734 



In a commiinicadoji which appeared in Volume XIY of these 

 Proceedimjs, I have (p. 255) considered the bihnear congruence [i>^], 

 which arises if the quadrics of two pencils are made to intersect. ^) 



It is not difHicult to understand that no bilinear congruences 

 of curves of a higher order can be produced by two pencils of 

 surfaces. For, if these pencils are of the degrees m and n, they 

 intersect an arbitrary line in two involutions of the degrees m and 

 11 and these have in common k=z[m — 1) (yi — 1) pairs; so we find 

 a congruence [(>'""] of the tirst order, and the class {m — l){n — 1) ; 

 only for m = ?z = 2 we find k=zl. 



2. In order to arrive at another group of bilinear congruences, 

 I consider a net of cubic surfaces [<ï*']. Through an arbitrary point 

 P pass 00^ surfaces 'f>^ which form a pencil included in the net, 

 of which pencil the base curve in the general case will be a twisted 

 curve Q^ of ,i»;enus JO. All the curves (/ included in the net conse- 

 quently form a congruence of order one. On an arbitrary line the 

 net determines a cubic involution of the second i-ank; the latter 

 possesses as we know a neutral pair iV,, N^; all the q^ through 

 iV"i pass through N^ as well, consequently the congruence is also 

 of the first clas.^, therefore bilinear. 



If all the <?»' have a curve in common, the curves (>' degenerate 

 into an invariable and a variable part, and a bilinear congruence of 

 curves of a lower order is found. We shall now consider the case 

 in which we have to do with a congruence [9"]. 



3. Let Q^ be a twisted curve of order five, and let the genus be 

 2, so the remaining section of a «f»^ and a ^/>'-, which have a straight 

 line in common. Any surface «f»^ passing through 14 points of (>* 

 contains this curve '^); consequently the 0" passing through q^ and 

 three arbitrarily chosen points H,, H^, H^, form a net. Two of these 

 surfaces have besides q\ a 9" of the 1^^ species in common, which 

 intersects q^ in eiglit points^). With a third ♦?>', ('•* has 12 points in 

 common, of which 8 lie on o% the other four, and to them belong 

 of course H^, H^ and H^ lie apparently on all '/>\ therefore on 

 all q\ 



1) If the bases of the two pencils have a straight line in cominon, one of the 

 two congruences [o*^] found by Veneroni arises. 



-) R. Sturm, Synthetische Untersuchungen über Flachen dritter Ordnung 

 (1867, p 234\ P. H. Schcute, La courbe d' intersection de deux surfaces 

 cubiques et ses degenerations (Archives Teyler 1901, t. VII. p. 219). M. Stuyvaert, 

 Cinq études de geometrie analytique (Mem. Soc. Liége, 1907, t, VII, p. 40). 



3) Schoute, (1. c. p. 241), Stuyvaert, (1. c. p. 41). 



