146 PEOCEEDINGS OF THE AMERICAN ACADEMY. 



Si, which is an f - j-tuple curve on *5„_i. There are not in general sta- 

 tionary points on ^S^, for the « + 2 equations 



. . 9A . 9'^' A 



ii+i 

 9B ^ 9^B 



^='''3J = ''---'^='>' 



have not in general any common solutions at all. 



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

 k — 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 i^„_2's intersect in an i^„_4, three in an F„_^, r in an 



Fn-iri — H — ^"^ ^^ -^1' ^^ ^ 's ^^^1 <^'' o ^" ^'^ -^0 if " is even. Tliere is a 



1-fold infinite system of each kind of flats. The i^„_2's are generators 

 of Sn_x, the i^„_4's of *S'„_g, the F^_2,'!?> of »S'„_2,.+i. Let us consider the 

 case where n is odd. Through any F^^_^ pass two consecutive i^„_2's, 



w — 1 



through any F^-^t pass r consecutive F„_2S, through any Fi pass 



consecutive F,^_2^- Any F,,^^ contains two consecutive F„_iS, three con- 



n — 1 . 



secutive F„_eS, — - — consecutive i^i's. Any i^„_2r contains two consecu- 



tive i''„_2(r+i)'s5 any two consecutive i^„_2/s determine one -F„_2(r_i)'s. "We 

 may then reverse the process and start with S^-, which lies in the space 

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

 two consecutive i^i's of this surface pass three-flats i^g's, these i^s's will 

 generate a four-spread «S'„_4. Through each two consecutive i^g's pass 

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

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

 consecutive F^_^9, pass i^„_2's ; these F,,_2^ generate an (w — l)-spread 

 S„_i. If we start with the system of (ti — 2)-flats we come down finally 

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

 system of (n — 2)-flats. 



If n is even, through any F,^_^ pass two consecutive F„_2S, through any 



Fn-2.r pass T cousecutive i^n^o's, through any Fq pass - consecutive i^^a's. 



