( 576 ) 
Mathematics. — “On curves which can be generated by projective 
involutions of rays.’ By Prof. JAN pe Vries. 
1. By the symbol 
(a,@, + dgag)™ 
we shall indicate a homogeneous form of order 7. 
By the projective involutions of rays 
(agen + as) 4 A (age, Has) = 0, 
(be, = b,x) +4 (Biz, + Bv) — 0 
a curve Co, is generated in which o' 2n-sides are described pos- 
sessing in OV, and Q, n-fold vertices. For brevity I call such a 2n-side 
bisingular. 
QO, and 0, are n-fold points of the curve. The tangents in 0, 
form a group of the first involution which is conjugated to the group 
of the second containing the ray O,0,. These two groups determine 
a singular 2n-side, where (, replaces }7(n-+-1), and Oy replaces 
4 n(n—1) vertices. 
If we can describe in a Cx, with two n-fold points one bisingular 
2n-side it bears an infinite number of those figures. 
For, if the indicated 2n-side is represented by the two groups of rays 
(az, + agt) el | age (ie -|- bz) ml | 
~ 
and if «,—= ma, is one of the rays of the first group, then the sub- 
stitution must furnish w‚” (bie, + b,ma.)”) = 0, wv, = me,, O, being 
an n-fold point. Hence the equation of (>, must have the form 
(a,2, == at) (3e, se Bw) = (bie, ie be) (a2, + ae) : (1) 
But then the equation can be formed by elimination out of 
(asv, + a,a,)™ + A(a,e, + a,x,)™ = 0, 
: : 2 
(bie, ln boe) ar (3,2, = Boe) == G 
and the curve contains the oo! bisingular 27-sides which can be 
indicated by these two equations, 4 varying. 
2. We shall now investigate under which condition two projective 
involutions of rays will generate a curve Co with three n-fold 
points O;,, so that n? points of intersection of two conjugate groups 
of rays are vertices of three different bisingular 2n-sides having each 
two of the points O7 as n-fold vertices. 
In that case we must be able to bring through the points of inter- 
section of 
