MORENO. — ON RULED LOCI IN W-FOLD SPACE. 155 



The equations of the locus of the intersections of r consecutive F n _ k s 

 are found by eliminating the parameter from the equations 



9 A 9^ A 



*=o>ff=o> 



case we come 



This gives a restricted system equivalent to r k — I independent equa- 

 tions. The locus is an (n — rk -f l)-spread ruled by the F tl _ rk %, it is an 

 r-tuple spread on S n _ k+1 . 



There are k cases according as n = (mod k), n = 1 (mod £), . . . , 

 n = k — 1 (mod k). In the first case we come finally to a curve S t 



which is an ( y- j-tuple curve on S„_ k+l . In the second 



down finally to a system of lines F^s which are generators of a ruled 

 surface S 2 . In the last case we come down finally to a ^-spread ruled by 

 (k — l)-flats. There are on S k in general special points where two con- 

 secutive F k _i's intersect. 



13. Multiple loci on the spread; mutual relations of the system of 

 spreads. 



S n -k+\ nas on it m general multiple loci that arise from the intersection 

 of non-consecutive F n _ k s. Any F n _ k intersects every other F n _ k in an 

 (n — 2 £)-flat ; there is in general a 2-fold infinite system of such 

 (« — 2 £)-flats constituting a double (n — 2 k + 2)-spread 2„_ 24+2 on 

 Sn-k+i- Evidently S n _ 2k+1 lies on 2 n _ 2fcf 2- Any three F n _ k 'B intersect in 

 an (n — 3 £)-flat ; there is a 3-fold infinite system of such (n — 3 k)- 

 flats, they constitute in general a triple (n — 3 k + 3)-spread 2 n _3 A . +3 on 

 'S'n-A+i- S n _ Sk+1 nes on %n-sw Any r consecutive F n _ k 's intersect in an 

 (n — r k)-i\at ; there is an r-fold infinite system of such (n — r£)-flats 

 in general, constituting an r-tuple (n — r k + r)-spread 2„_ rjbfr on S„_ k+U 

 on which lies S a ^ rk+V 



Finally the locus of the intersection of any a F n _ k 's where a is the 



n 



greatest integer in T is an a-tuple [n — a (k — l)]-spread l n _ a ll _ 1) on 



<S tl -k+i ; it is ruled by the a-fold infinite system of (n — a £)-flats. 



The question arises, When, in general, do these double loci cease to 

 exist? The double spread is in general an (n — 2 k -f- 2)-spread 2 n _ 2it+ 2. 

 To have a continuous locus of double points we must generally have 



