801 
the double points of the involution cut by this pencil into A, A, 
Also the degenerations in this pencil, as the pair of straight lines 
A, A, + SA,, cut a,, in a pair of points of this involution, in other 
words also these points lie harmonically with respect to P and P’. 
The point S", the intersection of A, A, and 4, S, can therefore be 
found by determining tbe fourth harmonical point to P,, (A, A,, A, A,) 
and P’. In this construction it is easily seen that if by means of 
P’ we had determined an S, out of S,, the same point S", hence 
the same point S, would have been found. 
The two points P, and P, corresponding to the same points S of c, 
lie therefore always harmonically with respect to A, and A,. 
Each conic of the pencil (A,, A,, A, S) defines on the straight lines 
P,A,, P,A,, P,A, three points and also on the straight lines P,A,, 
P,A,, P,A, three points, which form together with A,,....,A,, P 
nine base points of a pencil in which appear the degenerations P,A,,P,A,, 
P,A,, resp. P,A,, P,A,, P,4,, with completing conics and where all 
non degenerate curves have a point of inflexion in P, resp. P,. 
4. Out of a complex $,* with 5 base points A,,....,A, a point 
P of A, A, defines a net S, contained in the complex S,* with 
base points A,,.....A,,P. If in S, there is to be a pencil with 
the above mentioned properties, the failing three base points B,, B,, B, 
must be cut into PA,, PA,, PA, by aconic of the pencil (S, A,, A,, A,), 
where S is the point of theconic c, through A,,...., A, belonging 
to =S,". 
By $,° a biquadratie surface ®, with a double conic is represented *), 
where the cubics correspond to plane sections of ®,. The straight 
line A,A, is the image of one of the 16 straight lines of the surface ; 
the plane sections through this straight line p,, correspond to conics 
through 4A,,A,, A, and a fourth fixed point Q. This proves that 
the conic through S, A,, 4,, A, must also pass through Q and we 
must try to find the conic cutting PA,, PA,, and PA, in the points 
B, B, B, among the conies of the pencil with Q, 4,, A,, A, as 
, in a point S to which 
two points P on A,A, correspond. Each curve of the pencil arising 
in this way, must belong to S,°, hence also the degeneration PA, 
with the conic through A,, A,, A,, B,, A,, B, must be a curve of it. 
Now to each conic through A,, A,, 4,, A, corresponds one definite 
straight line through A,, detining a point P’ on A,A,. Between the 
base points. A conic &, of this pencil cuts c, 
1) See among others Sturm, Die Lehre von den geometrischen Verwandtschaften. 
áter Band, S. 309. 
