( 579 ) 
involution (O,) having O,O, as threefold ray. We now make the 
two involutions projective in such a way that the threefold rays 
correspond, that the group (77s) is conjugated to the group of the 
three tangents and that finally the groups are assigned to each other 
which are determined by the rays to an arbitrary point of C,. The 
two involutions then generate a C, having with the given C, ten 
points in common, thus coinciding with it. 
In each general cubic curve we can thus describe op* bisingular 
heaagons. 
Their threefold points lie on the curve of Hrssx. 
5. If the ray O,O, counted double belongs to corresponding groups 
of the cubic involutions (O,) and (0), these involutions generate a 
C, which has O, and 0, as points of inflection the tangents of which 
meet each other on the curve. For, out of 
a,ty° + Sa ag a, + daga,a,7 + agt, + Avg? = 0, 
bet, + 36,2,72, + 3bo¢,¢,7 + bz,’ + Awr, = 0 
we find 
(avg? +a, rg, HBagrgers Hasan), = (ber, 43, 2,7@,43b97,2,'+b,2,')ao, 
and this is satisfied by 
Pee ande BO, aa =), 
According to the rule found in § 3 0,0, is harmonically divided 
by (,. 
Inversely, when two stationary tangents of a C, intersect each other 
on the curve whilst their points of contact are harmonically separated 
by C,, then those points are threefold vertices of op* bisingular 
hexagons described in C;. 
For, in that case the equation of C, has the form 
(em, + 225) EL + (fe, + fate + favs) 7,027, + (9,7, + goa,)a,* = 0. 
If we replace it by 
: 2 id iy Ghd id id 3 id 
{c,v,° + AF Ze + (Bs + 02,4," + 9420 es + 
bew + frr, + (47, — v)a,2,? + Git, ta, = 0} 
it is evident, that the curve can be generated by the pencils 
en + f,a,7@, + (4f, + @) 2;2,7 + 9,2,° + Az,z,7 = 0, 
Ct, + fits %s BE Dit ER Q) Ty,” he RCN ma Ax yx,” = 0. 
Here we can still replace (4+ 09) by u. 
39* 
