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point of which radiates a cubic cone of singular rays s*. Since these 
rays are at the sume time coincidences ¢=1', the image of the sheaf 
[7] has at M a singular point of order five. 
Let u be an arbitrary plane, ® the plane through M and au. 
To each ray ¢ of (MZ, P) correspond in u six chords of the particular 
curve of which meets ¢ twice; their points of intersection Q with 
au we conjugate to the point of intersection, P, of ¢ and au. The 
line-complex fu}* which is conjugated to the pencil (Q,u) has 18 
rays ¢ which belong to the pencil (J/, >) and therefore determines 
18 points P. Since Q coincides 24 times with P, u contains an equal 
number of rays ¢’ which are in (ft!) conjugated to rays of [M]. 
Hence the class of the congruence which constitutes the image of a 
sheaf of lines is 24; it is a congruence (22, 24). 
The total of the rays [u] of a plane is transformed into a con 
gruence of which the order evidently is 24. In order to find its 
class we have to bear in mind that a plane @ can only contain 
such rays ¢’ as pass through the point of intersection, Q, of the 
planes a,u and &. The cone with vertex Q of the complex {2}* 
which corresponds to the pencil (Q, «), breaks up in the pencil (Q,u), 
the cubic cone which projects the curve of passing through Q, and 
a cone of degree 14. The last mentioned cone contains the additional 
chords which are sent through Q by the curves of meeting rays of 
(Q,u) twice. The three rays ¢’ in ® which are furnished by the 
cubic cone, are each conjugated to each of the three generators 
lying in w, and are therefore to be counted thrice. It follows from 
this that the class of the congruence is 23. So the image of the total 
of lines of a plane is a line-congruence (24, 28). 
