152 



PROCEEDINGS OF THE AMERICAN ACADEMY, 



In case n is even we have finally the (- j-tuple curve whose equations 

 are found by eliminating the parameters from the equations, 



n-2 



n-4 



9X'''9fJi 



n-2 n-2 



9'^' A _ 9'^' A 



— ^, • • • t „-2 ■ 



9 [X^ 



n-2 



9'^' B 



n— 2 n— 2 



n-2 — ^' « 4 — *-'? • 



9X^' 



0, 



0. 



5 A ^ (9/x 



n 



The order of the restricted system is ~ (I + m — n + 2), the order 



of «Si. 



We find the equation of the double spread 2„_2 on /S',,^!, by imposing 

 on the equations of the generating i^„_2 the conditions that they have two 

 common roots in the parameter. These conditions are,* 



(H) 



a, h 



a, h, 



a', b>, c', 

 «/, b', 



= 



where there are I -\- m — 2 rows and I -\- m — 1 columns. This is a 

 restricted system equivalent to two independent equations ; the order of 

 the system is h (l + rn — I) (I + m — 2). On 2„_2 must be aS„_3. We 

 find the equations of 2„_3 by expressing the conditions that the equations 

 of the generating flat have three common roots in the parameter.! The 

 result is a restricted system similar in form to (II), in which, however, 

 there are only Z + ?« — 4 rows and I + m — 2 columns. This restricted 

 system is equivalent to three independent equations, and its order is ^ 

 {I -\- m - 2) (/+/« — 3) (I + m — 4). 



The equations of 2„_^ are found by expressing the conditions that the 

 equations of the generating (n — r)-flat have r roots in common. By an 

 extension of the previous method we derive a restricted system of the 

 same form as (II), in which, however, there are only I + m — 2 (r — 1) 

 rows and I -{- ?n — (r — 1) columns. This is a restricted system equiva- 



* Salmon, Higher Algebra, Art. 275. 



tibid., Art. 285. 



