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latter four points lie thus with the former on four straight lines through 
S and on four others through 7’; let us now suppose e.g. that 
O,S, passes through S. The tangential plane along the line SS,O, 
to the cone [S| has then as trace with r the tangent in O, to &,’, 
i.e. the line OO,, and from this ensues that the four points S,, S,, 7,, 7, 
are nothing but the points of contact of the tangential planes through 
O of the cones [S] and [7 with r*; however, we know now that 
these four points lie on two lines through B, so in projection 
the points of contact S’,,S’,,7’,, 7’, of the tangents out of the 
cusps not passing through R’ lie on two straight lines through R’, namely 
S’, with T’,, and S’, with T’,.') That they lie also on two lines 
through H follows moreover again from the harmonic position 
with respect to H and A. 
The polar plane of R with respect to the remaining three cones 
is simply the plane STH; it intersects the plane AST in the 
line ST (whose central projection falls on A), and the two lines 
of intersection with u, and u, lying in this plane in two points 
R*, R**, lying on ST, and situated harmonically with respect to S and 
T: R* is then the point of intersection of O,R,, O,R,, R** that 
of S,T, and S,7,. If we transfer these results to the projection, we 
find: of the complete quadrangle O,O, R', R', two of the diagonal 
points are R'; R*' (the third is H), and of the quadrangle S',T',T'S, 
likewise R', R**'; the points R* and R** lie harmonically with respect 
to S' and T'. A similar property of the same points holds if 
we combine S’,,.S', with the cusps and then regard the other four 
points; or finally if we couple 7’, 7’, to O,, O, and join the other 
four to a complete quadrangle. And finally all six points #’,... 7", 
lie on a conic in consequence of the harmonie position with respect 
to h and H, whilst the points of intersection not lying on / of the 
six tangents out of the cusps lie likewise on a conic and at the 
same time in pairs on three straight lines through /. 
4. Besides the group of 8 points just considered consisting of the two 
cusps and the points of contact of the six tangents passing through 
these points, there lie on 4* still an infinite number of other such- 
like groups which have with respect to &', S', 7',H the same 
properties. Let us for instance suppose that the plane u, ($ 3). is 
still passing through RA but for the rest arbitrary, and let us 
suppose uw, again as harmonically conjugate to u, with respect to 
HRS, HRT, then on 7* a new group of 8 points is generated 
1) J. pe Vries, |. c. p. 498. 
dE 
