( 847 ) 



lhat intersection may take place of — = with - = 0, has 



dx* dxdv 



been proved on pages 834 and 835. 



We saw before that one (/-line may possess 2 or 4 points of 

 contact with /;-lines, but now we have a case in which the number of 

 points of contact can rise to 6. In fig. 14 has been drawn : 1. the curve 

 dp dj> 



— =0 and — = 0, 2. the loop-^-line, 3. a (/-line to which horizontal 



tangents may be drawn in 4 points, and a vertical tangent in 1 point 

 and 4 portions of 6 />lines touching the q-\i\\Q. The pressure in 

 point! is much larger than in 2, rises then, has a maximum in 3, descends 

 again and reaches in 4 its lowest value. The greatest pressure is 

 found in point 5, and in 6 the pressure has been drawn lower than 

 in 5, but it may be higher than in point 1. Consult tig. 1 for the 

 direction of the 7?-lines in the points of contact. These 6 points of 

 contact are again points of the spinodal curve. So on the right there 

 is again a portion of the spinodal curve which follows closely the 



dp 

 line — = 0'in its course, also on the left a portion that does not 



dv 



move far away from this line. But between these two portions the 

 spinodal curve must have been strongly forced back towards smaller 



c/ 2 ip 



volumes to avoid the line = 0. 



da 3 



dp d*ib 



In the points where — = intersects the curve - = the 



dx die* 



d 2 v>Y 



d*ty [dxdv J 



spinodal curve touches this curve, because — - must be == 



dx* d 2 xp 



dv 2 



for the points of the spinodal curve, and so it must remain in the 



^> . . . d*ib 



region where — - is positive, except when — — = 0. It may then 

 dx 2 dxdv 



even be doubted if v > b is found for all the points of the spinodal 



curve. 



Values of v <^b would mean that the left part and the right part 



of the liquid branch of the spinodal line would remain separated 



from each other; and this would imply for the miscibility or non- 



miscibility of the components that at the temperatures for which this 



is the case, even infinitely large pressure would be insufficient to 



bring about mixing. Already in my Theorie Moleculaire I raised this 



problem, and I showed, that if I, is a linear function of x, cases 



58 

 Proceedings Royal Acad. Amsterdam. Vol. IX. 



