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by the point on its axis with coordinate z equal to the power of 
the origin O with respect to the circle. 
All circles with radius zero are represented by the points of a 
paraboloid of revolution G (limiting surface) and the images of two 
orthogonal circles are harmonically separated by G. Two reciprocal 
polar lines are the images of two systems of coaxial circles ortho- 
gonal to each other. 
The systems (a), (8), (y) are represented by three involutions 
(A,,A,), (B,, B), (C,,.C,) situated on three straight lines a,b,c. 
The image D, of the circle d,, which intersects @,,8,,7, orthogonally 
is the pole of the plane A, B, C,. So we have now to consider an 
involution (D,, D,) of the points of space, which involution is 
characterized by the property that the polar planes A, and A, of 
D, and D, meet the given lines a,b,c in the pairs (A,, A,), (B,, B,), 
(C,, C,) of three given involutions. 
3. It is now easy to find the singular elements of the involution 
of circles again. In the first place we observe that A, becomes 
indefinite as soon as A, passes through a; for A, now any plane 
may be chosen which contains the points B, and C,, hence for 
D, any point of the polar line a’, of the straight line a, = B, C,,. 
If A, is made to revolve about a, then D, moves along the polar 
line a’ of a, and a, describes a ruled quadric. The line a’, also 
describes a ruled quadric (a',) of which the polar lines 6’ and c’ of 
hb and c are directrices. It is obvious that to every point of a’ a 
definite straight line of (a’,)? is conjugated. Similarly to the singular 
lines 6’, c’ correspond the ruled quadrics (6',)’, (c',)?. 
Secondly D, becomes indefinite as soon as A,, B, and C, are 
collinear and therefore situated on a transversal s of a,b,c. When 
s is made to coincide successively with the generators of the ruled 
quadrie having a,b,c for directrices, then A,, B, and C, describe 
three projective ranges, so that A, osculates a twisted cubic 5°, of 
which the lines a’, 6’ and c’ are bisecants. To every point S= D, 
of this singular curve 0* evidently is correlated a line s’ viz. the 
polar line of the corresponding line s. The lines s’ form a ruled 
quadric (s’)* with the directrices a’, 6’, c’. 
4. If D, describes the line /, then A, revolves about the polar 
line /, so that A,, B, and C, describe projective ranges. A,, B, and 
C, then also describe projective ranges; hence A, osculates a 
twisted cubic 4%, of which a’, 6’ and c’ are bisecants. Consequently 
D, and D, are conjugated in a cubic correspondence. « 
Since 7 has two points in common with (s’)’, 2’ rests on o* in 
two points. The rays of space are in this way transformed into the 
