( 632 ) 
the four points of intersection of 7* with 9, whose tangents pass 
through FR, and also the points harmonically conjugate to A with 
respect to the pairs 0,,R, and Q,, R, lying in the polar plane 
O,0,R of O with respect to the cone | F]. 
By passing to the projection we find out of this the following 
property of k*: through the points of contact of the four tangents 
out of R' not passing through the cusps, a conic r* can be brought 
touching the two tangents which do pass through the cusps in the 
points harmonically conjugate to R' with respect to the cusps and the 
corresponding points of contact; k* now hes harmonically with respect 
to R' and r® in such a sense that the four points of intersection of 
k* with an arbitrary ray through R' arrange themselves into two 
pairs, each lying harmonically with R' and one of the points of 
intersection of that ray with r*. 
Of course also the points S’ and 7” possess such a conic, but the 
point H likewise, namely the line 4 counted double; for, the polar 
plane of H with respect to r* is the plane RST, and the section 
with the cone [MZ] lying in this plane is projected out of O into 
the line A to be counted double. 
With the aid of the conic 7” it is easy to deduce out of one point 
of a group the seven other ones. If point 1 is taken arbitrarily, we 
find four others by the harmonic position of 4* with respect to the 
points &',S', 7’, H and the corresponding conics; on the line R1 
e.g. are lying besides 1 still 3 points of £*, but among these is only 
one forming with 1, R', and one of the points of intersection of R'1 
with 7? a harmonic group. The three then still missing points we 
find by simply combining in proper fashion the already found one 
Withee. is oor i, 
Just as in §2 by rays out of H now, too, a fundamental involution 
is generated on &* by those out of R' (or S, or 7") in such a 
manner that on each ray lie two pairs, originating from the two 
pairs of points of 7* on two generatrices of the cone [Zi] lying with 
O in one plane; each pair is harmonically separated by A’ and one 
of the two points of intersection of the indicated ray with #°. If 
now in two points of 7* lying on a straight line through A we draw 
tangents, then these intersect each other in a point of the plane 
STH, and the locus of this point of intersection is a plane curve 
of order four, double curve of the developable of r*, with double 
points in S, 7, H and having with r* the four points of intersection 
of r* with the plane S7'H in common. 
1). J. pe Vries, |. c; p. 498. 
