43 



three straight lines F^F^, F,F,, F,Fi, three trisecants, consecutively 

 passing through F^, F^, F,. 



The three straight lines t, meeting in an arbitrary point P, are 

 nodal lines on the surface /7", containing the points of support of 

 the chords drawn through F of the curves of the [?']. With the 

 cone which projects ihe q" |)assing through P, n' has, besides 

 this Q>S only straight lines passing through P in common ; they 

 are the three trisecants out of P, which are nodal lines for botli 

 surfaces, and the seven singular bisecants PFjc- From the con- 

 sideration of the points which /Z' has in common with an 

 arbitrary 9* follows that this surface has nodes in the seven funda- 

 mental points. 



For a point S of the singular quadrisecant 11' passes into the 

 monoid 2'. 



Mathematics. — "Bilinear congruences of elliptic and hyperelliptic 

 tioisted quintics." By Prof. Jan de Vkies. 



(Communicated in the meeting of April 23, 1915). 



1. We consider a net of cubic surfaces «ï»' of which all figures 

 have a rational quartic, 0', in common. Two arbitrary «/'' have 

 moreover an elliptic quintic q^ in common, resting on öSn <e?i points. 

 A third surface of the net therefore intersects 9', outside o\ in Jive 

 points Fk ; they form with 0' the base of the net. As a *' passing 

 through 13 points of 0" wholly contains this curve, only four of the 

 points Fk may be taken arbitrarily for the determination of the net. 

 The base-curves q' of the pencils of the net form a. bilinear con(/rue}ice, 

 with singular curve 0* and jive fundamental points Fk. 



The singular curve a* may be replaced by tlie figure composed of 

 a 0" with one of its secants, or by the figure composed of two conies, 

 which have one point in common, oi' by the figure consisting of a 

 conic and two straight lines intersecting it. 



2. The curves q\ which intersect 0' in the simjular points S, 

 form a cubic surface 2£', with node <S, which belongs to the net ; 

 .S' is therefoie a singular point of order three. The monoids 2* 

 belonging to two points S have 0* and a curve p' in common ; 

 through two points of 0* passes therefore in general one curve (/'. 

 The groups of 10 points which n* has in common with the curves 

 of the congruence form therefore an involution of the second rank. 



