310 



(or v) is iiull-raj- lo any of its points; in connection witli this llie 

 curve (/) degenerates for l=^u or l^v. 



3. If a plane r continues to pass tlirougii the point P, its null- 

 points describe a surface {P) of order (/> + 7 + -l)- ^'or a straight 

 line / passing through P bears (/> + (/) points xV, which send their 

 null-plane through P. 



The straight lines ?< and v, which intersect in /*, lie on (/^y, for 

 each of their points sends its null-plane through /*. 



On (/-*) lie further the (/>7-|-l) singular rays s aiul the singular 

 curves (d', i^', while the singular straight line n is evidently a 

 j>-i'ok\ line, the singular straight line b a 7-fold line. The surfaces 

 {P) and (Q) have, in connection with this, the singular lines s, a, 

 b, a and ,^ in common and intersect further along the curve (/), 

 which belongs to /^ PQ. 



4. As the straight line / intersects the ruled surface (j'*nu /> (7 + ^) 

 points, the curve (/) contains evidently />(7H- 1) singular null-points 

 A* and thus </(// + 1) singular null-points B*. 



There are further (7 + I) planes passing through /, which bear 

 a />-fold null-point .1* each, and consequently (/>+!) planes each 

 with a (/-fold null-point /i^. 



Let li be a point outside the straight line /. To the intersections 

 of the suiface {R) with the curve (/) belong in the first place the 

 ƒ>(/ null-points of the plane III. Further the p[<i-\-\) points .4* and 

 the r/(/)-l-l) points B*. The renminiug common points to the 

 number of (/> + 7-f J) (/> + 7 + />7) - pq — p{q ^\)- -q{V ^-^ i-e. 

 y/(^^_|_ l)_|_^^r/' (/;-)- 1) must be lying in the (7 + !) points A^ and 

 the (/j-4-i) points Z?*. As a on [H] is a />-fold line each of the 

 {q-\-\) points A^ must lie a />fold point of the curve (/). Analogously 

 has (/) in each of the (/) + !) points B-^ a (^-fold point. The curve 

 av is rational, sends consequently 2 (/;—!) tangent planes through 

 /. In each of these tangent planes two rays u coincide, so there are 

 q double null-points, so that the plane is 7-fold tangent plane of 

 (/). Analogously /i'/ sends through / 2(7 — 1) tangent planes which 

 are /)-fold tangent planes of the curve (/). As / is intersected by (/) 

 in (/)-}- 7) points, the rank of / is equal to 2 (/> — 1)7-|-2(7 — 1) 

 V +2(/) + 7). i-e. 4/97. 



5. Let us inquire in how far the results arrived at are altered 

 when the congruence of rays (1,7) is replaced by the congruence 

 (1,3) of the bisecants v of a twisted cubic ,5*. 



