( 34) 



da dh d^a 

 passes really into (1) after substitution of the values of — , — , -— 



QfiC (XtC O/tJC 



d^b 

 and — in accordance with the above hypotheses. 

 da^ 



But the equation (2) may be said to have been derived here for 

 the first time in the above simple form. It is true that van der Waals 

 gave a differential equation of this curve ^), and derived an approx- 

 imate rule for its shape ^), but he did not arrive at a general final 

 expression. Nor has Korteweg arrived at it in his very important 

 papers : "Sur les points de plissement" and "La theorie générale 

 des plis, etc." ') In his final equation (73) (I.e. p. 361) there occur, 

 besides T, still several functions y (u), ?(t'), tp(v) and x(^)> which 

 have been given respectively by the equations (37), (38), (40) and 

 (74) (I.e. p. 350 and 361). Korteweg's equation is one of the 9^^ 

 degree with respect to v, but it is easy to see that it may be reduced 

 to one of the 8^^» degree (1. c. p. 361). It appears from our deriva- 

 tion that this degree may be reduced to the 4**^. In a later paper ^) 

 Korteweg confines himself to a full discussion of the plaitpoints in 

 the neighbourhood of the borders of the ip-surface. 



I think that one of the reasons for failure in this direction is due 

 to the intricate form of the differential equation of the plaitpoint 

 curve, when we use the t|j-function. The S-function on the other 

 hand leads to simpler expressions. Already the differential equation 



for the spinodal line at given temperature , viz. I ^-^ I =0 or 



~\ z=zO, is much simpler than the corresponding expression in ijj. 

 ^^ Jp,T 

 And to get the plaitpoint curve, we have only to combine 



^^ ^ with f ^1 = 0. 



2. We shall now examine the shape of the curves given by (1) 

 and (2) more closely, and specially for the case that £=_0, i. e. 

 6j = 6, = h. The calculations are rendered very simple in this way, 

 and it is obvious from the adjoined figs. 1 — 4, that when 6, is not 

 = èj, so (5 not = 0, the results will be modified only quantitatively, 

 but by no means qualitatively. We shall come back to this in a 

 following paper. 



1) Verslagen Kon. Akad. Amsterdam, 4, p. 20—30 en 82—93 (1896). 



2) Id. 6, p. 279-303 (1898). 



3) Arch. Néerl. 24, p. 57-98 en 295-368 (1891), 

 ^) These Proc. Jan. 31, 1903, p. 445. 



