END 
( 633 ) 
Let us consider in particular the points O,, R, (§ 3), lying on a 
straight line through A, and let us remember that the tangent in O, 
to r* passes through OQ; we shall then find on OO, a point of the 
double curve of order four, whilst the tangent in that point is the 
line of intersection of the osculating planes of 7* in O, and R,; 
that one of O, passes through QO and furnishes in projection the cus- 
pidal tangent in O, to &*; so the central projection of the double 
curve will contain the cusps of 4*, and will touch the cuspidal 
tangents here. 
So, summing up we find: the point of intersection of the tangents 
to k* in two conjugate points of the involution generated by the pencil 
(B) generates a curve of order four with double points in S', T', H, 
situated harmonically with respect to h and H and having with kt 
m common the tangential points of the four tangents out of R' not 
passing through the cusps and likewise the cusps and the tangents 
in these. 
The new curve of order four cuts £* besides in the cusps (count- 
ing together for six points of intersection, and the tangential points 
of the four tangents out of 2’ not passing through the cusps, in 
six points more, of course again situated two by two harmonically 
with respect to h and H; now these six points lie twice with two 
of the four above mentioned points of contact (those namely whose 
connecting line passes through //) on a conic. For, the conic through 
two of those points of contact touching in QO, and QO, the euspidal 
tangents of £*, contains 2 XX 3 + 2 = 8 points of intersection of 
the two curves of order four; so the other eight must also lie on 
a conic. 
Also to the points S’ and 7” belongs such a locus of order four ; 
as to the point H we can observe that in space the locus belong- 
ing to H lies in the plane ARS 7’ and passes through O, so it 
passes by central projection into the line A counted three times, 
because each line passing through O and lying in the plane RST 
intersects the curve besides in O in three more points ; indeed, through 
each point of / pass three pairs of tangents io /*, whose chords of 
contact pass through #7. The central projection of O itself however is 
undetermined, and so / is covered with points for the fourth time; 
the locus belonging to AH consists thus of the line 4 counted four times. 
Remark. The properties found in this § hold in a some- 
what more general form also for £* with two nodes, because they 
have been simply arrived at by centrally projecting the complete figure 
of the tetrahedron of Poncerer; the points #’, S', 7", H have then 
however not such a simple position as for the bicuspidal 4. 
43* 
