890 
We will conclude this communication with the schematic P7- 
figure of the phosphorus; the connection between the unary and the 
pseudo-binary system will be treated in a following publication. 
When the calculated critical temperature 422° for liquid white 
phosphorus is correct about 18 atmospheres follows from the vapour 
pressure line for the critical pressure. The critical point is indicated 
by &,, in the drawing. The vapour pressure line of molten red phos- 
phorus exhibits probably a peculiarity that has never been met with as 
yet, viz. two critical points /, and £,, the former of which is metastable. 
Jt is of course also possible even probable that unmixing takes place 
in the pseudo-system between p and q, so in the metastable region. 
The point /, might, therefore, lie at even lower temperature and 
pressure than the point /,. Possibly the continued investigation may 
give an indication with regard to this too. 
It may finally be pointed out that when we apply VAN per Waars's 
De Tz 
log i. ay (= — 1) 
Pp ; 7 
equation, 
= 
and write 
3,94 is found for the value of f. 
This equation does not represent the observed vapour pressure 
line as well as the former, the cause of this may be that f is not 
constant as has been found indeed with several substances. 
Anorg. Chem. Laboratory of the University. 
Amsterdam, Nov. 29, 1912. 
Mathematics. — “On loet, congruences und focal systems deduced 
from a twisted cubic and a twisted biquadratic curve’. I. 
3v Prof. HENDRIK DE VRIES. 
(Communicated in the meeting of November 30, 1912). 
17. If we assume that the line / itself is a ray of the complex 
without however belonging to the congruence deduced from @°, then 
the two surfaces 2°" and @* undergo considerable modifications. 
The surface °° has no lowering of order; instead of the regulus, 
namely, which is the loeus of the rays s conjugated to the points of 
Ll we now have a quadratic cone (passing likewise through the cone 
