274 
If P lies on the curve 7%, S& degenerates moreover into the plane 
pencil of the trisecants, lying in the plane (Pe) and a quadratic cone &°. 
This cone contains the bisecants belonging to the second system 
of generatrices of the hyperboloids (6). For they are indicated (ef. § 1) by 
aa’, ai pb" aim ves Siting aa! 4. po", + Ye ln 
a 
they are therefore the bisecants of the curve 7’, which according 
to (7) is determined by 
° ' Ni ' 
ay b wn Cy 
The edges of %* therefore project 7? out of P as centre. 
§ 4. An involution /s, which is produced by the intersection of 
a pencil of curves on a curve of genus g, has 2 (y + s — 1) groups 
with a double point’). In an arbitrary plane lie therefore 14 touching 
bisecants; in other words the tangents r of the curves 9° form a 
congruence of class fourteen. 
If the plane y passes through ec, ten of those tangents belong to 
the plane pencil of trisecants lying in it; they are the tangents at g* 
from the vertex of the plane pencil. The tangents in C, and C, at 
y* are therefore to be counted twice. 
In order to be able to determine the order of the congruence ||, 
we consider the twisted curve containing the ends Q, Q’ of the 
chords lying on the complex cone of a point P. As this cone is of 
order 9, the order of the curve in question amounts to 18; this 
yg’ has evidently nodes in the ends of the triple chord, lying on 
that cone (§ 3) and triple points in C, and C,. 
The planes connecting Q and Q’ with the arbitrary straight line 
/ agree in an involutorial correspondence (18, 18), of which the plane 
(PI represents an 18-fold coincidence. The remaining 18 arise from 
pairs (Q’ (Q, consequently from tangents 7; hence the order of 
[| is eighteen. 
The points C, and C, are singular points of order one. 
1) Ibid. p. 322. 
