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If (c) has the base point B, and B, also the circles c’ pass through 
two fixed points and the image r of (c) lies outside 3. 
If on the contrary (c) has two point-cireles P, and P,, 7’ is the 
intersection of the planes touching 8 at 7”, and 1’, and the image 
r cuts the sphere. 
The image of a pencil of concentrical circles is evidently a straight 
line # through the point N. 
A parabolical pencil of circles is represented by a tangent of B. 
Any two circles of such a pencil touch at a point P; the images 
of two touching circles are therefore joined by a tangent. 
/ 
4. A net of circles [c] is transformed by inversion into a “net” 
[c’]; the planes y’ pass through a fixed point S, consequently the 
images C lie in a plane o, the polar plane of S. 
The image of a net with base-point Pis the plane touching 8 at P’. 
All the circles c cutting a circle c, at right angles, form a net 
[c]; to this belong the points P of c,. As these points may be 
considered as circles touching c,, they have their images in the points 
of contact of the tangents of 8 meeting in the image C,. Consequently 
the net is represented by the polar plane of C,. The images of two 
orthogonal circles are therefore harmonically separated by B. The 
sphere 8 plays here the same part as the paraboloid in the above 
mentioned representation. 
All the circles intersecting c, diametrically also form a net, [c*]. 
As [c*] has no circle in common with the net of the circles inter- 
secting c, at right angles, the image o* is parallel to the plane o of 
c,. To [c*] belongs also the circle c,: hence o* passes through C, 
5. An arbitrary conic d* contains the image of a system (c),, with 
index two; for the tangent plane o of a point A’ has two points 
in common with d and these are images of two circles e through 
the point R. The system (c), belongs to the net [c] which is repre- 
sented by the plane of d’. °) 
If the plane @ touching 8 at FR’ also touches J’, LF’ is the central 
projection of a point R of the curve enveloped by (c),. Now let Z 
be the image of the straight line / in ®; the enveloping cone of 
8 the vertex of which is Z, has four tangent planes @ in common 
N 
1) The orthoptical circles of a pencil of conics form a system (c),. For through 
a point of the straight line at infinity pass the degenerate circles consisting of 
Il, and the director lines of the two parabolas. The point-circles of the system 
are found in the double points of the three pairs of lines and in the centra (the 
orthogonal circle of the net to which (c), belongs) having its centre in the point 
of intersection of the two director lines. 
