( 574 ) 



To find reversely liow many points Q| 2 ...„ correspond to Q n +\ 

 we take arbitrarily oji / the points Qi+\, Qi + i, Q«-f 3> ■ > Qn-fi and 

 we bring through those points respectively a Pi-|-i, Fi-f 2, Fi-f-3,... , 



l^n-j-i. We now put the question how many points Q123...; He on 

 / in such a way that the varieties mentioned PÏ4.1, Fi-f2>--> Fn-fi 

 and the varieties P\, F,, .., F|- passing through Q123...1 have a com- 

 mon point not lying- on the base-varieties. For i<^n the answer is: 

 "1 ''1 + a a t\ -f- — -f «;?•,•■ 



To prove this we begin by noticing that the correctness is imme- 

 diately evident for i = 1 . If we now assume the eorrectness for / =j, 

 we have only to show that the formula also holds for i=zj-\-l. 

 Given the points Qj+z, Qj+3 , ■ • • , Qn+\- To determine the number 

 of points Q123...J+1 xv e take on / an arbitrary point Qi28...j, we 

 bring through it varieties V v V„ . . . , Vj and then through each of 

 the <ij-\-\ |)oints of intersection (not lying on the base- varieties) of these 



V lt V it ...,Vj and the varieties V~j+z, F}+s, . . . , V n +\ resp. passing 

 through Qy+jiQj+Si • • • »Qn+i we bring a variety Py + i : theseay.fi 

 varieties Py+i cut / in o/-fi /7+1 points Qj 4.1. So to Qios...j corre- 

 spond a,+i ?',_j_i points Qy-j-i and (according to the supposition that the 

 formula holds for i=j) reversely to Qj-\-\ correspond a l r 1 -{-a i r i -\- 

 -j- . . . -f-",/'/ points Q123...,;- So there are 'Vi "f "j r 2 + •••"!" 

 -\- <t t r f -\- <t l J r \ i- } + \ coincidences Q123...J Qj+M (hese are the points 

 Q12...J j + i belonging to the given points Q/+2, Qj+s, ■ ■ ■ , Qn+\ ; 

 in this way the correctness of' the formula has been indicated for 

 /=/+!. ' 



When asking after the number of points Qi2...« corresponding to 

 Q n -\-\ we have i = n, so that the formula furnishes aj\ -J- a s r* 3 -j- 

 -|- • • • • -j~ <( u >'n for it. This number must however still be diminished 

 by r,,_|_i, as each of the points of intersection of / with the V n -\-\ 

 passing through Q,,-j-i is a point of coincidence Q123....U— 1 Qn but 

 not one of the indicated points Q\2...„- 



So on / there exists between the points Qi2...« and Q n +\ an 



(a„+i r„_|_i — r„ + 1, a^ + ny 2 -f + a n r» — r„+i) correspondence. 



The a^ -f- 'V2 -f- • • • + «n+i '«+1 — 2r H +i coincidences are the 

 points of intersection of / with the locus L to be found and the 

 points of intersection of / with the (n — J)-dimensional variety of 

 contact R Fi2...n of the pencils ( V x ), ( T 2 ), . . . , ( V„); we understand 

 by that variety of contact the locus of the points, where the varieties 

 V x , V a , . . . , V» passing through them have a common tangent, so 

 where the (n — l)-dimensional tangential spaces of those varieties 

 cut each other according to a line. 



