( 497 ) 



2.1 + 3.2 -f 4.3 f . . . + {n \ 1) n ^- -i n {n \ 1) {n f2) = 2 (« | 2), 



and find for p the value 7i' (/i+3) — 2 {n-{-2)^ and therefore for the 

 order p -{- 2n of {P) 



1 2 



nM« f 3)— — n(Mf l)(». + 2) + 2« = -- n (;i -|-l)(n + 2) = 4 (n ^ 2), 

 o o 



So we have got ^) : 



Theorem I. "The locus of the point P emitting transversals Ijing 



on a cone C' to [n-{-2)^ arbitrarily given pairs of lines is a surface 

 (P) of order 4:{7i-{-'2)^ , of which the given lines are lines of niulti- 

 plicitj 71 and the pairs of transversals of the given pairs taken bj 

 two single lines." 



For ?z = 1 this result is contained in the paper quoted above ; 

 for n = 2 it admits of a simple check. In the special case of six 

 pairs of intersecting lines any quadratic cone of the system mnst 

 fulfil with respect to the combination of the point of intersection 

 Ai and the connecting plane «/ of each pair {ni,cii) one of two 

 conditions, i.e. either pass through Ai or have a vertex lying mai, 

 in which latter case the cone is to be counted twice, once for each 

 of the two edges lying in «/ . So we find in this case the generally 

 known surface of the vertices of the cones passing through six given 

 points Ai and besides this surface 0* with 25 straight lines the six 

 planes ai connted twice, i. e. an (9^' as the theorem requires. 



Inversely we find by means of the corresponding case for an 

 arbitrary n, i.e. of the case of (?z-{-2)., pairs of intersecting lines: 



Theokem II. "The locus of the vertices of the cones C passing 

 throngh {n-\-2)^ arbitrarily given points is a surface of order (7z-|-2), 

 of which the given points are points of mnltiplicity n." 



In the special case of {n-\-2)^ pairs of intersecting lines the (>-^("+2)3 

 of the vertices of cones C' consists of the {n-\-2).^ connecting planes 

 (li counting n times and of the surface of the second theorem. So 

 the order of this surface is 



4 («+2)3 - n (^+2), = 4 («+2)3 - 3 («+2)3 = {n^2\ . 

 As the lines connecting the («+2)^ points of intersection Ai by two 

 lie on 0'."+-)i each of these points must be an «-fold point of this surface. 



1) We remark that the number of points of tlie locus lying on an arbitrary 

 line can be found quite as easily by means of the method used above: in that 

 case the determinant itself, with its (n + 2\ rows each of order 2n in A, would 

 have been of order 6 («+2)3 in a, and diminution with 2 (^+2)3 would have 

 given the same result i{n^2\. This confirms that the given lines are w-fold 

 lines of the locus. 



