( 577 ) 



If the systems are pencils, then 



pi =1 , n — 2 (r,— 1) ; 

 thus the order of tlie variety of contact RV\2...n is: 

 2(r l + r i +... + n,)-n-l. 



4. Returning to the correspondence between the points Q12...B and 

 Q„-\-\ we find for the number of coincidences which are points of 

 intersection of / with the demanded locus L, i. e. for the order of L : 



a x r i + a % r 2 + • • • + «n+ 1 r n +1 — 2 (r, --f- r s + . . . -f r„_|_ 1) -f 



i = n + 1 

 i= 1 



+ n + l = 2{(ai-2)n + lj. 



It is easy to see that a base-variety B[ of the pencil (Fi) 

 is an (ai — l)-fold variety of L. The tangential spaces Sp n ~ ] of L 

 in a point P of J3; are the tangential spaces in P of the varieties 

 I r {, which are laid successively through one of the a- t — 1 points of 

 intersection (not lying on P and the base-varieties) of the varieties 

 V lt V„ . . . , Vi—u Fï+ii • • : i V n +\ passing through P. 

 So we find : 



Given n -f 1 pencils (Vi) (t = 1, 2, . . . , n + 1) of (n — l)-dimen- 

 sional varieties in the space of operation Sp". Let T{ be the order 

 of the varieties of the pencil ( Vi) and a; the number 0/ the points 

 of intersection (not lying on the base-varieties) of arbitrary varieties 

 of the pencils ( }\), ( ]\), . . . , ( Vi- 1), (7i + 1), . . . , ( V»+\). The locus 

 proper of the pairs of points lying on varieties of each of the pencils 

 is an (n — l)-dimensional variety of order 



v=n + 1 



2 [(«,— 2) tv+ lj, 



haring the (n — ^-dimensional base-variety of pencil (Vi)as(cti — 1)- 

 fold variety. 



If n ^> 3, then also in the general case the base-varieties of the 

 different pencils will intersect each other. In like manner as we 

 have dealt with pencils of surfaces x ) we can also determine the 

 multiplicity of common points, curves etc. of base-varieties. 



Sneek, Jan. 1907. 



l ) See page 555. 



39 

 Proceedings Royal Acad. Amsterdam. Vol.. IX, 



