( 13 ) 



rirjlit lines of which each cuts three tangents of R" belonging to a 

 same gi'oup of //', form a coii'/r/icncr of irhich (inlcr mid chiss <irr 

 ('(/mtl to {n-2) {/>—!) (/>— 2). 



3. lid MS coiisi(h^i' more closclv lli(> nroiip w Ikm'c (t is a (|()iil)|(' 

 element and ((' one of llie oilier clcincnls. To the jnst-nicnIioncMl 

 congrnence evi(l(Mitl_v belongs Ihe pencil ofravs in llie plane (.1,^/') «,, 

 Avitii vertex A and the pencil of ravs in the osculating |)lanc u 

 Avilh verlex {c(,a^)'^A^. So the congruence conlains al llic Icasl 

 4 (/> — J)(/> — 2) jïeiirils of ra.i/s; each of the 2 (/> — i) sioijuhtr jhiIhIs 

 A is the vertex of (/> — 2) pencils placed in diffei-ent planes; each of 

 the 2 (/> — 1) .WKjiilor iihiiics a bears [p — 'l] pencils with dilferenl 

 Acrlices; on the olliei- hand the 2 {p — 1) (/> — 2) suKjuhd- /ki/'n/s J, 

 and tiic 2 (/> — 1) {/> — 2) .siiKjnlar ////no's a^ om'\\ bear a pencil. 



The complex curve is as appears from the above of genus 

 \ [(2/i— 3) (/>— 1)— 1] [(2/^—3) (y>--D— 2] — (y/,— 2) {2n—T>) (/>-!)*- 

 3(/?, — 2) (/; — J)(/> — 2). For /> =: 3 this bec(n)ies equal to zero which 

 could be foreseen; for, to each point I* of Ihe curve ('"' Iheconnccl- 

 ing line P' P" can be made lo correspond, b\- which Ihe tangents 

 of the complexcurve coincide one bv one wilh I he points of a 

 rational cnr\T. 



In a plane <p through a langent c/' the com[)lc.\cur\(> degeneralcs, a 

 ]iencil of rays the vertex of which lies on Ihe tangent a separating 

 itself from the whole. 



In a plane « evidently {p — 2) pencils of i-ays separate (hemselves, 



4. We shall consider moi-e closely the simplest case, wjiere the 

 complex is determined by a quadratic involution of Ihe tangenis of 

 a skew cubic; /y=3,y>=2. 



If A and B are the points of contact of the tangenis d and l> 

 forming the double rays of the involulion. an<l if a and (i are llie 

 corresponding osculating planes, we assume as planes of coordinates 

 ,j?j = 0, .i'^ = 0, .t\=z{), ,v^=z() successively the osculating plane a, 

 the tangent plane [a,B), the tangent plane (A, /I), the osculatijig j)lane 

 /i. The curve R^ is then represented by 



d\ : ,/■, : ,r, : ,r^ == f -. t' : t : \, 

 and foi- its tangents we have the rclalion 



/>.. W'l. :/>,., :/'.. : 7'.,. :/'.:, — '' ■ -'' ■ '^'^ ■ 1 : -l^^ : /'■ 



The |H)ints .1 and />' being indicated by the parameters / = () and 

 t =:= re, the parameters / and t' of the points of contaci of two 

 conjugate tangents satisfy the relation f -\- I' ^= 0. 



