117(1 . 



ill which this line is cut by the Qx — 1) rays />", with which b^p' 

 forms i)airs of the quadratic involution. 



As ^''^~' has apparently («/ — 1) points in common with r, b is a 

 ((Lt — J>fold tangent of the curve (/>)„. Hence b, as common tangent 

 of the curves {p)n and (/?)'„ must be taken into account (fi — 1)- times. 

 The number of singular lines b satisfies therefore the relation. 



:^{ii~iy = n^-i (1) 



The singular lines b are apparently fandamental lines of the 

 birational correspondence {P,p). 



The curves (P)" belonging to the pencils that have R and R' 

 respectively as cent)-es, pass through the point P, which has been 

 associated to the common ray of those pencils. Each point B, which 

 they have further in common has been associated to two different 

 rays p, is consequently a sim/ubn- point of (P') and at the same 

 time a fundamental point of {P. p). 



The [)airs of points (P, /-*'). forming triangles A with B lie on 

 a curve [B], which has H as {ni — l)-fold point if its order is ni ; 

 then we call B a singular point of order m. On this yv/Zw/za/ curve, 

 the pairs (P, P") form a (/uadratic involution, in which B belongs 

 to (//I — 1) pairs; the line p^P F' envelops therefore a curve of 

 involittion of class {m — J). 



From this ensues that />' in the intersection of two curves 

 (/-*)" must be counted for {ni — 1)'' points, so that the number of 

 poijits B has to satisfy the equation 



^ {m—iy — «'—1 (2) 



4. The involution {l^) may also have singular points A, t'ov which 

 the pairs of points (P, P') form an involution P on a line a; the 

 latter is then sinijular for the involution (//) and the pairs {p' , p") 

 belong to an involution of rays with A as centre; a and A we 

 call singular of the first order. The pairs [A, a) are apparently not 

 fundamental for the correspondence {P, p); we indicate their number 

 by a. If ;/=l, as for the involution of Reye, (cf. § 1), then there 

 are only singular points and lines of the first order ; for now ?i^— 1=0. 



Let us now consider the curves ^"+^ and ö"+3 belonging to the 

 lines r and s. A point of intersection P of r with a determines a 

 triangle of involution of which a second vertex P' lies on s ; P" is 

 therefore a point of intersection of s with q. The third vertex P 

 lies therefore on the two curves q and a. They have also in common 

 the pair of points that forms a triplet of the (P') with the point rs. 

 The remaining points of intersection of q and a lie in singular 



