( 33 ) 



The points of contact of the three-yolntecl tangents of (i^") form 

 a surface of order 2(?i — 4) (?z* + 2n' -f 10?z — :12). 



7. Through the tangent .< in 5 to a we can make to pass four 

 tangent planes to the cubic cone of the principal tangents (§ 1). So 

 /S is a parabolic point on four surfaces of the pencil. Therefore ais 

 a fourfold curve on the locus of the parabolic points. 



As the parabolic points of an i^" lie on a curve of order 4?2 (?z — 2) 

 the locus under consideration is cut by each of the surfaces F" in 

 a curve of order 4?i(n — 2) -|- 4?2^ = 8?i (?z — 1). 



The locus of the ])araholic points of the surfaces of a pencil {F'-) 

 IS a surface of order S{n — 1). 



Chemistry. — "On the shape of the plaitpoint curve for mixtures 

 of normal substances." (Second communication). By J. J. van 

 Laar. (Communicated by Prof. H. A. Lorentz). 



1. In a previous paper ^), starting from van der Waals' equation 

 of state, in which b is assumed to be independent of v and T, I have 

 found for the equation of the spinodal curves at successive tem- 

 peratures (1. c. p. 690) : 



^ x{\ — x)6'' + a{v—h) 



RT 



(1) 



and for that of the plaitpoint curve in its v, x projection (I.e. p. 695): 



x{l-x)S' 



(1— 2.r)t' — 3.r(l— .r)/? 



-\-\/a(v—hy 



^x{l—x)d{d—^V^a)-\- 



-f a{v — h){v—U) 



■= 



(2) 



In this <9 = .TT -f- « (^' — b), rr = h^ ^/a, — 6, \/a^, a = ^/'a, — V^a^, 

 and i3 = ^, — ^r 



The equations (1) and (2) hold for the so-called sijmmetrical csLse, 

 where not only b^^ = 7^ (^i + b^) is assumed, but also a^^ = [/a^a^. 

 These hypotheses lead to : 



6 = (1 — x) 6, 4- xb, ; a = [{l — x) [/a^ -\- x |/aj'. 



The equation (1) had been given already before by van der Waals 

 in implicit form '), for after some reduction his general equation 



1) These Proc. April 22, 1905, p. 646—657. 



2) Gont. II, p. 45 ; Arch. Néerl. 24, p. 52 (1891). 



Proceedings Royal Acad. Amsterdam. Vol. VIII. 



