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tangents 7,,8,,t, are projected out of the point H into the tangential 
moms Res Ss de 07 whestangentsnns sat. 
If S’ corresponds as a double-point of the (2,2) to S=s,s,, then 
it follows from 
O(RR'SS') = O,(RR’SS’), that we have O,(RR’SS’) = O,(R’ RS'S). 
From this follows that the points R,, R,, S,,.S, are connected with 
ORO mby a. conte. Also “the (groups 05, O.R, Roden 
OPO Tar hie on Jconies: 
If K=hk we find out of 5 
OO int ONO MC REN OF CK OF RA) 
that through A, and A, passes a conic which is touched in O, and 
0, by the cuspidal tangents. The pairs of points SS, and 7’, 77. 
procure two analogous conics. 
If two arbitrary points X and Y of h are projected out of O, 
and O,, then the points (O,X, O,Y) and (O,Y, O,X) lie in a right 
line through #. 
From this follows that A bears three right lines which contain 
successively the pairs of points 
aa | o's i. | 
> =r 
En 
Lik 
Above we found that these six points lie on a conic and form 
two hexagons having 0, and QO, as point of BRIANCHON: it is now 
evident that they determine a third hexagon, having A as point of 
BRIANCHON. 
d. From (Ar, st) = (4r,5,7,) follows 
(Ar,s,t,) = (r‚kt.s,) == (s,t,47,) = (245,74). 
So we can bring through O, and O, three conics Q,, 6,,T, with 
respect to which the line & has as poles the points &, S, T, whilst 
containing successively the pairs of points 3,6; 2,5 and 1,4. 
On these three conics the pencils (O,) and (O,), arranged in (2,2) 
determine, just as on A, involutory correspondences (2,2); for, the 
two systems of points generated on them have again the branch- 
points in common. 
If M,, M, is a pair of the (2,2) determined on g,, then the points 
(O,M,, O,M,) and (O,M,, O,M,) lie on C, and in one line with 
the point R, namely on the polar line of the point (M/‚M,, O,O,) with 
respect to @,. 
The pencils with vertices A, S and 7 generate therefore on C, three 
more fundamental involutions of pairs of pomts where again each 
