154 PROCEEDINGS OF THE AMERICAN ACADEMY. 



S'„_s. By means of an (?i — 3)-way developable lying in an (w — 2)-flat 

 and two arbitrary curves we can generate a ruled (u — Ij-spread by 

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

 (« — 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 (?? — 2) -way developable of the same nature, so 

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

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

 spread. 



III. Loci derived from an (n — ^')-flat whose Equations 

 INVOLVE a Single Ahbitrary Parameter. 



12. Description of the derwed loci. 



We shall comj^lete the general theory by considering the locus of the 

 1-fold infinite system of (« — ^-j-flats, where 2 ^ k whose equations all 

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



A = 0, B = 0, . . . , (7=0. 



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

 parameter from these equations. The result is a restricted system 

 equivalent to ^^ — 1 independent equations. 



The locus is an {n — k -{- l)-spread /S^.j^, ruled by the F„^^'s. Any 

 two consecutive i^„_t's intersect in an (n — 2 k)-^-a.t i^„_2i whose equa- 

 tions are 



9A 9B 



A = 0,^ = 0,B=0/^ = 0,.... 



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

 svstem equivalent to 2^—1 independent equations. The locus is an 

 (71 — 2 /t + l)-spread S„^2k+i ruled by the i^„_2i's ; it is a double spread 

 on S„_^. 



Any three consecutive i^„_2*'s intersect in an {n — 3 X-)-flat F,^^ whose 

 equations are, 



The elimination of the parameter from these equations gives a restricted 

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

 (ii — S k + l)~spread ruled by the F,,_o^'s.. aS^.^j+i is a triple spread on 



