1260 
and lies in the plane (/’,s,"). Both component parts are variable. 
There are apparently five more systems equivalent to this, each 
determined by a pencil (d*) and a pencil (d). 
The locus of the conics d° is therefore also here of order ten. 
The surface A appears to be of order eight; it has d,, as quadruple 
straight line, each of the six straight lines s as nodal lines. For if 
the complete intersection of two surfaces A is considered, it appears 
that the order w is to be found from the equation 2? — 3u — 40 =0; 
hence «= 8. 
In a plane ® a cubic involution possessing one singular point of 
order four and six singular points of order two is determined by 
this [o*|. It has been described in $ 14 of my paper quoted above. 
\ 
Chemistry. — ‘“Hquilibria in Ternary Systems” XVIII. By Prof. 
SCHREINEMAKERS. 
In the previgus communications here and there some equilibria 
between solid substances and vapour have been brought in discussion 
already ; now we will consider some of these equilibria more in 
detail. We may distinguish several cases according as /’ and G are 
unary, binary or ternary phases. 
I. The equilibrium /+G; F is a ternary compound, G a 
ternary vapour. 
The equilibrium /#’+ G is monovariant (P and 7’ constant), this 
means that the vapours, which can be in equilibrium with solid F, 
are represented by a curve. In order to find this curve we construct 
a cone, which touches the vapourleaf of the S-surface and which 
has its top in the point, representing the & of the solid substance /. 
The projection of the tangent curve is the curve sought for, viz. 
the saturationcurve (P and 7’ constant) of the substance £. From 
this deduction it is apparent also, that this curve is circumphased 
and that we cannot construct from / a tangent to it. 
The equilibrium + G is determined by: 
Wy 
0Z, OZ, 
Z, +-,(a—«,) ane + (B) Oy, = Gai ne Si (1) 
When we keep P and 7’ constant in (1), it determines the vapour- 
saturationcurve (P, 7’) of #. When we assume that in the vapour 
the compound # is completely decomposed into its components and 
that the gas-laws are true, (1) passes into: 
aloga, + Blog y, + (l—a—8) log (lL—#,—y,)=C - - (8) 
