975 



lilies of the plane arc Ihen moreover arranged in a cubic involution. 

 It is further supposed that the points of a triplet are never collinear, 

 the lines of a triplet are never concurrent. 



2. If each point P is associated to the opposite side p of the 

 triangle of involution A which is determined by P, a hiratlonal cor- 

 respondence {P,p) will arise. Let n be the degree of that correspond- 

 ence ; then the points F of a line r will correspond to the rays 

 p of a system with index n, in other words to the tangents of a 

 rational curve {p)„ of class n; the rays /; of a pencil with centre 

 R pass into the points P of a rational curve (P)" of order n. 



Between the points P of /• and the points P*, where r is cut by 

 the lines p, exists a correspondence in which each point 7-^ deter- 

 mines one point P* while a point P* apparently determines n 

 points P. So i/i -f- I) points Plie on the corresponding line p = P'P". 



In that case one of the points P' has coincided in a definite 

 direction p with P, while p has joined with p'. The coincidences oï 

 the involution (P^) form therefore a curve of order (n -\- 1), which 

 will be indicated by 7"+' . In a similar way it is demonstrated that 

 the coincidences of the involution (p^) envelop a curve of class (^??. -f- !)• 



When P describes the line r, the points P' and P" describe a 

 curve of order (n-[-3); for this curve has in common with r the 

 two vertices of the triangle of involution, of which one side falls 

 along r, and the {n -\- 1) coincidences Pz^P', indicated above; we 

 indicate it by means of the symbol p"+^. 



Analogously there belongs to a pencil of rays with its centre 

 in E a curve of class [n -f- 3), which is enveloped by the lines p and p" 

 of the triangles A, of which one side p passes through" R. 



3. The two curves (p)« and {p)'n belonging to the lines r and r' 

 have the line p, which has been associated to the point of inter- 

 section (rr'), as common tangent. Each of the remaining common 

 tangents b is the side of two triangles A, of w^hich the opposite 

 vertices are respectively on r and r' ; h therefore bears a quadratic 

 involution P of pairs {P',P'). 



The pairs {p',2)"), which form triangles «of involution with a 

 singular straight line b, envelop a curve (6). If it is of the class (i, 

 then it has b as {pi — Ij-fold tangent, for through a point b passes 

 only one line p'. We call b a singular line of order fi. The pairs 

 {p',p") form a quadratic involution on the rational curve {b). Its 

 curve of involution /?, i. e. the locus of the point P'=^p'p", is a 

 curve of order (f* — 1) ; for it has with b only in common the points 



