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fourfold infinity of twisted cubics, which intersect each of the lines 
a’, 6’, ec’ and the curve 6? twice. 
A plane ® is transformed into a cubic surface passing through 
a’, b’, ce’ and o*. The images of two planes have these four lines 
and the image 2° of their line of intersection in common. 
9. A tangent plane of the limiting surface G is the image of 
the circles which pass through a given point. The involution (d,, d,) 
therefore (by § 4) has the following property: A system of coaxial 
circles is transformed into a class of circles with index three. 
This class contains sta circles with radius zero and three straight 
lines. The singular circles form three coaxial systems (§ 1) and a 
class with imdeax three (§ 3). 
To each singular circle a system of coaxial circles is conjugated ; 
these systems form four classes. 
The image of a system of coaxial circles contains eight singular 
circles. 
6. Evidently the representation of the field of circles on the 
points of space enables us to deduce from each involution in the 
latter an involution in the field of circles and vice versa. 
A particularly simple involution is obtained as follows. On every 
ray h which meets OZ at right angles the paraboloid G determines 
an involution of conjugated pairs (P,P’). In the field of circles the 
analogon hereof is the correspondence which conjugates to each 
other two circles intersecting orthogonally and having the same 
power with respect to a fixed point O. 
The point P’, conjugated to P, is the intersection of the ray h 
with the polar plane a of P. If P lies on OZ, then for A may be 
taken any perpendicular to OZ passing through P. Since 2 now is 
perpendicular to OZ, to P will be conjugated every point of the 
line of infinity of z—0. 
A point of G lies in its own polar plane and therefore consti- 
tutes a coincidence of the correspondence. When P reaches the 
vertex of G or the point at infinity of OZ, then P’ is an arbitrary 
pomt of c= 0 or of 2 = ow. 
If P moves along a line /, then 4 describes a ruled quadric g? 
and 2 a pencil of planes projective with 97, so the locus of P’ is 
a twisted cubic 2°. The polar line /’ of / meets 9’ in two points 
P’; each plane through / contains besides these two points still 
another point P’ not lying on /’. Hence / is a chord of 4*. So is 
/, for its points of intersection with G are ¢oincidences. 
7. To the points P of a plane U” correspond the points P’ of 
a cubic surface ¥*. Two such surfaces in the first place have the 
