132 PROCEEDINGS OP THE AMERICAN ACADEMY. 



where there are 2 (m ■— 2) rows and 2 m — 3 columns. This restricted 

 system is of order | (2w — 3) (2 m — 4). The double spread repre- 

 sented by these equations consists of two distinct parts, S n _ 2 and 2 n _ 2 . 

 The order of 2 n _ 2 must be, 



J (2 m — 3) (2 m — 4) — 3 (m — 2) = 2 (m — 2) (m — 3). 



A triple point on S n _i must lie on three F n _ 2 's. We may find the equa- 

 tions of the total triple spread on S n _ 1 by expressing the conditions that 

 the equations of the F n _ 2 have three common roots. These conditions 

 are expressed by means of a rectangular system similar in form to (I), 

 in which however there are only 2 (m — 3) rows and 2 m — 4 columns. 

 The order of the restricted system is 



~ (2 m - 4) (2 m-b) (2 m- 6). 



This triple spread consists of two distinct parts, S n _ 3 and 2 n _ 3 . The order 

 of 2„_ 3 must be 



1 2 



-^(2m-4) (2m-5)(2m-6)-4(m-3)=-(m-3)(m-4)(2m-l). 

 o I o 



In like manner we can find the equations of the total &-tuple spread 

 on S n _ u by expressing the conditions that the equations of the JF n _i have 

 Jc roots in common. These conditions are expressed by means of a 

 rectangular system similar to (I), in which, however, there are only 

 2 (m — k) rows and 2 m — h — 1 columns. This is a restricted system 



equivalent to k independent equations, of order -r~j (2 m — k — 1) 



(2 m — h — 2) . . . . (2 m — 2 k). This spread consists of two parts, 

 S n _ k and % n _ k \ the order of the latter is 



JL (2 m — k—\)(2m-k-2) (2 m — 2 k)- (k + 1) (m — k). 



The total (n — l)-tuple curve on #„_! is given by means of a restricted 

 system similar to (I), in which, however, there are only 2 (m — n + 1) 

 rows and 2 m — n columns. We have then a restricted system equiv- 

 alent to n — 1 independent equations whose order is 



(2 m — n) (2 m - n — 1) . . . (2 m - 2 n + 2). 



(n - 1) 



