(578 ) 
3. The projective involutions of rays 
(a7, + a) + Avg’ (a,x, + Ge) 
(biz, + be) + Ark (8,2, + Bee) A @ 
generate evidently a Cor, which has O, and O, as (n—k)-fold 
points and as equation 
(asc, + a,2,)™ (Biz, + Bor) D= (be, boz) (a,z, Ha, . (5) 
The two multiple points are for £>>1 of a particular kind. For 
the tangents in QO, are represented by (ar, + 0) = 0, Zand 
each of them has as is evident from substitution (4 + 1) points in 
common with the corresponding branch of the curve. 
For 2, = 0 we find 
atk an—k (a,8,7,*—b,a,0,") = 0. 
Therefore the curve is intersected by O,Q, in a group of the invo- 
lution J; which has O, and QO, as Z-fold points. 
If we can deseribe in a Co 7 with two (n—k)-fold points a bisin- 
gular 2n-side having those multiple points as n-fold points it has an 
equation of form (5). But then it can be generated by two involu- 
tions of form (4) and it bears therefore oo bisingular 2n-sides. 
4. For k=n we find a C, which will in general not possess any 
singular points. Yet it is in general not possible to generate a C, 
by two involutions of rays of order 7. For, the centres O, and 0, 
of the involutions must be n-fold points of an involution T,, of which 
the points of intersection of C, with 0,0, form a group. But then 
the polar curve of 0, would have to have (x—1) points in Q, in 
common with the right line OO, and this is not possible for a 
general Cn. 
But each cubic curve can be generated by two projective cubie 
involutions of rays. Their centres O, and Y, are conjugate points of 
the curve of Hessr, for the two double rays which Q, possesses 
(besides the threefold ray 0,0,), bearing each of them the points of 
contact of three tangents out of O,, form the polar conic of O,, 
whilst the rays which complete the two double rays to groups of 
the involution form the satellite conic of 0, 
Let us now take inversely O, and Q, as two conjugate points of 
the curve of Hesse. We regard 0), as centre of a cubic involution 
which has 0,0, as threefold element, whilst a second group is formed 
by three tangents the points of contact of which lie in a line 7, so 
that their points of intersection with C, are situated on a line s. The 
line counted double and the line s we unite to a group of a cubic 
