( 61 ) 



first doiible-plaitpoint case of Kortkwkc; '). Tlio second case for a 

 doiihle-plailpoiiit, i. e. 4 t^f^j — c/3" rz= 0, does nol occur on llie i|'-siirtace. 



16. Application to a particular ('quatioH. 



In a comuHuiication |)ul)li8lied in the Pi-<)ceedi]i,ü,s of the Academy 

 for 31 Jan. J903, Kortkweg lias determined the })hiit|)oint and critical 

 point of contact for niixtnres with a small proportion of one coniponcnl, 

 but on the assumption that these mixtures satisfy van der Waals' 

 ecpiation of state 



RT rt, 



P — J ^ , 



V — Ox V 



where 



a, = a, (1 -.ry + 2 a,, x (1 - .v) + a, x' 

 and 



h, = h, {l-xf + 2 h,, X {l-.r) + h, .v\ 



The formulae found by Korteweg can be immediately deduced 

 from my formulae, when we introduce the special forms \vliich my 

 coefficients will then assume. 



First we may note that, in this case, the critical constants for the 

 homogeneous mixture are 



8 «i- 1 a,- 



27 1>,R -27 b\r 



1) 1. c. p. 1166. hi using the same method with Korteweg's e(|iuilioii {2), as I 

 have used to determine the critical constants, IJiave found the following expression: 



■'- + ^- = 4..(rf.,-4o,.,) •■"" + ■'•■* ' 



and 



4:d^e\ — 2d,e,e,-}-d\f 



■'■' - ■^■- = 2..(.f.-4c..,) - <"■' + ■'■■> '^' - ■"■' • 



where x^, Xc,, t/i and t/o are the coordinates of the ends of the langenl-chord. 



hi the special case when f/3 = we get 



I e, 1 d^ 



I/.-\- 1/1= —-r- ('^'2 + '^'i) ' -''^ - '^^ = — V ~ (■'■•^ + ■''■1^ ^•^' ~ '''^ 

 4: e, 2 Cj 



and 



1 /d\ 3 e\\ 



<•"■-•"■>' = 2^(7: -^' + ^7:} (■'•'+;«■'• 



By the introduction oi' the above values for the coelUcienls, my expressions 

 for *, ip and f are again found. The first approximation for (/o, Ci and e^ will 

 then be certainly /^vsiitficient in the last expression. 



