( 503 ) 



1''^. the vertices of the four cones through the eight points Ai, 



2'"^. in each of the eight planes ai the vertices of the si.v cones 

 througli the seven points Ai not lying in that plane, counted tivice, 



3'^^ in each of tiie tiücnty eight lines of ijitersection of the eight 

 planes «/ by two the vertices of the four cones through the six 

 points Ai not lying in either of the two planes, connted four times, 



4. each of the ffty six points of intersection of the eight planes 

 «i by three, counted eight times. This gives 



1.4.1= 4 



8.6.2= 96 

 28.4.4 = 448 

 56 . 1 . 8 = 448 



996 



7. We unite the results found for ?i = 2 in : 



Theorem III. "The locus of the point P emitting transversals lying 

 on a quadratic cone to six arbitrarily ^) given pairs of lines is a 

 surface 0^\ This surface passes twice through the 12 given lines and 

 once through the 15 pairs of transversals of the six given pairs by 

 two; moreover it contains the 15 twisted curves (^^^ forming the 

 locus of the point for which four of the six transversals are complanar. 

 These 15 curves cut each other by five in 120 points for which 

 five of the six transversals are complanar; each of these points is a 

 node of 0^^ with a tangential cone determined by the tangents of 

 the five curves 9^° passing through that point." 



''The locus of the point F emitting transversals lying on a quadratic 

 cone to seven arbitrarily given pairs of lines is a twisted cnrve^'"^ 

 cutting each of the 14 given lines in 16 and each of the 42 trans- 

 versals of the seven pairs by two in 1 2 points ; it passes through 

 the nodes of the surfaces 0^" corresponding to six of the seven pairs 

 and touches in these points the tangential cones of these surfaces." 



"The number of points P emitting transversals lying on a quadratic 

 cone to eight arbitrarily given pairs of lines is 996." 



In following communications we hope to extend these consider- 

 ations to the cases n = 3, 4, etc. and to give polydimensional 

 generalisations of the problem. 



1) We do not wish to enumerate different special cases here. It may only be 

 pointed out that the surface O'" becomes indeterminate if in order to obtain six 

 pairs of lines we borrow three pairs of reciprocal polars of each of two linear 

 complexes. 



34* 



