( 629 ) 
O,,0, the cuspidal tangents and the one passing through D,, D,. 
The locus of the points of intersection of corresponding elements 
consists of k* and the line PP 
We can finally allow P,, P', to coincide with the cusps, we can 
thus consider all the conics touching the cuspidal tangents in O,,0, ; 
these too are paired involutorily by the pencil (/7) whilst the branch- 
rays are again represented by d and O,0,; to d is again conju- 
gated the polar conic of H, to 0,0, a conic having in each of the 
two cusps four coinciding points with &* in common; so it can be 
nothing but the line O,O, counted double. As the locus of the points of 
intersection of corresponding elements of both pencils appears this 
time besides £° still the line O,0,. 
3. Among the planes of the pencil (O,0,) are of special impor- 
tance those containing the cone-vertices R, S, or 7’; the former 
e.g. is the polar plane of O with respect to the cone [ 2] and contains 
therefore the two generatrices RO,, RO,. Each of these cuts r‘ in 
one point more, eg. Rk, and A; the tangential planes along these 
generatrices to [A] pass through O however; and from this ensues 
that the central projections of RO, and RO,, cutting each other in 
the projection R’ of A lying on A, must touch £* at the points 
R',, R',. We can, however, also bring through O two tangential 
planes to the cones [S|] and [7’], so: out of each of the two cusps 
three tangents can be drawn to k*, the traces of the tangential planes 
through O to the cones [R], |S], [7]; these tangents intersect each 
other in pairs in three points Rk’, S’, T’ of h*) (this also follows 
from the harmonic position of the whole figure with respect to 
Hand h), the projections of the three cone-vertices R, S, T. 
Remark. If we take for 4%, £,? (see $ 1) two concentric circles, 
and if we then project the figure in space on a plane zr (§ 2) which 
is parallel to t, the oval of Descartes is generated and R’, S’, 7” 
pass into the foci. 
The twisted curve r* can be generated in six different ways as 
intersection of two of the four cones; let us take in particular [R 
and | /7]. If we make to pass through the line RH two planes u, u, 
harmonically separated by the planes RHS and RHT, the points of 
the two quadruples of points lying in these planes are situated two 
by two on four straight lines through each of the four cone-vertices ; 
we now take for u, the plane passing through the points O,, O,, R,, R,, 
and we shall call the four points in the other plane S,, S,, 7,, 7,. The 
1) J. pe Vries, l.c. p. 501. 
