1325 
with the cone projecting d out of Z. From this it ensues that the 
system (c), is enveloped by a curve of the fourth order. 
The points of intersection of d’ with 2 are the image of four 
point-circles belonging to (c),; the points of intersection of d? with 
the plane » represent the two straight lines of (c),. 
6. A twisted cubic d* is the image of a system (c), with inder 
three. At such a system we can arrive in the following way. Let 
us consider three projective pencils of circles (c,), (c,), (c,) in ®; 
let c be the circle intersecting the homologous circles c,, c, and c, 
at right angles. The image C of c is the point of intersection of 
the polar planes y,, y,, 7, of the images C,, C,, C,. These planes 
revolve round the polar lines 7,,7,,7, of the straight lines 7*,, 7*,,r*, 
containing the images C,, C,, C,; the locus of the point C is accord- 
ingly a twisted cubic, d*. Apparently we can inversely choose for 
”,7,,7, three arbitrary chords of a given curve d*; their polar 
lines with respect to 8 define in ® three projective pencils of circles, 
which in their turn define the system (c), which has d* its image. 
7. A plane curve d* is the image of a (c), belonging to the net 
that is represented by the plane d of d*. A tangent plane o of 8 
intersects d* in three points of the straight line dg; as a second 
tangent plane to 8 can be passed through this straight line, the 
system (c), is characterized by the property that the three circles 
through a point P have another /* in common. If do is a tangent 
to 8 the three circles touch in a point P; this point of contact lies 
evidently on the orthogonal circle (diametrical circle) of the net. 
In a special case the orthogonal circle can be replaced by a straight 
line, containing in this case the centres of the circles of the system (c),. 
8. Let O be the centre of the sphere 9. If the images C, and 
C, of two circles c, and c, are such that “OC,C, is a right angle, 
c, is intersected diametrically by c, ($ 4). If C, is fixed C, remains 
on the sphere 7” having OC, as diameter. This sphere is apparently 
the image of the twofold infinite system of circles c that are inter- 
sected diametrically by the fixed circle c,. The intersection of two 
tangent planes of 3 has two points in common with J"; hence through 
two points there generally pass two circles of the system. A pencil 
of circles contains also two circles of the system. 
9. We arrive at another representation of the field of circles in 
the following way. In the plane ® of the field there are assumed 
