272 SCIEXCE PROGRESS. 



presenting the co-existing phases. Points on the i/- x v curve 

 between those points of contact correspond to points on the 

 theoretical isothermal inside the border curve. 



For a mixture ^/^ is a function of v and x and may be 

 represented by a surface the co-ordinates of which are ;//, v 

 and X. Van der Waals proves that co-existing phases are 

 now given by a double tangent plane. The surface presents 

 a plait and in rolling over this plait the double tangent plane 

 traces out on the surface the curve which represents the 

 co-existing phases. This curve is called the connodal 

 curve. Points inside this curve on the surface represent 

 unstable or half-stable conditions of the mixtures, analogous 

 to the points inside the border curve. If the corresponding 

 points of contact approach and reach each other on that 

 part of the surface which is inside the two planes x = o and 

 X = I the plait is said to have a real plait-point ; a plait-point 

 is a point at which the co-existing phases become identical. 

 If we project this connodal curve with its plait-point upon 

 the V X plane we obtain the border curve in the v x 

 diagram, in other words fig. 3. This diagram appears 

 therefore in an entirely new light. P is now the projection 

 of a plait-point and the curve the projection of a connodal 

 curve. P^rom the meaning of P as a plait-point it is 

 clear that we may expect the critical phenomenon to be 

 characteristic of that point. In general it appears that the 

 study of condensation and critical phenomena may be identi- 

 fied with the study of the shape, position, changes and dis- 

 placements of plaits and plait-points on the ;// v x surface. 



We are now able to sketch the method by which to find 

 the border curves say in the / v diagrams for the different 

 mixtures. We must suppose the isothermals of a given 

 temperature for a number of mixtures to have been de- 

 termined. The equation of an isothermal may be written 



in the form p = f {y). As -^ = — /> we can calculate, as is 



shown by Van der Waals, the ?// and draw the ;/- v curves for 

 each of the mixtures. These -^ v curves form together the 

 ^ V X surface. We then apply the double tangent plane and 

 make it roll over the plait of the surface. The double 



