dv.dxj 



From equation (1), which can easily be transforntied into-. 



( 



(4) 



and from — J = — p= constant and ( — I = q= constant, for 



\dv /x \dx J„ 



which the following relations are valid : 



(dv^\ dv^dx^ /üfwj\ dx^* 



dx^Jp d*^) \dxjjg d*\p 



dv^* dv^dx^ 



follows that equation (4) for a parabolic point reduces to: 



'dv\ f'^^\ f^^\ 



dxjiin \dxjp \dxjg 



Hence the two binodal lines touch the p- and the <2'-linöS, and 

 accordingly they are in contact with each other; the two nodal 

 lines form arbitrary angles with the binodal lines and also with 

 each other. 



Wli£n one of the phases participating in the three-phase equilibrium 

 corresponds with a parabolic point on the \]>surface, there is contact 

 between the two binodal lines ; the binodal lines either lie partly inside 

 or entirely outside the three-phase triangle. 



c. . <^ ) . Hyperbolical point. 



dv* ' dx^ \dvdxj 



In a hyperbola |)airs of conjugate diameters do not separate each 

 other. Hence: 



Moving in a definite direction round a hyperbolical point, the order 



of binodal lines and nodal lines is: 



^1 ^1 "i "i Of '''i "» ^t ^*i • 



54 



Proceedings Royal Acad. A.msterdara Vol. XXI. 



