( 575 ) 



2. To determine the order of R F12... n we must observe thai R V\2... n 

 is the locus of the points of contact of the varieties V n with the 



curves of intersection C12.. „ — 1 of the varieties I", F„ , f",_i. 



So the question has been reduced to that of the order of the variety 

 of contact of a system of cc 1 (n — l)-dimensional varieties and a 

 system of oo" — x curves. That order can be determined nut of the 

 points of intersection with an arbitrary line /. 



In a point of intersection of / with a variety of the system we 

 bring the (n — l)-dimensional tangential space Sp n ~ ' and in a point 

 of intersection of / with a curve of the system the co"— 2 tangential 

 spaces Sp n ~ l . If we act in the same way with all varieties and 

 curves of both systems, then the tangential spaces of the varieties 

 furnish an 1-dimensional envelope E 1 (i. e. a curve) of class \i -\- v (as 

 is evident out of its osculating spaces Sp n ~ ' through an arbitrary, 

 point of /) with v osculating spaces Sp 1l ~ x passing through /; here 

 ft is the number of varieties of the system passing through an 

 arbitrary point, and v that of the varieties touching an arbitrary right 

 line. The tangential spaces of the curves in the points of intersection 

 with / have an (n — lydimensional envelope E» of class <p -f if? with 

 I as ty-fold line, where <p is the number of curves of the system 

 passing through an arbitrary point and if? that of the curves touching 

 an arbitrary space Sp il ~ l in a point of a given right line of that 

 space; for, if we bring through a point Q of / an arbitrary Sp r, ~ 2 , 

 then each of the <p curves of the system passing through Q furnishes 

 a tangential space Sp n ~ [ passing through this Sp n ~- whilst the space 

 Sp n ~ x determined by / and Sp' l ~- (just as every other Sp"— X passing 

 through /) is \\> times tangential space of the envelope. 



Both envelopes have thus (ft -f 1?) (<p -j- if?) common tangential spaces 

 Sp n ~ ] . Each of the v osculating spaces Sp n ~ l oï E x passing through 

 I is a if'-fold tangential space of E iy so it counts for if? common 

 tangential spaces; so that fty -f- f«p + v<p common tangential spaces 

 not passing through / are left; these indicate by their points of 

 intersection with / the points of intersection of / with the variety of 

 contact, so we find : 



The {n — lydimensional variety of contact of an oc' system of 

 (n — lydimensional varieties of which (i pass through a given point 

 <t ml v touch a given right line, and an oo" - 1 system of curves of 

 which <p pass through a given point and if? touch a given space 

 Sp r '~ ] in a point of a given right tine of that space, is of order 



fiif? 4- v(p + (i(f. 



3. With the aid of this result it is easy to determine the order 



