391 



rajs p, which pass through the cardinal points 0^ T contains 

 evidently oo* scrolls {pY- 



If r is a ray of 7', ;■' and ;■" cut each other, so that the projective 

 pencils of polar planes produce a quadratic cone which has. the 

 point r'r" as vertex. From tliis follows that the complex cones of 

 2' are i-epresented by the point ranges (P) lying on complex rays. 



3. The rays of T which lie in a plane (p (and which accordingly 

 envelop the complex conic 7'), are represented by the points /'' of 

 a twisted curve which passes through the cardinal points Ok- For 

 the intersection of the planes 'f and ojj. is a tangent of f/i' and is 

 represented in 0/,. As to^ can only conlain the images |)oinls Oi, 

 Oin> 0,„ the image of the system of the tangents of (/' is a twisted 

 cubic <f' circumscribed to the tetrahedion Oi 0, 0, (^4. 



4. The complex T cuts a linear complex A in a congruence (2,2) 

 which has singular points in Ok, singular planes in o>k- For Oj- is 

 the vertex of a plane pencil belonging to both complexes, hence to 

 (2,2). The polar lines p' and //' of the rays of this plane pencil 

 with resjiect to «' and ^i'' form two projective |)lane pencils in tojt 

 and these produce a conic circumscribed to Oi U,n 0„. The image oi' 

 the congruence (2,2) is therefore a quadratic surface il" circum- 

 scribed lo (>, O, 0, O^. 



As A does not generally contain any of the sides OkOi, iP will 

 not generally contain any of these sides either. ') 



The qd' siirfages ii' are the images of oo' congruences (2,2) 

 contained in T. To these belong od* axial (2,2) defined by the oo' 

 axial linear complexes. 



5. The rays of 7' belonging to two complexes yf, and A^, form 

 a scroll {pY of the fourth order; this scroll belongs of course at 

 the same time to all complexes A of the pencil defined by //, and 

 A,, hence also to both axial complexes of this pencil. Their axes 

 are director lines of ( pY and moreover double director lines, for 

 tiie complex cone of a point lying on one of these axes, is cut 

 twice by the other axis. 



1) If A is defined by Ed p =0, il- has for equation 



6 



Ed^,c y y = 0. 



g kl mn^m" n 



Inversely the surface Ef.,y, y, = is the image of the (2,2), which is defined 



6 *' * ' 



fkl 

 by the complex T, — p„„ =0. 



6Ci/ 



