586 
of the pairs of points (P, P) lying in the plane V, in three points; 
consequently this locus is a curve of order three. The plane V con- 
taining as a rule no coincidences, this curve is the complete inter- 
section of V with the surface (V), which for this reason is a surface 
of order three. 
The surface (V)* contains the lines a,, a, and a,, for each point 
of one of these lines is associated to a line }, which cuts the plane 
V in one point. In the same way the surface (V)* passes through 
the curve 6’. 
Let Q be a point of the plane V, / a straight line of V passing 
through Q. This line contains three points P, associated to a point 
P' in V. If we connect these points P' with Q and associate these 
lines of connection to the line /, we get in the pencil of the rays 
through Q a correspondence (3, 3) with siz coincidences. These must 
originate from the rays (P, P') passing through Q and each of these 
rays furnishes two coincidences, as the correspondence (P, P’) is 
involutory. Through Q pass therefore three rays PP’ which lie in 
plane V and accordingly the lines PP’ as a rule form a cubic 
line complex. 
§ 9. Singular straight lines of the involution (t, t’). We now proceed 
to the consideration of the involution (¢, ¢’) and first investigate its 
singular rays. 
The line A, A, is bisecant of all the curves 9%. On an arbitrary 
curve @° each of the two supporting points A, and A, coincides 
with its associated point; in this case the line A, A, is associated 
to itself. 
According to § 7 the line A, A, is a component of one degenerate 
e® and as such contains two singular points D,’ and D,'; to these 
points correspond all points of the two transversals 6, and 5, of the 
lines A, A,, a,, a, and a,. If we consider the points D,' and D,' as 
supporting points, there is associated to the line A, A, a belinear 
congruence of rays which has the lines 6, and 5, as directrices. If 
we consider one of the points D,' and D,' and one arbitrary other 
point of the line A, A, as supporting points, we find that to the line 
A, A, there are moreover associated two fields of rays lying in the 
planes which connect the lines 6, and 6, with the line A, A. 
Also the line a, is bisecant of all curves 9’. The supporting points 
EF, are each time the two points of intersection of the line a, 
with a surface g?,,. The points 4’, and F”’, associated to these, lie 
on the generatrice 4 and u of the surface w’ corresponding to /, and F,. 
Through each pair of points (£,, /,) pass o'* curves g°; the cor- 
