(12) 

 Mathematics. — ^'On comjtlc.rcs of r(iys ill fcldtioii to (I rdtivnal 



.s7iV//" CHir<\" \\\ Prof. J. DK VlUES. 



(Communicated in the meeting of April 24, i;X)3). 



1. Siipposiii.ii' llic tangents of a rational .skew cni-ve /i" of degree 

 // to l)e ari-anged in gron})S of an involution //' of degree j^i, let ns 

 consider tiie complex of i-ays formed Itv the common 1 ransversals of 

 each pair of tangents belonging to a group. So this comjdex contains 

 each linear congruence the directrices of which belong to a group of 

 //'. If these directrices coincide to a double ray a of //' the con- 

 üTuence evidently dejicnerates iido two svstems of ravs, viz. the sheaf 

 of ravs with the poijit of contact ..1 of (( as vertex and tlic field of 

 rays in the corresponding osculating ]»lane a. 



To find the degree of ti»e complex let us consider Ihe involution 

 //' of the intersectiojis of the tangents with an ai-bitrary plane r/. 

 The surface of the tangents intersects cf according lo a curAC 6'"' of 

 degree /ii =z 2 {ii — 1) and the comjilex curve of (/ envelopes the lines 

 connecting the pairs PP' of //'. This involution having (/// — 1) (/> — 1) 

 paii-s in common wilh the inxolution forming ihc intersection with 

 an arl>itrar\ pencil of rays, f/ic coiiip/t'.r /s of diujrci' {'1 it — 3) (y>— 1). 



2. We then consider the correspondence between two j)oints 

 (>, U '»f 6'"' situated <»n a right line l\l*' . As Q lies on the lines 

 comiecling any of {ni—2){/t — 1) pairs, there are (//i — 2) {p — 1) {in — 3) 

 points (/. The correspondence {Q, Q') has (//i — 2) (/// — 3) (/> — Impairs 

 in common with fi>, so the com pi even rr<' has 



i {in— 2) {in—Z) {p—tf = {n—2) (2 y/— 5) {p—lY 



dinihi' l>rii(/t'iits, the coinph'.i-roni' as many doulde cdtjCd. 



Kvidentiv these double rays form a coiit/ruence comprised in the 

 com[»le\, of \vliich oi-dcr <ind vhi.ss arc equal to {n — 2)(2yi — 5)(y;— 1)"'. 



The complexcurve also j)ossesses a number of threefold tangents, 

 each containing three points of Ii> belonging to one and the same 

 "•roup. To tlnd this number we make each point of intersection /S' of 

 O" wilh the right line ]*P' to correspond to each point P" of the 

 group indicated by P. To each point P" belong h {[> — 1) {p — 2) pairs 

 P, P', so i (/> — 1) {p — 2) {in — 2) points >S'; each point .S" lies on 

 (^fi, — 2) {p — 1) connecting lines PP', and therefore it is conjugate to 

 („, — 2) {p — 1) {p—2) points P". Every time P" coincides with >S', three 

 }M)ints P lie in a right line and each of those points is a coincidence 

 of the correspondence {P",S); so we iind ^ {in— 2) (/>— i) (p — 2) 

 threefold tangenU. From this appears at the same time that the 



