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



developable. In general any r-flat where r > n — k + 1 cuts any S k in 

 a developable (k + r — w)-spread. 



Any F n _ x of the system cuts the S n _ x in an (ft — 2) -spread, and the 

 F n _ 2 that it has in common with the consecutive F n _ l appears twice in the 

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

 than the order of S n _ v This (ft — 2)-spread is also a developable. 



An F n _o is met by the consecutive F n _ 2 in an F n _^ ; it is met by any 

 other F n _ 2 in an (n — 4)-flat. In general, where 4 < n, there are a 

 2-fold infinite system of these (ft — 4)-flats and their locus is an (n — 2)- 

 spread which is a double spread on S n _ v In the case of cones and 

 conoids this double spread may be of fewer than n — 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 S,^ consists in 

 general of two parts, S n _ 2 and 2„_ 2 , where 2„_ 2 is the locus of the 

 2-fold infinite system of (ft — 4)-flats arising from the intersection of 

 non-consecutive F^s, while S n _ 2 is the locus of the 1-fold infinite 

 system of (ft — 3)-flats arising from the intersection of consecutive F„_ 2 s. 



Any three non-consecutive F n _ 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 S n _ 2 . Any (ft — G)-flat is the 

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

 tains a 1-fold infinite system of such (ft — 6)-flats. This 1-fold infinite 

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

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

 spread on S,,^ consists in general of two parts *S, ( _ 3 and 2„_ 3 where 2„_o 

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

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

 S n _ v The spreads S n _ 2 , S n ^, . . . are developable, but 2„_ 2 , 2„_^, . . . arc 

 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 (« — l)-flat, 



a t m + mb r- 1 + i m (m — 1 ) c P~\ -f . . . . = 0, 



9 



