828 



When equation (1) is written in the form 



T-^ + y^, = 0.. (3) 

 av,x, dec. 



it appears from (2) and (3) that the binodal line and the nodal line 

 represent conjugate diameters in the indicatrix, which has been 

 demonstrated by Kortrweg '). 



2. The Relative Situation of Binodal Lines and Xodal Lines. 



A great number of inferences which are of importance for the 

 treatment of the more intricate cases of heterogeneous equilibrium, 

 which may present themselves for binary mixtures, may be made 

 from the above mentioned conclusions from the theory of binary 

 mixtures, which have been known already for a long time. 



We shall imagine tino binodal lines going through a point of the 

 tf>-plane; each binodal line with the nodal line belonging to it is a 

 set of conjugate diameters in the indicatrix. Depending on the form 

 of the indicatrix we now get the following cases: 

 d*^ 



dv^ dx* \dv dx 



Elliptical point. 



From the well-known thesis of the ellipse that two pairs of con- 

 jugate diameters separate each other, follows when we indicate the 

 direction of the tangents to the binodal lines, by 6, and />,, that of 

 the nodal lines by 7i^ and ?i, : 



Moving in a definite direction round the elliptical point, the succession 

 of binodal and nodal lines is: 



b, b, M, 7/,. 



NoAv the two phases coexisting with A (fig. 1) can : 



1. form a three-phase triangle ABC, so that 

 the two binodal lines lie entirely outside the 

 triangle, and 



2. form a three-phase triangle ABD, so that 

 the two binodal lines lie on one side of A 

 within the triangle. 



As, however, binodal lines within the triangle' 

 indicate two-phase coexistences which are metas- 

 table with regard to the third phase (the three phase equilibrium is, 

 namely, stable inside the three-phase triangle'), it appears tiiat the 

 above mentioned conclusion can also be expressed in the following words: 



1) KoRTEWEG. Arch. Néerl. 24, 57 and 295 (1891). 



2) We assume that there occur no points on the I^-surface where the surface 

 seen from below is concave-concave. 



