B04 
These points are the images of those points of ®, where there 
passes through each of the principal tangents a plane containing a 
cubic of one of the systems of these curves. 
If we take into consideration that the intersection of ®, with 
the tangent plane at P’ is represented by the cubic that has a double 
point in P, we have here a new proof for the algebraically proved 
auxiliary proposition. 
The point of intersection of m with the common harmonical polar 
line / is the image of the point of contact Q’ witb the tangent 
drawn from P’ to c,”. The double point D’ of c, is represented 
as a pair of points on the straight line m, i.e. as the two points 
on m associated to the curve corresponding to the double curve 
o, of ®,. This pair of points is cut into m by a curve of the pencil 
and lies therefore harmonically with P and (/, m). 
Besides P’ c,” has 2 more points of inflexion, which lie with P’ 
on the same straight line; they are therefore cut into c,” by a curve 
of the pencil (P’,8,’). It appears from this that the corresponding 
points in the image lie also harmonically with respect to P and 
(/,m). The curves of the. net which have double points in these 
two points, must have m as one of the nodal tangents‘). The same 
holds for the straight line n. 
6. We return now to the complex $,* of cubics with 5 base 
points A,,...., A, and suppose that the point P has been construct- 
ed on A, A, in such a way that the straight lines PA,,...., PA, 
are parts of degenerations belonging to the same pencil. 
One of the other two straight lines, parts of degenerations through 
P, always coincides with A, A,; the other passes through a fixed 
point O. The curve which has a double point in P, splits up into 
A,A, and a conic through A,, A,, A,, P and a fixed point Q*). 
According to the above mentioned auxiliary proposition the last 
mentioned straight line through P touches this conic. 
If we suppose that in each point of intersection of A,A, with a 
conic of the pencil (A, 4, A, Q) the tangents to that conie are 
drawn, these straight lines envelop a curve of the 3" class *) to 
1) 3 points have therefore been found on the straight lines m and n, each of 
which straight lines is one of the nodal tangents of the curve of Sz having a 
double point there. Generally five of these points can be found on an arbitrary 
straight line. To the three points mentioned we can add here the two points of 
intersection of m or m with their corresponding conics. 
2) Sturm, l.c. S. 306. 
5) SPORER, , Ueber eine besondere mit dem Kegelschnittbüschel in Verbindung 
stehende Curve’, Zeitschrift für Mathematik und Physik, 38 Jahrgang, 1893, S. 34. 
