122 PEOCEEDINGS OF THE AMERICAN ACADEMY. 



These {n — 3)-flats may be cousidered as arising from the intersection 

 of two consecutive in — 2)-flats of the system of {ii — 2) -flats. Tlae 

 elimination of the parameter from these equations gives a restricted sys- 

 tem equivalent to two independent equations. The system represents an 

 (/i — 2)-spread, »S„_2, which is ruled by the (?i — 3)-flats. 



In like manner r consecutive (ii — l)-flats of the system intersect in 

 an {n — r)-flat whose equations are 



. . dA ^ 9'--' A ^ 



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



Any of these (n — r)-flats may be considered as arising from the inter- 

 section of two consecutive (« — r -\- l)-flats of the system of (?2 — r + 1)- 

 flats that are the intersections of r — 1 consecutive (w — l)-flats of the 

 system. The elimination of the parameter from these equations gives 

 a restricted system equivalent to /' — 1 independent equations. These 

 equations represent an (ii — r + l)-spread, *S„_,. + i, which is ruled by the 

 1-fold infinite system of (n — r)-flats. 



The locus of the intersections of n consecutive (n — l)-flats of the 

 system is a curve, while n -\- 1 consecutive (w — 1) -flats do not in 

 general have any common intersection. 



We will use S^ to denote that one of the related spreads of this system 

 that is of k ways. It is geometrically evident that each one of these 

 spreads is a developable spread.* 



Considered in this light we see that the (n — 2)-spread is a double 

 spread on »S'„_i corresponding to the cuspidal edge or edge of regression 

 in ordinary threefold space. f 



The *S'„_o is a double spread on a^„_2, etc., and *S'i on /S'2. We see also 

 that ^„_3 is a triple spread on S„_i ; Killing calls it doubly stationary. 

 Finally, Si is an (w — l)-tuple curve on S^-i ; it is a multiple curve on 

 all the other spreads of the system. J 



If the equation 



A = 



contains k arbitrary parameters connected hjk—1 equations 



</, = 0, X = 0. 'A = o» 



* Killing, Nicht-Euklidische Raumformen, p. 195 et seq. 



t Puchta calls the Sn—i the most general developable spread in n-fold space. 

 Puchta, Ueber die allgemeinsten abvvickelbaren Raume, ein Beitrag zur raehrdi- 

 uiensionalen Geometrie. Wien. BerichtCj CI. 



t Killing, loc. cit. 



