( 835 ) 



BiBjn, by which a (1, l)-coiTespondence is determined between the 

 points of Q^ and the points of a plane section of the scroll /?. 



As each point of t evidently bears 5 bisecants, whilst a plane 

 through t contains 3, the scroll ^ is a scroll of order 8. A plane 

 section must now show singularities equivalent to 19 nodes. Now 

 the intersection of t is a 5-fold point whilst the 6 intersections of p" 

 furnish as many nodes ; the missing three nodes are evidently sub- 

 stituted by a threefold point which is the intersection of a trisecant 

 resting on t. 



So on the scroll r of the trlsecants these are arranged in pairs of 

 an involution. 



Furthermore follows from this that the scroll t is of order 12. 

 For, if X is the oi-der of r, then one of the {x — 1) points which t 

 has in common with the remainder section in a plane laid through t 

 is to be regarded as intersection of t ; the remaining {x — 2) are 

 derived from multiple curves. Now t is cut outside (>" by one tri- 

 secant and in each of its points of support by three trlsecants, so 

 .1' — 2 = 10 and x = 12. 



6. Out of a point C of q^ we find F^ projected on the curve in 

 the triplets of an ijivolution C^ 



For, if P is a point of q^ then the right line CP cuts the scroll 

 ^'' in an other point F, and the plane through C, F and the point 

 conjugate to it in F"^ determines on q^ two points P' and P' more, 

 forming with P an involutory group. 



The planes jt^PP'P" envelop a quadratic cone, namely the 

 tangential cone of <^"' having C as vertex. A right line / through C 

 is thus cut by two triplets of chords PP' situated in the two planes 

 jr through /; but moreover by the two chords connecting C with 

 the two connecting points C' and C". The involutory scroll of C^ 

 is therefore of order eight. 



As we conjugate P to the chord P'P" this scroll is also of genus 

 two. In a plane section the point of intersection with (j^ are nodes. 

 From this ensues that there must be (see § 5) a nodal curve of 

 order thirteen. 



The central projection of q^ out of C is a quadrinodal c^ upon 

 which each group of C^ is collinear to a pair of F\ If we regard 

 c^ as central projection of a q^ then C originates from the /' on 

 the trlsecants ; consequently C^ has like the last mentioned /^ eight 

 coincidences. 



7. If we bring a cubic surface tp^ through 19 points of q\ this 



