1064 
14. To each nodal c*‚ of which the node D lies on a we 
associate its tangents dd’, and consider the figure produced by those 
projective systems. As the ct passing through a point P form a net, 
3(n--1) nodal curves of the system in question pass through P. 
The index of the system [d‚d’] is, as appeared above, (4n—5). To 
the produced figure the straight line a belongs six times. So the 
order of the locus of the points /, which c” has moreover in 
common with d,d’, is a curve of order 7 (4n—5) + 6(n—1) — 6 = 
= 4n?-+n-—12. 
For n=3 we find 27; in this case (É) consists apparently 
of 27 straight lines. If I has six base-points, this result is confirmed 
as follows. Each c* passing through 5 base-points intersects a in 
two points D; the lines connecting these points with the 6'h base- 
point form each with c? a c* of T, and belong to (/); in this way 
12 straight lines are found. The connecting line 4 of two base-points 
cuts a in a point D, which determines with the remaining four 
base-points a c°; the 15 lines 4 belong apparently also to (/). 
The curve (B) euts a in (42—7) groups of (n—3) points / arising 
from the nodal curves which have a for tangent in their nodes. In 
each of the remaining intersections a nodal c” has a three-point 
contact with one of its tangents d. From this ensues that the locus 
(I) of the jlecnodes is a curve of order (20n—883). 
In the above case n=3 this figure consists of six conics and 
fifteen straight lines. 
15. The tangents d‚d’ in the nodes of the nodal curves of a 
net envelop a curve of class 3(2—1)(2n—3)"). If the net has a base- 
point B there is a c* having a node in 5. Through B pass then 
3(n—1) (2n—8)—6 tangents d of nodal curves of which the node 
does not lie in B. In order to understand this we consider a net 
of cubics with seven base-points. Through the base-point B pass no 
tangents of proper nodal curves. But the straight line connecting B 
with another base-point 5’, forms with the c° passing through the 
remaining base-points a binodal c*; the straight line 55” represents 
therefore two tangents d. For n= 3 we have 3(n—1)(2n—3) = 18; 
as the 6 straight lines BB’ represent 12 tangents d, the tangents d 
of the nodal c* having its node in B are each to be counted thrice. 
We now consider the system of the nodal curves ec”, which send 
one of their tangents d through P. Any ray passing through P is 
tangent d for (4m—7) curves (§ 12) and is moreover cut by those 
1) Cf. for instance my paper “On nets of algebraic plane curves.” (These 
Proceedings VII, 631—633). 
