( iÖö ) 



are directrices. The quadric bearing tliis regulus cuts .t in a conic 

 cr and on this conic the regulus itself marks a series of i)oints, in 

 projective coj'respondence with the scries of points P on t^. So we 

 get in .T a system of {n-{-2')^—l series of points on conies, in mutual 

 projective correspondence, wilh the parlicuhirity that tiie point of 

 intersection B of 4 and rr is a common corresponding point of all. 

 As often as a point P of /.. diiferent from B furnishes {n-\-'2).^ — 1 

 points Pi of (hese series on conies Iving on a curve c' of order n 

 passing through B, as often t/> cuts the surface {P) in a point not 

 lying on one of the lines ö, f/ \ in oilier words, if the first number 

 is ]), the order of [P) is p-{-'2/i. Now the number p can be easily 

 deleriniiied. If we assume in .t a triangle of coordinates of which 

 B is the vertex .i\ = 0, ./■, = ().. the {n-{-2)._ — i series of points can 

 be represented by 



where h-2j and A.-?^, will have to disappear for all the values of i if 

 we stipulate that P. = corresponds to the common point B. So the 

 equation of the curve r" through the ,^^+2), — 1 points Pi corre- 

 sponding to ). is obtained by [iiitling a determinant of order (??-[- 2), 

 equal to zero, of whicli 



11 2 ' 1 3 1 >2 I '2 3 3 



is the first row, whilst the other rows can be deduced from this one 

 by substituting for j\ , x^ , d', successively the quadratic forms in ?. of 

 .iCO, ,r('), .?!^') corresponding to the diiferent values of z. Substitution of 



1,0,0 for d\ , ,i\ , n\ in the first row furnishes then the equation of 

 condition determining /. If L is the minor of the determinant with 

 respect to x'' , the equation of condition is L = 0, the substitution of 



1,0,0 in the first row annulling all the elements of tliis row with 

 exception of the first. The order of this minor in / would be 



— n (/i+3) times 2n, or ri' {n-\-^), if h-yj and A3,, did not disappear 



for all values of /. But on this account the order has to be lessened, 

 as we can divide the elements of columns 1 and 2 of the minor 

 by ?., those of the columns 3, 4 and 5 by V, those of the columns 

 6, 7, 8 and 9 by /.', etc. and those of the last n -\- 1 columns by 

 P. , whilst the value zero of ?, corresponding to the point of coin- 

 cidence B of the series, has to be discarded. So we have to diminish 

 ?f {n-\-'S) by 



