392 



Tlie scroll (p)" is represented by the twisted ctirve a* whieli is 

 tlie interseclion of tlie two surfaces 12' that are the images of the 

 congruences defined b}' A^ and A^. 



If tiie axes /•, and r, of two axial complexes cut each other, the 

 congruence (2,2) which these complexes have in common with T, 

 degenerates into the system of the complex rajs /j through the point 

 R^i\>\ and the complex rajs in the plane (.) ^ >-ir,. In connection 

 with this the image surfaces SI' defined by ?',r, cut eacli other in 

 the twisted curve y' representing liie complex rays in y, and in 

 the |)olar line r of R (the image of the complex cone of R); 

 evidently r is one of the bisecants of y'. 



If (»' is an arbitrary twisted c/ihic circtumscribed to 0,0, 0,0^, 

 there pass cc' surfaces ü' through y' of which any two have also 

 in common a bisecant of (>' ; evidently they represent two axial 

 complexes of which the axes cut each other, su that the corresponding 

 (2,2) splits again up into a fovi/ile.c cone and a c<in)/)/e.P conic; the 

 latter is lepresenled by (j'. 



G. A conic (Rj' has four points in common wilh the suiface i2' 

 belonging to an axial complex ./; it is accordingly the image of a 

 rdtiomd scroll (/>)^ Any ray *• of T lying in the |)lane of {Ry, 

 contains two points of {RY ; the image S of .s carries therefore two 

 rays of {pY- Hence llie curve (N)', representing the rays .v, is the 

 douhle cttrce of (/>)^ 



If (/-")■ passes through (>,, it is the image of a cubic scroll (pY 

 of which the double director line passes through O^ ; for the points 

 of intersection of (P)Mvith u>, are the images of two rays /> through 0,- 



If [Ry passes through C, and through (>,, it is the image of a 

 tpiadratic scroll (/*)^ Inversely a scroll (/>)'^ has two rays in common 

 with an axial complex; its image cuts accordingly' the corresponding 

 surface iP outside Ok in two points. Hence this image is either 

 a strnight line (^ 2) or a code through two cardinal points O. 



7. The points R of a plane <f represent the rays of a congruence 

 [/;]. The polar planes « and [i of R with respect to two quadratic 

 surfaces «' and ^' of the given pencil form two projective sheaves 

 of planes round the poles of if. Their iiilerseclions with a plane i^> 

 form two projective fields of rays, hence ^\) contains thiee rays 

 p = «iJ. 



The planes u through a point Q form a pencil; one plane of 

 the corresponding pencil (/J) passes through Q, hence Q carries one 

 ray p. 



