For the net N+! of the curves c’+! lying in the plane 3 the 
locus of the nodes H breaks up into the right line «3 and a curve 
of order (Bn —1). For, «8 forms with each curve J” a degenerated 
curve c*t!, 
The locus of the nodal edges of the cones of the complex is a 
cone with verter A of order (8n—1) having the n? right lines 
AB, as nodal edges. 
§ 5. The tangents in the nodes of a net Ne envelop a curve Z 
of class 3(p—1)(2p —3) '), the curve of Zeurnen. It breaks up 
for the net N+! indicated above; for, the tangents to the curves 
br in their points of intersection with @ envelop a curve, which 
must be a part of the curve Z The pencil (/”) is projective to the 
pencil of its polar curves p”~! with respect to a point V; the points 
of intersection of homologous curves form a curve of order (2n—1); 
in each of its points of intersection S with ag a curve 4” is touched 
by OS; so these tangents envelop a curve Z' of class (2n — 1). 
So for N+! the curve of ZeurHeN consists of the envelope Z/ 
and a curve Z" of class 3n (An —1)— (2n —1)=(8n—1) (An —1). 
The pairs of tangents in the nodes of the genuine curves of 
N+! determine on a right line / a symmetrie correspondence with 
characteristic number (22 — 1) (Bn — 1). To the coincidences belong 
the points of intersection S of / with the curve H; to such a point 
S are conjugated (2n — 1) (8n —1)— 2 points distinct from S; so 
S is a double coincidence. The remaining 4 (n — 1) (Bn — 1) co- 
incidences evidently originate from cuspidal tangents. 
The locus of the vertices of cones of the complex, possessing a 
cuspidal edge consists of 4 (n— 1) (Bn — 1) edges of the cone A. 
A general net of order (7+ 1) contains 12 (7 —1)n cuspidal 
curves, thus 4(2—1) more;- therefore each of the 2(mn—1) 
figures consisting of the right line «ap and a curve 4” touching it 
is equivalent to two curves c’t+! with eusp. Evidently the nodes of 
these figures form with the point C, the section of «8 with the 
curve H. 
§ 6. On the traces of a plane a with the planes « and @ the 
pencils (a) and (/") determine two series of points in (/, 1)-corre- 
spondence; the envelope of the right lines connecting homologous 
points is evidently a curve ef class (7+ 1) touching ez in its point 
of intersection with the ray a conjugate to the curve 6" through 
1) This has been indicated in a remarkable way by Dr. W. Bouwman (Ueber 
den Ort der Beriihrungspunkte von Strahlenbüscheln und Curvenbiischeln, N. 
Archief voor Wiskunde, 2nd series, vol. IV, p. 264). 
