393 



Tlie field of points [/'*] is tiierefore I he image of a congruence 

 (1,3). Tliis consists of the chords of a tivisted cubic r>' which passes 

 throiigii tlie poinls (); for llie range of points (P) in w^. is the 

 image of tlie generatrices /> of a quadratic cone wiiicli has (4 for 

 vertex. 



8. If the twisted cubic {PY passes throngh three cardinal points, 

 it is the image of a Cïcbic scroll (/>)'. For an arbiti'ary snrface «I»' 

 representing an axial complex cuts {P)' in three more points; on 

 the axis of this complex there rest therefore three lines of the 

 scroll. One pencil («l*^) can be passed through (/•*)'; foi' throngh any 

 fonr points of IfJ^)' oo' <I>^ can be passed, each of which contains 

 se\en points of (P)'. Tlie corresponding complexes ./ form also a 

 peticil ; the axes of both axial complexes belonging to this pencil, 

 cut all rays of the scroll and are therefore the director /ines of the 

 cid>ic scroll {pY- 



If (/■*)' passes throngh two cardinal /loints, it is the image of a 

 scroll of the fourth order. In this case one *' passes through (Z-*)' : 

 the scroll belongs to the congruence (2,2) which the corresponding 

 complex J has in common with 7'; as it is rational, it has a 

 double cubic. 



9. A .surface [/-"]" is the image of & concjruence w'nh sheaf degree 

 n, for its intersections with a ray / of T are the images of n rays 

 through the vertex of the complex cone represented by t. The jleld 

 degree of the congruence is generally 'on for each point of inter- 

 section of [/"*]" with the cubic 7' representing the rays t lying in 

 a plane <( , is the image of a ray of the congruence in (/. If [PJ" 



passes sk times through Oi„ the field degree is evidently 3// — 2isk- 



4 

 A twisted curve (P)" is the image of a scroll of the order 2n, 



for the image surface [P]' of an axial complex cuts (P)" in 2n 



points, which are the images of as many rays t cutting the axis 



of the complex. 



10. If the base of a pencil of quadratic surfaces consists of a 

 cubic (>' and one of its chords, the |)olar lines of the points of 

 space form a quadratic complex which is represented in the same 

 way as the letrahedral complex. 



We can always represent this pencil by 



The polar planes of the point y relative to the cones « ^= and 



