MORENO, — ON RULED LOCI IN W-FOLD SPACE. 129 



developable. In general any r-flat where r ^ n — ^ + 1 cuts any ♦S'^. in 

 a developable (k -\- r — K)-spread. 



Any i^„_j of the system cuts the >S'„_i in an {n — 2)-spread, and the 

 Fn_2 that it has in common with the consecutive i^„_i appears twice in the 

 intersection, so that the proper (ii — 2)-spread is of order less by two 

 than the order of 5'„_j. This {n — 2) -spread is also a developable. 



An i^,_2 is met by the consecutive i^„_2 in an F,^_?, ; it is met by any 

 other i^„_2 in an (n — 4)-flat. In general, where 4 ^ w, there are a 

 2-fold infinite system of these (« — 4)-fliits and their locus is an (« — 2)- 

 spread which is a double spread on Sn-v I"^ the case of cones and 

 conoids this double spread may be of fewer than » — 2 ways. - Thus in 

 four-fold space the planes which join a line to the successive points of an 

 irreducible conic form a three-way developable. This developable is a 

 conoid and the one-way head is the only multiple locus on the conoid. 

 In three-fold space cones are the only developable surfaces that do not 

 possess a proper double curve, if we call the cuspidal curve a double 

 curve. In general there is a double curve distinct from the cuspidal 

 curve. We will assume that we have the general case of a developable 

 and not a cone or conoid. The total double spread on aS'„_j consists in 

 general of two parts, aS^.s and 2„_2i where 2^2 is the locus of the 

 2-fold infinite system of {ii — 4)-flat3 arising from the intersection of 

 non-consecutive i^„_2's, while -S'„_2 is the locus of the 1-fold infinite 

 system of {n — 3)-flats arising from the intersection of consecutive i^„_2'3. 



Any three non-consecutive i^,_2's intersect in an (n — 6)-flat ; there 

 are in general a 3-fold infinite system of such {n — 6)-flats whose locus 

 is an (n — 3) -spread, a triple spread on aS„_2' Any {n — 6)-flat is the 

 intersection of three {n — 4)-flats of 2„_2 and any such (n — 4)-flat con- 

 tains a 1-fold infinite system of such {n — 6) -flats. This 1-fold infinite 

 system of (n — 6)-flats does not, in general, fill out the {n — 4)-flat, for 

 this would require a 1-fold infinite system of them. The total triple 

 spread on »S',i_j consists in general of two parts /S„_3 and 2„_3 where 2„_3 

 is the locus of the 3-fold infinite system of (w — 6)-flats. We can supply 

 a similar mode of reasoning to the spreads of higher multiplicities on 

 S,^_^. The spreads *S'„_2, aS^.^, ... are developable, but 2„_2, 2„_o, ... are 

 not developable. 



5. Special case where the parameter enters rationally. 



Let us illustrate this theory by the case of the developable which is 

 the envelope of the (ri — l)-flat, 



a f" -f m h f"-i + \m {in — 1 ) c T'^ -1- . . . . = 0, 



9 



