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net of which all the circles have in (w,,y,) the power z,° + y,? — z,=0. 
Two pencils of circles are in general represented by two skew 
straight lines. If, however, they have a circle in common their 
images lie in a plane and their four point-circles lie on a circle; 
the pencils belong to a net. 
§ 4. For two orthogonal circles we have d? =r,° + r,°, so 
(a, —a;)" al (b,—6,)’ = (a;*--6,°—+,) > Cae a, 
2a,a, + 26,6, =c, + ¢,. 
For the images we have consequently 22,7, + 2y,y, = 2, + 2,, 
i.e. the images of two orthogonal circles are harmonically separated 
by the limiting surface. 
To the connection between pole and polar plarfe corresponds the 
fact that all circles intersecting a given circle orthogonally form 
a net. 
To the relation between two associated polar lines corresponds 
the fact that pencils of circles may be arranged in pairs, so that 
any circle of a pencil is intersected orthogonally by any circle of 
the other. 5 
To a polar tetrahedron corresponds a group of four circles that 
are orthogonal in pairs. (Of them only three are real). 
or 
§ 5. If the circle C intersects the circle C, diametrically we 
have @ —7r*? — r,? or 
(a,—a)? + (6,— 6) = (a#+-6?—c) — (a,?+6,?—¢,). 
We consequently have for the images 
2x x = 2y iY TI 2,” = 2y,° ae i" 
The circles that intersect a given circle diametrically form a net. 
According to §3 this net has as radical centre }¢—=.2,,;8=%, 
i. e. the centre of C, (which was to be expected), and in that point 
2 
2 
the power 2, — 27, y= TE 
§ 6. The circles touching at a given C,, have their images on 
the enveloping cone of G, which has the image of C, as vertex 
$2). Three enveloping cones have, eight points in common; they 
are the images of eight circles which touch at three given circles. 
The circles touching at two circles C, and C, are represented by 
a twisted curve of of the fourth degree, a net of circles conse- 
quently contains four circles that touch at C, and C,. The enveloping 
cones that have the images of C, and C, as vertices touch at G 
along conics that have two points in common, viz. the images of 
the intersections of C, and C,,. 
me he ee 
