735 



Plere we have eonse(]iiently a bilinear congruence [(/] with four 

 cardinal points B^ and a singular curve (^/ ; i.e. all (>* pass through 

 the four cardinal points and rest in 8 points on (>* ^). 



4. Let t be a triseeant of q' ; the pencil of net surfaces determined 

 by a point of t has for base the complex of (/,t and a plane cubic 

 y\ which has a point T with t in common, and 5 points with q'\ 

 This v' must contain the four cardinal points H; consequently the 

 cardinal points are situated in a plane <( . 



Any curve y' connects the 4 cardinal points and the 5 points 74, 

 in which q^ cuts the plane (p, with the intersecting- point 7' of the 

 triseeant belonging to it. As the trisecants form the quadratic ruled 

 surface <P\ on which (>' lies, the points R, together with T may 

 be connected by a conic r^ 



The curves y'' form a pencil with base {Rk, Hk) ; any y" intersects 

 x" in the point 2\ through which the straight line t passes, which, 

 considered together with f belongs to the congruence \j^*'\ '). 



The locus of the degenerate figures (y"' -\- t) is apparently the com- 

 plex of *P^ and (p, and consequently belongs to the net [r/»']. 



5. Let b be one of the four bisecanis of (>', which pass through 

 the cardinal point Hie. All the ^P"^ which contain b, have moreover 

 a Q^ in common, which has /; as bisecant and rests in 6 points on ^*. 



Consequently there are sixteen figures (9^ -f- b) in [(»^J. 



A third group of complex figures is formed by pairs of conies 

 («^ /i^). Let «^ be a conic passing through i/j, H^, intersecting ^'^ in 

 4 points, the </*" passing through cr and q^ have an other conic pMn 

 common, which intersects «^ in 2 points, q"" in 4 points and passes 

 through //a, H^ 



The number of <r we deduce using the law of permanency of the 

 number. We replace q^ by the complex of a 0^ and a 0*, which 

 have three points in common ; through a point P pass consequently 

 3 straight lines, which rest on 0^ and 0'^ ; with the bisecant of 0' 

 they form the 4 straight lines which replace the 4 bisecants of 9* ; 

 consequently (0^ -\- 0^) is to be considered as a degeneration of (>^ 

 In any plane passing through H^ and H^ lies a conic (f"^ connecting 

 these points with 3 points of 0^ ; as the straight line H^H^ cannot 



1) If the base of the net consists of a curve p*', of genus 3, antl a cardinal 

 point H, the second bilinear congruence [o^] is formed. 



-) That the figure (y^ 4. ^) is a special case of a ^'^ appears from the fact that 

 through an arbitrarily chosen point F, two straight linos may be drawn which 

 intersect , ^ and t ; they replace the bisecants which ;,. •• sends out through P. 



