585 
that to every point of such a transversal a definite other point of 
the same transversal is associated, no matter of which degenerate 
o* we consider the transversal to be a component. 
From $ 5 follows further that on each degenerate g° of the first 
series there lies one singular point D’. We shall determine the locus 
of these singular points. 
It appeared in § 2 that to this series belongs a ov? consisting of 
the straight line A,A, and the two transversals 6, and 6, of the 
four lines A, 4,, a,, a, and a,. AS we can combine each of the 
two transversals with the line 4,4, to a degenerate conic £°, there 
lie on this conie “ve singular points D', and D',. 
A plane = through the line A,A, contains one conic &? and con- 
sequently it intersects the locus in question, besides in the points 
D', and D',, in one more point D’; the locus is therefore a twisted 
cubic o°. 
A point D’ is associated to a straight line d, which intersects 
the three lines a,, a, and a,. To the points associated to D’ belong 
therefore three points which lie on the three lines mentioned; con- 
sequently the point D’ must lie on the three surfaces w? found in 
the preceding §. All these three surfaces pass therefore through the 
curve o°. 
§ 8. If a point P describes a straight line Z, the point P’ asso- 
ciated to P describes a curve (/). As the line / has two points in 
common with each of the three surfaces w?, the curve (/) has two 
points in common with each of the three lines a,, a, and a,. A 
surface ,,’ intersects the line / in two points and contains both 
the points of the curve (/) associated to this line, so that in all this 
surface y,,? has stv points in common with the curve (é). For this 
reason (/) is a twisted cubic. 
In general the line / and the curve (l)? have no points in common, 
for as a rule no two associated points of the involution (P, 2’) lie 
on /; for the rest this involution has only a finite number of coin- 
and the points D, found in $ 5, 
etc. intersect the corresponding planes 
cidences, viz. the points A, and A, 
in which the transversals 4,,, 
a,, ete. As a rule therefore the line / does not contain any coinci- 
dences either. 
If a point P deseribes a plane WV, the point P' associated to P, 
describes a surface (V). In order to find the order of this surface, 
we draw in the plane V a straight line /. The curve (/)* associated 
to this line /, intersects the plane V in three points, each of which 
is associated to a point of £. The line / intersects therefore the locus 
