ON RULED LOCI IN w-FOLD SPACE. 

 By Halcott C. Moreno. 



Presented by W. E. Story, May 8, 1901. Received June 1, 1901. 



The present paper is a discussion of those loci in n-fold space that 

 can be generated by flats whose equations involve a single arbitrary 

 parameter. The ruled loci of space of three dimensions can be repre- 

 sented in this way. 



I. Loci derived from an (n — 1)-flat whose Equation involves 

 a Single Arbitrary Parameter; Developables. 



1. Description of the derived loci. 



Let us consider the loci derived from the equation 



A = 0, 



the equation of an (n — l)-flat involving a single arbitrary parameter A. 

 If the parameter enters rationally, we suppose it to enter to as high a 

 degree as n, the number of ways of the space. If the parameter enters 

 rationally to the degree m where m < ?i, the locus is of a special kind to 

 be discussed later. As the parameter varies continuously we have a 

 1-fold infinite system of (11 — l)-flats. 



Two consecutive (« — l)-flats of the system intersect in an (n — 2)-flat 

 whose equations are 



If from these equations we eliminate the parameter there remains a 

 single equation of an (n — l)-spread, S n _^ which is ruled by the 1-fold 

 infinite system of (n — 2)-flats. 



Three consecutive (n — l)-flats of the system intersect in an (n — 3)- 

 flat whose equations are 



a a ^ A 9-A 



