144 PROCEEDINGS OF THE AMERICAN ACADEMY. 



we derive the equation of an (ti — l)-spread S n _ v which is ruled by the 

 system of (n — 2)-flats, F n _ 2 s.* 



Two consecutive F n _ 2 's intersect in an (?i — 4)-flat, whose equations 

 are, 



^ = 0,^ = 0^=0,^ = 0. 



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 n _f&. S n _s is a double spread on S^_ x . 

 Three consecutive F n _ 2 's intersect in an (ti — G)-flat F n _ 6 , whose equa- 

 tions are, 



. 9 A 9" A 



A = °>-9^ = °> 9X>=°> 



9B_ 9*B_ 



If we elimiuate 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 F„^s. S n _ 5 is a triple spread 

 on S n _ 1 and a double spread on S n _ s . 



Similarly r consecutive F H _ 2 s intersect in an (n — 2 r)-flat F n _ 2r , whose 

 equations are, 



A A 5 A A 9 r ~ 1 A A 



„ A 9B A 9 r - x B A 



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

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

 spread, S n _ 2r + V ruled by the F n _ 2r , s. S„_ 2r + i is an r-tuple spread on 

 aS^j ; 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 F n _ 2 s intersect in a line, F 1} whose equations are, 



* From now on we shall use S k to denote the ^-spread of this system. 



