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three arbitrary points K,L,M; the powers of the circle c with 
respect to these points are considered to be the coordinates «, y, z 
of a point C with respect to an orthogonal system of axes. 
The plane «=O contains the images of the circles passing through 
K. As a pencil of circles (c) sends one circle through A, the image 
of (c) has one point in common with «=O and is therefore also 
in this case a straight line. As further a pencil (c) has one circle 
in common with a net [c], a net is represented by a plane. 
A pencil (c) has two pointcircles; the locus of the images of the 
pointeireles P is again a quadratic surface d°. We find its equation 
by making use of the well known relation between the sides of the 
complete quadrilateral PAK LM.) Substituting there AL* =A, LM*= f, 
Mk? = g, we find after some reduction, 
per + gy? + he? + (h—f—g) HMD) (fg) (ye + fe) + 
+ (g—h—f) (ze + gy) + fgh = 0. 
The plane «0 contains only the image of the point-circle K; 
from this follows that ®* touches the coordinate planes. 
Any circle concentrical with the circle KLM, has equal powers 
relative to K, L and M, is therefore represented by a point of the 
straight line &=y==z; as a concentrical pencil contains only one 
point-cirele at finite distance, ®? must be an elliptical paraboloid 
the diameters of which make equal angles with the three coordinate 
axes. | 
_ If we choose K,/, M in the angular points of an equilateral 
triangle, so that f=g=h, ®° becomes apparently a paraboloid 
of revolution. 
1) See e.g. SALMON-FiepLeR, Anal. Geom. des Raumes | (1879) p. 74. 
