132 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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

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

 sented by these equations consists of two distinct parts, »S„_2 and 2„_2. 

 The order of 2„_2 must be, 



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



A triple point on S„_i must lie on three i''„_2's. We may find the equa- 

 tions of the total triple spread on *S'„_i by expressing the conditions that 

 the equations of the F„_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 (?« — 3) rows and 2 m — 4 columns. 

 The order of the restricted system is 



^ (2 m - 4) (2 m - 5) (2 m - 6). 



This triple spread consists of two distinct parts, S„_s and 2„_3. The order 

 of 2„_3 must be 



1 2 



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



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

 on *S„_i, by expressing the conditions that the equations of the F„_i have 

 k 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 — k — 1 columns. This is a restricted system 



equivalent to k independent equations, of order ^ (2 m — ^- — 1) 



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

 /S„_j and 2„_t; the order of the latter is 



_L (2 m - /5; - 1) (2 m - ^ - 2) (2 m - 2 ^^ - (^- + 1) (m - k). 



The total {n — l)-tuple curve on S^_i is given by means of a restricted 

 system similar to (I), in which, however, there are only 2 {in — n ■{- 1) 

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

 alent to w — 1 independent equations whose order is 



^— -: (2m-n) (2 m - w - 1) . . . (2 m - 2 w -f 2). 



{n — 1) I 



