495 
We shall now consider a pencil of lines (7,1). The curve 0 
which has a ray ¢ of this pencil for one of its chords, has six chords 
in «; the points of intersection Q of these six lines ¢, with tr we 
conjugate to the point P=(t, a). To the pencil (Q, @) corresponds 
(§ 3) a line-complex of order 18, constituted by the bisecants of the 
curves of which meet the rays of (Q, a) twice; this complex has 
18 rays ¢ which belong to (7,r), so that to Q are conjugated 18 
points P. Since P thus coincides 24 times with Q, (7,r) contains 
24 chords of curves of, each of which chords meets one of the 
chords of the same of, which lie in «. The curve t°, corresponding 
to (Tr) (§ 3) determines on ear five points P,; the curve o* passing 
through one of these points conjugates three rays ¢, to the ray 7'P,; 
hence ZP, is to be counted thrice in the above-mentioned group of 
24 rays 4. It follows from this that by the transformation (tt) a 
plane pencil of lines is transformed into a combination of jive cubic 
cones and a regulus of degree nine. 
This regulus (¢’)*? has the line ar for its directrix. The ruled 
surface on which it lies has, in addition to the line at and the five 
rays TP,, three more lines ¢’ in common with the plane vr of the 
pencil (7,7). A confirmation of the foregoing result may be obtained 
as follows. In the plane r each curve of determines four points Zi; 
the chords u= Rk, Rk, and v=(R,, R‚) are reciprocally conjugated 
by a quadratic transformation’). If w describes the pencil (7,7), v 
envelops a conic; the point of intersection of w and v therefore will 
thrice reach a position on the line ar. Hence there are three rays 
t of the pencil, of which the corresponding ray ¢ lies in t (and 
does not coincide with 2). 
§ 5. A sheaf of lines with vertex M is transformed by (t, t’) 
into a congruence. A curve of of which one chord ¢ belongs to 
(M)\, has two chords w passing through the point N. To the point 
of intersection, P, of ¢ and « we conjugate the points of transit, 
Q, and @Q,, of the chords u, and u,, Similarly to each point Q 
correspond two points P. If P moves along a line ¢ describes a 
plane pencil; in the complex {uj{'*, determined by this pencil, w will 
then describe a cone of degree 18, Q a curve a'*. Hence P and Q 
are correlated in a (2,2) correspondence of degree 18. Since this 
correspondence in general contains 22 coincidences, the order of the 
congruence which corresponds to [|M | is 22. 
The curves o* which possess a chord passing through J/, consti- 
tute a surface of degree five. Hence in « there lies a curve «* each 
1) Vide my communication: “A quadruple involution in the plane and a triple 
involution connected with it.” (These Proceedings vol. XIII, p. 86). 
33* 
