494 
in common with s, the centre S of the involution will, if ois made 
to revolve about s, twice occupy a position on the line s at each 
complete revolution. Hence the chords ¢’ meeting s at a point P 
constitute a cubic cone with s for a double edge. The singular lines 
s are therefore also singular with respect to the involution (t, t'). 
§ 3. We shall now consider the locus W of the curves o*, each 
of which furnishes one ray of the plane pencil (Zr) by one of its 
chords. The curves og* which intersect 8* at B, lie on the hyper- 
boloid 7g? which passes through B. This hyperboloid intersects r 
along a conic, on which the curves op* determine an involution /*; 
the corresponding directing curve is of the third class. Hence the 
pencil (4) contains three chords of curves o* and from this it follows 
that the base-curves 3* and y* are triple curves on the surface ¥, 
The locus of the point-couples Q,Q’ at which the rays ¢ of the 
pencil (Zr) are twice intersected by curves of is a curve t° with a 
triple point 7, which curve contains the points of transit B, and 
Cr (kk: = 1, 2,3, 4) of the base-curves 8* and y*.’) 
A hyperboloid 3? has, in addition to the four points B, three 
point-couples Q,Q’ in common with t°. Hence the rays ¢ establish 
a (3,3)-correspondence between the quadrics of the systems (8*) and 
(y?); in consequence the locus W is a surface of degree twelve. 
Let (Z,4) be another pencil of lines, 4° the corresponding curve 
(analogous to t°). Of the points of intersection of 2° and ¥1? 8 x 3= 24 
coincide with the points Bz, C.; the remaining 36 form 18 couples 
of points, each couple common to a curve of and one of its chords. 
It follows from this that the bisecants of the curves v* of each of 
which a given pencil of lines contains one chord, constitute a line- 
complex of order 18. 
.§ 4. The curve of passing through a point P, assumed in «@, is 
projected from P by a cubic cone, the edges s* of which are suugular 
rays of the involution (é,¢’)..Hach edge is conjugated to every other 
generator of this cone and is therefore at the same time a ray of 
coincidence. The curve t° determined by a pencil of lines (Zr) has 
five points P in common with the line ar; hence each of these 
points furnishes one singular ray s* in the pencil. The rays s* 
therefore constitute a complex of the fifth order. 
In general a line ¢, in the plane a is chord to one of. All chords 
of this v* which meet 4, are in the involution (¢,t’) conjugated to 
t.. To this singular ray t, correspond therefore the rays of a quadric 
regulus and the rays of the two cubic cones which project o* from 
its points of intersection with ¢,. 
1) Le. p. 256. 
