799 
the above mentioned property; then it is easily seen that every 
time (besides the point P) two of the 9 base points of the pencil 
containing the degenerations, must lie on the straight lines PA,,... PA). 
For arbitrary situations of A,,..., A, p is therefore at most equal 
to 4; for p=5 P must lie on one of the joins A; Ar. If for 
p44 A,,...,A, are the base points of S,, B,,..., B, the other base 
points of the pencil, lying on the straight lines PA;, a straight line 
PA, must be completed by a conic through A; A; 4, B; B, Bn 
(7,4,mfFk); the polar straight lines of P relative to these four 
conics coincide in a straight line 7 and all the non degenerate 
cubics of the pencil are cut by P4A,,..., PA, in points lying har- 
monically with respect to P and /, in other words all the cubics of 
the pencil have P as an inflexional point and have a common har- 
monical polar line |. *) 
3. We shall now investigate the case p=5 more closely. With 
a view to this we shall start from the system S, with 6 base 
points P, A,,...,A,, where P, A, and A, lie on the same straight 
line. Now it will be possible that S, contains nets without other 
base points, so that the degenerations formed by PA,, PA, and PA, 
together with completing conics belong to one pencil. The situation 
of the other base points 4,, B, B, on the straight lines PA,, PA,, 
PA, can be determined. 
For the system S, represents a cubic surface ® with a double 
point O; PA,, PA, and PA, correspond to 3 straight lines p,, p,, 
ps of ®, which do not pass through O; a net out of S, without other 
base points corresponds to the plane intersections of ®, with planes 
of a sheaf the vertex Q of which does not lie on @,; the pencil 
to which the degenerations PA,, PA,, PA, belong, is the image of 
the intersections with a pencil of planes in (Q), which must also 
contain the planes (Qp,), (Qp,) and (Qp,). The axis of this pencil 
of planes must therefore cut p,,p,,p,, in other words, Q lies on 
the quadratic scroll R, having p,, ps Pp; as directrices. 
Generatrices of R, are among others the straight line p of ®, 
represented by the point P in the plane, and the straight line q 
corresponding to the conic c, through A,,...., A, 
If we project all the generatrices of R, out of O, there appears 
1) S. Kantor, , Ueber gewisse Curvenbtische! dritter und vierter Ordnung”, 
Sitz. ber. Akad. d. Wiss. in Wien, Bd. LX XIX (1879). See also H. J. van Veen, 
„Eigenschappen van bundels van vlakke kubische krommen by algemeene en by 
bizondere ligging der basispunten’, Nieuw Archief voor Wiskunde, 2e reeks, dl. 
XII, 1918, p. 279. 
