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line D,D,, because the plane through this line and ( isa tangential 
plane of [H]; so D,D, is the double tangent of £*, and H is the 
point of intersection of this double tangent with the connecting 
line O,O, of the cusps’). 
Each generatrix of [H] contains two points of r*, lying harmoni- 
cally with respect to the point H and the point of intersection with 
the polar plane RST of H; so if we call / the line of intersection of 
this plane with a, it ensues immediately that each line of x through 
H contains four points of k*, lying harmonically in two pairs with 
respect to H and h; each pair originates from two points on a gene- 
ratrix of [H]. 
If we consider 0,, O,, D,, D, as base-points of a pencil of conics, 
then for each curve of this pencil H is the pole of A; each curve 
containing the cusps and the points of contact of the double tangent 
of k*, it cuts this curve in two more points P,, P,, whose connecting 
line passes through MH. These pairs of points determine on £* a 
fundamental involution in such a way that on each ray through M 
lie two pairs, originating from the two pairs of points of 7* on two 
generatrices of {| situated with O in one plane; the conics of the 
pencil are thus arranged by the rays out of H in pairs of a 
quadratic involution, whose double elements correspond to 0,0, and 
the double tangent d; the former consists of the conic of the pencil 
touching in V, and O, the cuspidal tangents, a curve which together 
with A forms the first polar curve of H with respect to /*; the second 
must break up into the lines 0,0, and d, because this conic must 
touch the line d in D, and D,. By the pencil (H) and the pencil 
of conics (O,,0,, D,,D,) paired involutorily conjugated to it £* is 
generated as the locus of the points of intersection of corresponding 
elements, where besides £* also the line d appears. *). 
However, k* can be generated in still another way. Let us imagine 
through O,O, instead of 2 another plane; this will intersect r* 
besides in O,,0, in two more points P,, P,, whose connecting line 
passes through # and is divided harmonically by these three points 
and the plane RS 7’; so the central projections P’,, P’, are situated 
likewise on a line through M, and lie harmonically with respect 
to H and h. Let us now consider the pencil of conics (0,,0,, P, P’,). 
The different conics of this pencil are likewise involutorily paired by 
the pencil (H); the branchrays are again 0,0, and d, the double 
conics conjugated to them are the conic of the pencil touching in 
1) See the paper of Prof. Jan pe Vries (Proceedings of Amsterdam of Dec. 1908 
p. 499): “On bicuspidal curves of order four”. 
2) J. pe Vries, |. c. p. 500. 
