( nro ) 



to tlje piiirs {'ii,a'i) nine transveisaLs forming the base edges of a 

 pencil of cones C^ instead of* determining a single cubic cone. This 

 particuh\rity presents itself also for larger values of ii. So we can 

 say in general that for n ]> 2 a twisted curve occurs forming with 

 the two groups of lines corresponding to the value of n the locus 

 of the point P ïor which the ti-ansvei'sals to the (n-j-2), — 1 pairs {a;, a'{) 



determine a pencil of cones C". If the oi-der of this twisted curve 

 is found we also know tlie order of the locus <;) of the point emitting 



transversals lying on a cone C' to (?^-|-2),-|-l given pairs of lines. 

 Though the theoretical determination of the order of the first curve 

 implies no difficulties the practical execution requires more room 

 than we have at our disposal here; this is the cause why we restrict 

 ourselves now to the case n =^ 2. 



4. Now that we have experienced that the curve o^" of the case 

 ?i = 1 plays a part in the investigation of the case n = 2 we may 

 conjecture that the twenty points with complanar quintuples of 

 transversals (see the paper quoted) will do likewise. 



Let D be a point emitting five transversals t.^ lying in a plane 



J to the five pairs (a,, a/), and let / be a line through D not lying 

 in Ö. Then the method indicated in art. 2 furnishes in d with the 

 aid of the six reguli (/, a;, a',), (/, b, h') six series of points in mutual 

 projective correspondence of which five lie on lines r,- and the sixth 

 on a conic passing through D. So the number of points common 

 to / and 0,^ and different from D is equal to the order of the equation 



in P., i. e. 14. So D is a node ^) of 0!^. As the five curves ol^ 



corresponding with four of the five pairs {fn, a'i) pass through D 

 the tangential cone of 0,^ in D is determined by the tangents in 



D to these five curves; so not only the point D itself but also the 

 tangential cone of 0'-^ in D is entirely independent of the sixth 



pair {h, h'). So the surfaces 0]^ and O^*" admit in the common node 



D a common tangential cone But this implies that the complete 

 intersection of 0!^ and 0'^ passes through D with six branches. 



For, if D is the origin and tk a homogeneous form in x, y, z of 

 order k, the equations of the two surfaces assume the form 



1) This result could have been predicted by remarking tliat the quadratic cone 

 ^^^th vertex D is indeterminate, as it consists of ? and an arbitrary plane through 

 the transversal h' 



