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curve 2? in common, which is the image of the line of intersection 
of the two corresponding planes. In order to obtain a proper insight 
into the meaning of the figure which they have in common in 
addition to this, we observe that the involution (P, P’) is a particular 
case of the following correspondence. 
Let a quadric surface ®° be given and the pair of polar lines 
dd’. Through a point P the straight line ¢ is drawn which meets 
d and d’; the polar plane a of P defines on ¢ the point P’, which 
we conjugate to P. 
The points of intersection of d and # we denote by £,, /,, those 
of d' and @ by E’,, H',. The straight line H, #’', lies in #°; to each 
of its points P evidently is conjugated any of its points. To each point 
of d corresponds every point of d'. Thus all the edges of the tetra- 
hedron ZM, E', E', are singular, so that these six lines are conjugated 
to their points of transit through a plane YY. In addition to the curve 
A? two surfaces ¥* then have these six singular lines in common. 
If ®* now again is replaced by G, then d becomes the axis OZ, 
d' is the line at infinity of z—O and the other four singular lines 
are to be found in the imaginary lines along which G is intersected 
by 2==0 andes. 
8. If P is caused to move along a line /, which meets OZ, then 
h describes a system of parallel lines which is projective to the pencil 
constituted by the polar line of P with respect to the parabola in 
the plane through / and OZ. The points P' now are situated on a 
rectangular hyperbola which by the line at infinity of z= 0 is 
completed to a 2°. 
By the correspondence of the orthogonal circles, which is alluded 
to in $ 6 a system of coaxial circles is again transformed into a class 
with index three. The circles with radius zero are coincidences. The 
two cireles of a pair are real only if they have a negative power 
with respect to 0. When O lies without a circle, then the conjugated 
circle has an imaginary radius. 
