the system LH G@. The appearance of such a point has no influ- 
ence on fig. 8 unless this accidentally coincides with the point /’ of 
one of the previously examined figures. Such a singular point, that 
at each 7 occurs only at a definite P, proceeds in the component 
triangle along a curve which may happen to pass through /. If 
this should take place, and if this point is a statonary point, then, i 
the case of the correlated P and 7, the vapour and liquidum line 
of the heterogeneous region Li+ G and the theoretical liquidum 
vapour line pass through #'; if this point is a maximum or mini- 
mum one these three lines coincide in /. From this it follows that 
in fig. 3 the singular point must always lie simultaneously on the 
lines g’ Sg, e’ We and f Kf. The coincidence of a singular point 
with the point /’ therefore causes the above three curves of fig. 
to have one point in common; from other considerations it follows 
that they get into contact with each other. 
This point of contact may lie in the solid as well as in the liqui- 
dum-gas region; in the first case, the system liquid # + vapour F 
is metastable, in the second case it is stable. 
This point of contact may — but this is not very likely — also 
coincide with point S of fig. 3. The system solid / + liquid # + 
vapour /’ would then occur in the stable condition and the subli- 
mation and melting point curves would then continue up to the 
point JS. (To be continued). 
Mathematics. — “On complexes which can be built up of linear 
congruences’. By Prof. JAN DE Vries. 
(Communicated in the Meeting of December 28, 1912). 
§ 1. We will suppose that the generatrices « of a scroll of order 
m are in (1,1)-correspondence with the generatrices 6 of a scroll of 
order 7, and consider the complex containing all the linear congru- 
ences admitting any pair of corresponding generatrices a,b as direc- 
tor lines. The two scrolls admit the same genus p; as the edges of 
a complex cone are in (1,1)-correspondence with the generatrices 
a,b on which they rest, p is also the genus of all the complex 
cones‘). The rays of a pencil are arranged in a correspondence 
(m, n) by the generatrices of the scrolls (a), (6); so in general the 
complex is of order m + n. 
1) For m = n = 1 (two pencils) we get the tetrahedral complex. In a paper 
“On a group of complexes with rational cones of the complex” (Proceedings 
of Amsterdam, Vol. VII, p. 577) we already considered the case of a Pea in 
(1,1)-correspondence with the tangents of a rational plane curve. 
