146 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Si, which is an [ - J-tuple curve on S n _i. There are not in general sta- 

 tionary points on >$i, for the n + 2 equations 



n+l 



.9 A 9' 2 'A 



A = 0, ^— = 0, . . . , — ^r= 0, 



<?A 



9 A" 2 



3 i? 5~ 2 i? 

 R— -— — - — — 



9\ 



9 A 



have not in general any common solutions at all. 



If the equation of the (u — 2) -flat involve k parameters connected by 

 h — 1 equations, the properties of the derived system of loci is the same 

 as in the case just discussed. 



8. Mutual relations of the derived loci. 



Two consecutive F n _ 2 s intersect in an F n _ v three in an F n _$, r in an 



71 • 1 71 



F n -o_ri — 5 — m aD -^ij ^ n ' s odd, or - in an F if n is even. There is a 



1-fold infinite system of each kind of flats. The F n _ 2 s are generators 

 of S n _i, the F^s of *S„_3, the F n __ 2r J s of «S^_ 2r+1 . Let us consider the 

 case where rc is odd. Through any F n _ 4 pass two consecutive F n _ 2 s, 



n — 1 



through any i ?T „_ 2r pass r consecutive F n _ 2 's, through any F x pass — - — ■ 



consecutive F„_ 2 s. Any F n _ 2 contains two consecutive i^ n _4's, three con- 



n — 1 



secutive F n _QS, — - — consecutive i^'s. Any F n _ 2r contains two consecu- 



Li 



tive F n _ 2{r+1) , s, any two consecutive i^_ 2r 's determine one -^,_ 2 ( r _i)'s. We 

 may then reverse the process and start with S. 2 , which lies in the space 

 of n ways but in no flat space of a less number of ways. Through each 

 two consecutive FiS of this surface pass three-fiats F s 's, these F 3 's will 

 generate a four-spread S„_ 4 . Through each two consecutive F 3 's pass 

 five-flats ; this can be done as the i^_3's intersect consecutively in i^'s. 

 These five-flats will generate a six-spread S 6 . Finally, through each two 

 consecutive F H _Js> pass F n _ 2 s ; these F n _ 2 s generate an (n — l)-spread 

 S n _ i . If we start with the system of (n — 2)-flats we come down finally 

 to the surface, or starting with the surface we may work back to the 

 system of (ti — 2)-flats. 



If n is even, through any F n _± pass two consecutive F n _ 2 s, through any 



71 



F n _ 2r pass r consecutive F n _ 2 s, through any F pass - consecutive F^_ 2 s. 



