154 PROCEEDINGS OF THE AMERICAN ACADEMY. 



<S'„_3. By means of an (n — 3)-way developable lying in an (n — 2)-flat 

 and two arbitrary curves we can generate a ruled (a — l)-spread by 

 taking all the (n — 2)-flats that can be drawn through the enveloping 

 (n — 3)-flats of the developable so as to meet both curves. 



We have seen that the section of an (n — l)-way developable by an 

 (n — l)-flat gave an (?i — 2) -way developable of the same nature, so 

 here the section of an (n — l)-spread ruled by (n — 2)-flats by an 

 (n — l)-flat gives an (n — 2)-spread of the same nature as the (n — 1)- 

 spread. 



III. Loci derived from an (?i — &)-flat whose Equations 

 involve a Single Arbitrary Parameter. 



12. Description of the derived loci. 



We shall complete the general theory by considering the locus of the 

 1-fold infinite system of (n — &)-flats, where 2 < £ whose equations all 

 contain a single arbitrary parameter. Let the k equations of the flat be 



A = 0, B = 0, . . . , G = 0. 



The equations of the locus of these i^^'s are found by eliminating the 

 parameter from these equations. The result is a restricted system 

 equivalent to k — 1 independent equations. 



The locus is an (n — k + l)-spread 5„_ HI ruled by the F„_ k 'a. Any 

 two consecutive i^'s intersect in an (n — 2 £)-flat F n _ 2k whose equa- 

 tions are 



A-O.g-O.B-O.g-O, 



If we eliminate the parameter from these equations, we derive a restricted 

 system equivalent to 2 k — 1 independent equations. The locus is an 

 (n — 2 k + l)-spread *S , „_ 2 i+i ruled by the F n ^ 2k s ; it is a double spread 

 on S„_ k . 



Any three consecutive F n _ 2k 'a intersect in an (n — 3 £)-flat F n _ 3k whose 

 equations are, 



The elimination of the parameter from these equations gives a restricted 

 system equivalent to 3 k — 1 independent equations. Their locus is an 

 (,a _ 3 h -}- l)-spread ruled by the F^-^s. S n _o k+l is a triple spread on 



