. ( 576 ) 



of' the variety of contact (locus of the points with common tangent) 

 of n simple infinite systems (ft,, \\), (ft a , » a ), . . . . , (ft„, r„) of (« — 1)- 

 dimensional varieties. 

 This order is 



jte, n, . . -(Jul 1 h • • • H h n — 1 



VMi f* a M» 



as can be shown by complete induction. The formula holds for n = 2. 



We assume the correctness of the formula for n = i and out of 



this wo must find the correctness for n = i -}- 1. 



The variety of contact for / + 1 systems in ,Sy>' + ' is the variety 



of contact of the system of varieties (fi lt «\) and the system of curves 



formed by the intersections of the i remaining systems of varieties. 



So we have : 



(1 = 11, , v = r, , y = p a p a . . . pj+'i. 



The points of contact of the curves of the system with a given 

 space Sp { form the (i— l)-dimensional variety of contact of the sections 

 of Sp { with the systems (ft a , r a ), (ft,, r s ), . . . . , (fif+i, Pt-f i)i these sections 

 are likewise systems (fi a , u a ), . . . . , (ftj+i* Vi+i), but of (/—^-dimen- 

 sional varieties. The variety of contact mentioned is according to 

 supposition of order 



( V * l"« J _L 1,, '+ 1 _L,' 1 



VM, f's M' + i 



The points of intersection of thai variety of contact with a right 

 line / of Sp { being the points of / in which Sp' is touched by 

 Curves of the system, we have: 



n' = m 2 f*„ • • • M»+i [ — + — + ••• + — — + * — i 



Thus according to the formula ftip + r<p + ft</> * Me order of the 

 /-dimensional variety of contact of the /+1 systems of varieties 

 l)ecomes 



Mi f< 3 • • • f*t-H I - H h • • • + f i 



by which the correctness of the same formula for n = i-\-l has 

 been demonstrated. So we find : 



For n oo 1 systems ((i v i\), (ft a , r 3 ), . . . , (fin, r ?l ) <?ƒ (n — l)-dimen- 

 sional varieties the locus of the points where the varieties of the 

 systems passing through it have a common tangent is an (n — ^-dimen- 

 sional variety (variety of contact) of order 



ft, ft, ■ • (in - + -+... + -+»- 

 VMi Pi f» 



