122 PROCEEDINGS OF THE AMERICAN ACADEMY. 



These (ft — 3)-flats may be considered as arising from the intersection 

 of two consecutive (ft — 2)-flats of the system of (ft — 2) -flats. The 

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

 tem equivalent to two independent equations. The system represents an 

 (n — 2)-spread, S n _ 2 , which is ruled by the (ft — 3)-flats. 



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

 an (n — r)-flat whose equations are 



A 9 A n 9 r ~ 2 A 



A = , _ = „,... ^ = o. 



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

 section of two consecutive (ft — r + l)-flats of the system of (n — r + 1)- 

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

 system. The elimination of the parameter from these equations gives 

 a restricted system equivalent to r — 1 independent equations. These 

 equations represent an (n — r -f l)-spread, S n _ r + l , which is ruled by the 

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



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

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

 general have any common intersection. 



We will use S k 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 (ft — 2)-spread is a double 

 spread on S n _i corresponding to the cuspidal edge or edge of regression 

 in ordinary threefold space."} - 



The S n _ s is a double spread on S n _ 2 , etc., and S : on S»> We see also 

 that $„_g is a triple spread on *S', i _ 1 ; Killing calls it doubly stationary. 

 Finally, S t is an (ft — l)-tuple curve on £„_! ; it is a multiple curve on 

 all the other spreads of the system. J 



If the equation 



A = 



contains k arbitrary parameters connected by k — 1 equations 



<£ = 0, x = 0, ^ = o, 



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



t Puchta calls the S„—i the most general developable spread in w-fold space. 

 Puchta, Ueber die allgemeinsten abwickelbaren Riiume, ein Beitrag zur mehrdi- 

 mensionalen Geometric Wien. Berichte, CI. 



% Killing, loc. cit. 



