MORENO. — ON RULED LOCI IN W-FOLD SPACE. 143 



These equations, however, in general have only the point P' counted 

 a multiple number of times in common. In general, then, in a space 

 of more than three ways a surface is so twisted that there are no lines 

 that meet the surface three times at a given point. This proof is easily 

 extended to a surface in a space of more than four ways. 



d. The spreads that arise by considering the junctions of the 

 consecutive tangent flats. 



Consider now any surface in rc-fold space. Draw the 2-fold infinite 

 system of tangent planes. Pass a four-flat through every two consec- 

 utive planes and there is a 3-fold infinite system of four-flats, form- 

 ing in general a seven-spread. Each four-flat is met by the infinity 

 of consecutive four-flats in the same plane. We may pass six-flats 

 through every two consecutive four-flats. There is a 4-fold infinite 

 system of six-flats constituting a ten-spread. This system of ruled loci 

 in no wise resembles the system of developables we derived from a 

 curve. Starting with a surface we cannot derive a system of develop- 

 ables in the same manner as when we start with a curve. The same 

 is true if we start with any ^-spread where 2 < p. Only in case the 

 ©-spread lies in a (p + l)-flat do consecutive tangent p-flats intersect 

 generally in (p — l)-flats; the only exception is in the case the w-spread 

 is a curve. 



II. Loci derived from an (n — 2)-flat whose Equation 

 involves a Single Arbitrary Parameter. 



7. Description of the loci. 



Let us consider next the system of loci represented by an {n — 2)- 

 flat whose equations involve a single arbitrary parameter. The parame- 

 ter may enter rationally or irrationally. If it enters rationally we 



n 

 suppose it to enter to as high a degree as - iu each equation. Let the 



equations of the flat be 



.4 = 0, .5=0. 



In these equations we suppose further that the linear function of the 

 coordinates that appear as coefficients of the various powers of the param- 

 eter cannot be expressed in terms of fewer than n + 1 linear functions 

 of the coordinates. Eliminate the parameter from these equations and 



