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 w-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 (« — 1)-FLAT WHOSE EQUATION INVOLVES 



A Single Arbitrary Parajieter; 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 ?i, the number of ways of the space. If the parameter enters 

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

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

 1-fold infinite system of (?^ — 1) -flats. 



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

 whose equations are 



^ = 0, |i = o. 



If from these equations we eliminate the parameter there remains a 

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

 infinite system of (n — 2)-flats. 



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

 flat whose equations are 



, - SA . 5M . 



^ = *^' 5X^^' 51^ = ^- 



