727 



We consider now the surfaces S"+\ belonging to the rays /and 

 /' ; they have in common the y" lying in the plane iff), and intersect 

 further along a curve o of order (?i'-l-n+r;, which passes through i^'). 



Through a point S. of n pass two curves y", the planes of which 

 contain successively the straight lines ƒ and ƒ'. S is therefore a 

 singular point and lies consequently in oo^ curves y". The planes of 

 these y" form the pencil with axis FS; the curves themselves lie 

 on a ^"+1. which has a node in S; for a straight line passing 

 through S meets -5'"4-J in (??, — 1) points situated outside S. 



Let /" be an arbitrary ray through F, s = FS a bisecant of the 

 curve (J; y" in the plane (/".v) passes through S. The surface 2 

 belonging to ƒ" contains therefore the curve o and the latter is 

 base-curve of the net which is formed by the oo^ surfaces 2. The 

 y" which is determined by an arbitrary point P, forms with a the 

 base of a pencil belonging to the net. 



A y" can meet an arbitrary surface 2^"+\ in singular points S 

 only ; consequently it rests in ?i (;n -f- 1) points on the singular curve 

 ö«*+"+', while its plane cuts 0^ still in the pole F. 



A bilinear congruence [y] consists of the curves y", which cut a 

 twisted curve of the order {n^ -\- n -{-1) in (n -j- 1) points, and send 

 their planes through a fixed point of that curve''). 



The curve o may be represented by 



X 



6» 

 X 



^x 



yx 



=0, 



hence the [^"+1] by 



fyi 



x 



X 



= 0, 



«X ^x yx 

 and the congruence [y] by the relations 



= 0. 



2. The surface ^ formed by tJie y", which rest in a singular 



1) 0- is of the rank n (^n~ -\- n -\- 1) and the genus ^n{n — l)(2w + l); it sends 

 I n2 in" + 1) bisecants through one point. 



2) For' n = 2 this lias been pointed out by Montesano ("Su di un sistema lineare 

 di coniche nello spazio'\ Attl di Torino, XXVII, p. 660 — 690). Godeaux arrived 

 at the congruence [7'*] by inquiring into linear congruences of y» of the genus 

 V2 (w — 1) (>i — 2), which possess one singular curve, on which the >« rest each 

 in w (w + 1) points. ("Sulle congrucnze lineari di. curve piane dotate di una sola 

 curva singolare", Rend, di Palermo, XXXIV, p. 288—300). 



47 

 Proceedings Royal Acad. Amsterdam. Vol. XVI. 



