MORENO. — ON RULED LOCI IN W-FOLD SPACE. 149 



spread 2„_"„ a triple spread on S„_i. S„^s li^s on 2,^_3, and constitutes 

 but an infinitesimal part of it. If ti is sufficiently great there is a quad- 

 ruple (n — 4)-spread 2„_4 ruled by the 4-fold infinite system of (>i — 8)- 

 flats arising from the intersections of four i^„_2's. S,^_j lies on aS'„_5. 

 We can go on in this manner until we reach a limit due to the narrowness 



of the space. If n is odd we have finally an ( — - — j-tuple ( — - — J- 



/n — 1\ . . 



spread ruled by the ( — - — j-fold infinite system of lines that arise from 



n — I 

 the intersection of — - — generating F,^_2^- There may be further an 



— - — j-tuple I — - — j-spread made up of the I — - — j-fold infinite 



re -f- 1 

 system of points that are the intersection of — - — generating i^„_2's, an 



/« -f 3\ , /« — 3\ , , , , /2 — 3,,, . ^ . 



( — - — j-tuple ( — - — j-spread made up of the — - — fold infinite 



n + 3 

 system of points that are the intersections of — -- — generating i^„_2's, 



etc., but these spreads do not always occur. In special cases the 2,j_2, 



n ■\- \ 

 or some component of it, may be of greater multiplicity than — - — • 



ii 



In three-fold space a ruled surface generally has on it a double curve. 



This double curve, or some component of it, may, however, be of 



greater multiplicity than two. It is to be observed that aS'„_5 lies on 



2^2- lu three-fold space this means that consecutive generators of a 



ruled surface, if they intersect at all, must intersect in points of the 



double curve. If n is even we have finally an ( - j-tuple ( - j-spread 

 2„ that is made up of the ( - j-fold infinite system of points that 



71 



arise from the intersection of - generating i^„_2's. There may be an 



{—■\-\ J-tuple f - — 1 j-spread 2„ whose points are points of inter- 



•)\ f n \ ( n \ 



section of - -f 1 generating F,^_^s, an ( - -f 2 j-tuple ( - — 2 j-spread 



2„ whose points are points of intersection of - -f- 2 generating i^„_2's, 

 etc., though these spreads may not always be present. 



