23 



cuts t\A'o rays of each of the scrolls {ijciY, (>' lias the straight lines r/jt 

 as chords; it rests in two points on 0' because r meets two rays/. 



Let us now consider the bilinear congruence of rays [r] which 

 has two of the straight lines t an directrices. Throu^ii the cubical 

 transformation it is transformed into the congruence [(>^] of which 

 the curves ^' |)ass through two fixed points *S\ and «Sj and have 

 the three fixed straight lines a,,a^,a^ as bisecants'). 



Inversely any congruence [9*] with two base points *S\, *S, and 

 three fixed bisecants ak can be represented on a bilinear congruence 

 [r]. With a view to this we take two transversals t^, t^ of the 

 straight lines ajc and we define the involution of planes through ak 

 by associating the planes {auS^) and («/caSJ to the planes {aut^) and 

 [ükU). 



^ 2. The curve 9" degenerates as soon as the ray r rests on one 

 of the singular lines 0* or aic- 



If r passes through the point .S of 0* its image is composed of 

 the straight line t associated to S, and a conic 9" through ^Sx and aS', 

 cutting «!,«, and a^. The locus of the conies (>' is the dimonoid 

 of the fourth order, A^ which has threefold points in S^ and aS,, 

 contains the straight lines a^ and has a double torsal straight line 



The image of A' is the scroll {rY with directrices q\ ty and i^, 

 where t^ and t^ are threefold, which has the straight lines a^ as 

 double generatrices. This may be verified by combining (?')" with 

 a curve fi', which is the image of a straight line ni. 



If the ray r is to rest on «,, it must belong to one of the plane 

 pencils having the points B\ = a^ t^ or B\ ~ a^ t, as vertex and 

 belonging to the bilinear congruence of rays (J,l). The former 

 plane pencil lies in the plane B\t^; the image of this plane is the 

 scroll (/„)' combined with the plane S^a^. For [(>'] there is found 

 from this a pencil of conies which have aS, and the intersections of 

 a, and a^ with the plane aS, a^ ase base points. The fourth base 

 point is the intersection with the straight line 6',,, which, as a 

 transversal through aS, of a, and a,, is the image of the point B\. 

 Here we have therefore a group of degenerate figures each consisting 

 of the straight line 6',, and a conic of the pencil in question. 



1) This congruence has for the first time been investigated by M. Stuyvaert 

 (Dissertation inaugurale, Gand 1902). A different treatment of the "Congruence of 

 Stuyvaert" is found in the thesis for the doctorate of J. de VSies, Utrecht 1917, 

 where also the literature on bilinear congruences of twisted cubics is mentioned. 



