150 PROCEEDINGS OF THE AMERICAN ACADEMY. 



11. Special case where the parameter enters rationally. 



Let us consider the special case where the parameter enters rationally. 

 Let the equation of the generating (n — 2)-flat F n _ 2 be 



A = a t l + b t 1 ' 1 + c t l ~ 2 + = 0, 



B = a' r + b' r _1 + c> r~ 2 + .... = o, 



where a, b, c, . . . , a', b', c', . . . , are linear functions of the coordinates 

 that cannot be expressed linearly in terms of fewer than n + 1 linear 

 functions of the coordinates. If we eliminate the parameter from these 

 equations, we have the equation of the £„_! ruled by the -F„_ 2 's ; it is of 

 order I -\- m. It is more convenient in what follows to use two param- 

 eters, A and fx, that enter homogeneously into the equations. 



Two consecutive generators intersect in the F n _ 4 whose equations are 



9 X 9 jx 9 X 9 /x 



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

 stricted system equivalent to three independent equations the locus is 

 £ n _3, whose order is 



2 {I— 1) + 2 (m— 1) = 2 (Z+m — 2). 



The order is found by expressing the conditions that the four equations 

 have a common root. The locus of the intersections of three consecu- 

 tive F n _ 2 's is a locus of F„_ e 's ; the equations of this locus are found 

 by eliminating the parameters from the equations, 



3"-A ^L_ 3M_ 

 ix a [x 



9 A 2 ' ' 9X9fx ' 9r 2 



9 2 B _ 9 2 B 9 2 B _ 



9X 2 ~ ' 9\9fi~ ' 9fx 2 ~ 



This gives a restricted system equivalent to five independent equations ; 

 it represents S n _s, whose order is 3 (I + m — 4). 



The r-tuple spread S n _ 2r+i on £„_! is represented by the equations that 

 result from eliminating the parameters from the equations, 



9" A r*A 3^ 



ir-1 — U > CI vr-2 Cl .. — U ' * • * ' O ..r-1 U > 



9X"' 1 " ' 9k r ~ 2 9fx ' ' * "'5 



/• 



9 r ~'B 9 r ^B 9r ~ 1]3 -o 



9X^ ~ ' 9x r ~ 2 9,x - u ' • • • ' ^ " 



