144 PROCEEDINGS OF THE AMERICAN ACADEMY. 



we derive the equation of an (n — l)-spread S„_p which is ruled by the 

 system of (« — 2)-flats, F ^s* 



Two consecutive i^„_2's intersect in an (« — 4)-flat, whose equations 

 are, 



The elimination of the parameter from these equations gives a re- 

 stricted system equivalent to three independent equations. The locus is 

 an (n — 3)-spread ruled by the F„_iS. *S„_3 is a double spread on S„_i. 

 Three consecutive i^„_2's intersect in an (n — 6) -flat ^„_6, whose equa- 

 tions are, 



If we eliminate the parameter from these equations we derive a 

 restricted system equivalent to five independent equations. The locus 

 is an (n — 5)-spread *S„_5, ruled by the i^„_6's. S,,_^ is a triple spread 

 on *S'„_j and a double sjjread on S„_s . 



Similarly r consecutive i^„_2's intersect in an (n — 2 r)-flat i^„_2r5 whose 

 equations are, 



On the elimination of the parameter we derive a restricted system equiv- 

 alent to 2 r — 1 independent equations. The locus is an (n — 2 r + 1)- 

 spread, *S„_2r + i> ruled by the i^„_2r's. S„_2r + i is an r-tuple spread on 

 S„_-^ ; it is a multiple spread on other spreads of the system. 



Two distinct cases arise according as n is odd or even. If n is odd, 



n — 1 

 then — - — consecutive i^„_2's intersect in a line, Fi, whose equations are, 



* From now on we shall use 5^ to denote the A:-spread of this system. 



