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



spread 2„_3, a triple spread on S,^. S n _ 5 lies on 2„_ 3 , and constitutes 

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

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

 fhits arising from the intersections of four F n _ 2 's. S a _ 7 lies on S n _ 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 f — - — j-tuple ( — - — )- 



spread ruled by the f — - — Wold infinite system of lines that arise from 



n — 1 

 the intersection of — - — generating i^ 4 _ 2 's. There may be further an 



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



system of points that are the intersection of — - — generating F^s, an 



fn + 3\ , fn — 3\ , , , , n — 3 , , , . „ . 



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



system of points that are the intersections of — - — generating -F„_ 2 's, 

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

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



M 



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 S n ^ lies on 

 2„_ 2 . In 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 n that is made up of the ( - j-fold infinite system of points that 



n 



arise from the intersection of - generating F n _.?s. There may be an 



I - + 1 j-tuple f - — 1 j-spread 2„ whose points are points of inter- 



section of - + 1 generating i^,_ 2 's, an ( - + 2 j-tuple ( - — 2 j-spread 



n 



2 n whose points are points of intersection of - + 2 generating F n _ 2 's, 



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



