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



The equations of the locus of the intersections of r consecutive i^,_t's 

 are found by eliminating the parameter from the equations 



case we come 



This gives a restricted system equivalent to rk — 1 independent equa- 

 tions. The locus is an {n — rh ^ 1 )-spread ruled by the i^„_ri's, it is an 

 r-tuple spread on »S'„_4_^i. 



There are k cases according as « = (mod ^•), n = 1 (mod Ic), . . . , 

 n = yl- — 1 (mod k). In the first case we come finally to a curve Sx 



which is an ( , j -tuple curve on ^S^.^+i. In the second 



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

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

 {k — l)-flats. There are on Sj^ in general special points where two con- 

 secutive -F/,._i's intersect. 



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

 spreads. 



»S^,_i-i-i has on it in general multiple loci that arise from the intersection 

 of non-consecutive F„_,^'s. Any F„_,, intersects every other F„_f. in an 

 (« — 2 k)-i\at ; there is in general a 2-fold infinite system of such 

 (n — 2 Z,')-flats constituting a double (71 — 2 k + 2)-spread 2„_2i+2 on 

 S„_^^i. Evidently S,j_2k+i li^s on 2„_2t-i-2- -^"J three F„_iJs intersect in 

 an (n — 3 Z,')-flat ; there is a 3-fold infinite system of such (?^ — 3 ^)- 

 flats, they constitute in general a triple (n — 3 ^' + 3)-spread 2„_3i.|_3 on 

 'S,i-k-+i- '^n-3k-+i I'^s on ^nsk+r,' ^"7 '" consccutive F„_f.'s intersect in an 

 (« — r ^)-flat ; there is an r-fold infinite system of such (n — r ^^)-flats 

 in general, constituting an r-tuple (n — r k + r)-spread 1„_^f.^^ on -S^.^.+i, 

 on which lies S„_y,^_^i. 



Finally the locus of the intersection of any a F„_i.'s where a is the 



n 

 greatest integer in j is an a-tuple [n — a (k — l)]-spread 2„_a,j_i, on 



iS„-i:+i ; it is ruled by the a-fo!d 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 (w — 2 ^ -|- 2)-spread 2„_2i4.2. 

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



