723 
TTS, counted double, whilst the tangents to this conic degene- 
rated into a double line, thus the complex rays in this plane can 
only go through 7, and 7;*; the four planes IT; (i=1,....,4) 
touch 2* along the four lines T; T;* (i =1,....,4), and the 8 points 
fs bee Wir, are nottal. points of .2'. 
The nodal point 7,* lies in t, and is characterized by the property 
that its complex cone breaks up into the plane tr, and the plane 
TXL, so that each ray through 7,* cutting / is a complex ray. Let 
us assume e.g. the plane 17%, 7,*, 7,*; this cuts / in a certain 
point Z and according to the preceding the lines L7,*, L7,*, L7,* 
are complex rays. But if three complex rays lying in one plane pass 
through the same point, then the complex curve in that plane must 
degenerate into a pair of points, and this takes place only for the 
planes through the four cone vertices; so the plane 7,* 7,* 7,* 
passes through a vertex, in our notation 7’. And with this we have 
proved the following property: the eight nodal points of 2‘ can be 
Woede Tanto twargroups. Of jour Ti, suf, and. TE. ran Bn 
and the four tetrahedra having these points as vertices are simulta- 
neously described in and around each other. 
The surface &* is one of those already found and described by 
Prücker in his “Neue Geometrie des Raumes”, Part 1, $ 6, 
p. 193 ete, on the occasion of his general investigations of quadratic 
complexes. 
We shall now intersect the surfaces 2‘ and 2*° with each other. 
The section which must be of order 80 consists in the first place 
of the line / to be counted twelve times, because 7 is for @‘ a 
double line and for 2% a sixfold line; the residual section is thus a 
curve of order 80 — 12 = 68. Now there lie in a plane 4 through 
/ fourteen generatrices of 2*°, and these touch a conic lying on 2%; 
so the residual section is a curve having with a plane À through / 
fourteen points in common. However, we must keep in view that 
the two surfaces touch each other in every ordinary point which 
they have in common outside /; so the residual section must be a 
curve to be counted twice, from which ensues that its order must 
be 34; as it has outside / with a plane 4 only fourteen points in 
common, it must have with / itself 20 points in common. It then 
goes 9 times through each of the four points 7; (¢=1....,4), and 
5 times through each of the four points 7;*(¢—1,....,4) because 
these points are respectively 9- and 5- fold points of 2*° (§ 13) and 
nodal points of @*; the curve counted double has then 18- and 
resp. 10-fold points, as should. 
How does now a point of intersection of the curve found just 
47 
Proceedings Royal Acad. Amsterdam. Vol. XV. 
