716 
tetrahedral complex. Now according to one of the theorems of 
HaLPHEN a complex of order p and a scroll of order n have pn 
generatrices in common; the tetrahedral complex is quadratic, the 
surface of tangents of 4* is of order 4 and so the number of common 
rays is 8; so <o=8. From this ensues 2 ef = 8 -- 2% 6 S 20 
«3 = 10. Now e2 represents the class of a plane section of our scroll 
of order 6; by applying the first PLtcker formula for plane curves 
($9) we thus find d=16, a number we can control with the aid of 
op+eg+eB=gh ($9); viz. 
op + 6 41036 
6p == 20; 
and this is twice the order of the nodal curve as we proved in $ 9. 
Summing up we thus find: The nodal rays of our congruence form 
a scroll of order 6 with 8 torsal lines and therefore also 8 pinch 
points lying on a nodal curve of order 10 which is intersected by each 
generatrie in 4 points and having the vertices of the four doubly 
projecting cones of k* as threefold points. 
§ 13. We shall inquire in this $ into the scroll of the rays s of our 
congruence, resting on an arbitrary line / and in particular on a 
ray s. All rays s intersecting / form a congruence (2,2); for the 
quadratic - complex cone with an arbitrary point of space as vertex 
intersects 7 in 2 points, so that through that point 2 rays of the 
congruence pass; and an arbitrary plane contains of the complex cone 
of the point of intersection with / likewise 2 rays, so that in an 
arbitrary plane lie likewise 2 rays of the congruence. An exception 
is made by the points on /, which are vertices of quadratic cones 
of rays of the congruence and the planes through l containing an 
infinite number of rays of the congruence, which evidently envelop 
a conic because two of them pass through any point of the plane. 
Among these planes are four, which are distinguished trom the others, 
because the conic which they bear breaks up into a pair of points, 
and dualistically related to these are 4 points on / whose quadratic 
cone breaks up into a pair of planes; the p/anes are those through 
land the 4 cone vertices, the points are the points of intersection 
of / with the four tetrahedral planes. In the plane /7, eg. 
according to § 10 all the rays through 7, belong to the com- 
plex, so the conic in this plane must degenerate into 7, and one 
other point; or expressed in other words: of the two rays of the 
congruence through a point of this plane one passes through the 
fixed point 7, so the second must also pass through a fixed point, 
