( 631 ) 
Jying in pairs on four lines through each of the 4 vertices, and whose 
central projections thus possess the same property with respect to 
the points R’,.S’, 7’, H. If we divide the 8 projections into two 
quadruples, in such a way that one belongs to the four points of 
u,, the other to those of wu,, then the two quadruples form two 
complete quadrangles with the common diagonal points R’ and M, 
whilst the others, R*’ and R**’, lie harmonically with respect to S’ 
and 7”; the pairs of points R*’, R**’ on h form therefore a quadratic 
involution with the double points S’, T'. Similair properties hold 
for the two other possible divisions of the group of 8 points into 
two quadruples, namely with respect to the points S and 7’. 
A special group of 8 points is found by choosing for the two 
planes w the tangential planes through the line RH to the cone 
[H], for these are likewise harmonically separated by HRS, HRT, 
but they furnish instead of 8 points 4 pairs of coinciding points of 
r*, namely the points of contact of the 4 tangents out of A to 7“. 
These points of contact lie in the polar plane o($3) of A and on 
two generatrices of the cone [$S], and likewise on two of the cone 
[7']; the tangents themselves pass in projection into the four tangents of 
k* through R’ not passing through the cusps, so: the pomts of 
contact of the four tangents of k* through R’ not passing through 
the cusps are the vertices of a complete quadrangle whose diagonal- 
points are the points S’ 7’, H; the corresponding points A’, R**’ 
are the points of intersection of the two sides of that quadrangle 
passing through H with h. 
Another special group of 8 points is generated if we choose for 
the planes u the tangential planes through RH to the cone [ZR]; 
we then find the four points of 7* in the plane RST, and therefore 
in projection the points of intersection of r* with A, whose tangents 
indeed pass through H, in consequence of the harmonic position of 
k* with respect to h and H. 
5. A group of 8 points of &* must be determined by one of these 
points; for the connecting line of this point with O intersects 7* in 
one point, which determines with the line AA the plane u,; and 
by u, at the same time u, is determined. Planimetrically we can 
deduce out of one point of a group the other ones with the aid of 
the following property. The cone [RR] intersects the plane ge = STH 
in a conic 7’, and we find that 7* lies harmonically with respect 
to this and the point &, in that sense that the two points of r‘ 
on a generatrix of |R] are always harmonically separated by 
R and the point of intersection with 7’; in particular 7’ contains 
43 
Proceedings Royal Acad. Amsterdam. Vol. XI. 
