134 PROCEEDINGS OF THE AMERICAN ACADEMY. 



in which form the parameter do longer appears. Any p -f 1 consecutive 

 i^_i's intersect in the same F n _ p _ 1 as any other consecutive p -f 1 ; 

 i. e., all the -F n _i's of the system contain the same F n _ p _ x . Any p-flat 

 that does not meet this F n _ p _ l cuts S^_ x in a developable (p — l)-spread 

 of order 2 (p — 1). This developable (p — l)-spread of order 2 (p — 1) 

 lying in a p-flat is exactly similar to the case in n-fold space where 

 m = n. The curve at the base of this system is of order p ; it is the 

 rational normal curve of p-fold space. Hence we may derive this system 

 by joining by lines all points of a developable (p — l)-spread of order 

 2 (p — 1) in a p-fold space, to all points of an (n — p — l)-flat that does 

 not meet the p-flat that contains the (p — l)-spread. S n _ 1 is a conoid 

 of (n — 2)-flats with an (n — p — l)-way head. The generating F n _ 2 's 

 of S n _i arise from the junction of the (« — p — l)-way head with the 

 generating (p — 2) -flats of the (p — 1) -spread. The generating F n _ s 's 

 of S n _ 2 arise from the junction of the (n — p — l)-way head with the 

 system of generating (p — 3)-flats of the (p — 2)-spread, and so on. 

 Any conoid ruled by a 1-fold infinite system of <?-flats with a (q — l)-way 

 head is a developable spread, but not so if it has only an r-way head 

 where r < q — 2. The latter spread is a developable only when the 

 consecutive ^-flats have (q — l)-way intersection. Any conoid ruled 

 by a 1-fold infinite system of (n — 2)-flats that have an (n — 3) -flat in 

 common is a developable, but if they have only an (n — £)-flat in com- 

 mon where k < 4, the conoid may or may not be developable. The 

 cones and conoids with a 2-fold infinite system of generators are not 

 developables at all. 



The points of intersection of two consecutive generators are not in 

 general points of intersection of three generators. The equations of 

 a generator may be written 



e+(m - 1)d+ (*'- i y™- 2 > c+ — o, 



/+(m - 1)c + ("- 1 H°'- 2 ) rf+ .,.. = o. 



The points of intersection of three generators of the system are given 

 by the equations 



