40 



rational o/, with wiiicli it is coimected into a degenerate y\ For 

 tlirough any point of a straiglit line i\ pass oo' rnled snrt'aces R', 

 wliicii iiave /■, in common; so they pass all moreover throngh a 

 rational q\ of which </ is a trisecant. All pencils {li") which arise 

 when j'l is made to revolve round F^ have in common the degene- 

 rate rnled surface composed of the plane (F^q) and Ki\ These two 

 figures have in common, besides q, a straight line />,, which is 

 apparently the locus of the point ZA^'O'i-p/)- 



Tlirongh the live points Fk{i:='i to 7) a twisted cubic p-^.^ may 



l)e laid intersecting q twice. If R^ and R^ are its points of inter- 

 section with the planes {F^q) and {F,q), the straight lines ;•, ;s i*^,/?, 

 ■dud i\^^F,R.^ form with o^ ^ a degenerate q\ Apparently q'-^ forms 

 with q the intersection of the hyperboloids R,^ and R^'. 



The conynhence therefore contains seven si/steins of degenerate 

 curves {Qt,rk) and 21 degenerate f (fures {q^^, rk. ri j. 



3. Any curve ^)' intersecting the sinqular quadrisecant 5' in a 

 point S belongs to the base of a pencil of which all the surfaces 

 touch each other in ,S. In order to determine the locus of those 

 curves, I consider two arbitrary pencils of the net [/?*]. If to each 

 ruled surface of the lirst pencil the two ruled surfaces of the second 

 pencil are associated, which touch the first ruled surface in S, the 

 pencils are in a correspondence (2,2). To the figure of order 12, 

 which they produce, the common ruled surface lielongs twice. The 

 curves <>' passing through S form therefore a surface .2"". This sur- 

 face must be a monoid as an arbitrary straight line drawn tin-ough 

 5 is chord of one curve (>', consequently intersects ^' outside S 

 in one point only. From the consideration of a plane section it 

 ensues as a matter of course tliat 7 is a i/uadruple straight line of 

 the monoid. 



Tlirough the quintuple point S pass the seren straight lines FkS. 

 y\n arbitrary ()'' of the congruence intersects 2' only on q and in 

 the points F; from this it ensues at once that tiie monoid has 

 seven nodes Fk. 



If 2" is projected from S on a plane 7 , the system oc ' of the 

 curves in which tiie monoid is intersected by a pencil of planes 

 finds its representation in a pencil of curves q'. passing through 

 the images F'k of the points Fk- One of these curves has apparently 

 a node in Fk' ; the remaining curves will therefore have in Fk' the 

 tangent in common. 2'' has in that case the same tangent plane in 

 all the points of SFk- 'he monoid has seven iorsal straight lines SF^. 



