152 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



In case n is even we have finally the f — J-tuple curve whose equations 

 are found by eliminating the parameters from the equations, 



« 



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



of 8 V . 



We find the equation of the double spread 2„_ 2 on *S'„_ 1 , by imposing 

 on the equations of the generating F n _ 2 the conditions that they have two 

 common roots in the parameter. These conditions are,* 



a, b, c =0 



b, .... 



(") 



a, 



a' 



b', e', 

 < V, 



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

 restricted system equivalent to two independent equations ; the order of 

 the system is \ (J + m — 1) (I -\- m — 2). On 2„_ 2 must be S n _o. We 

 find the equations of 2„_g 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 I + m — 4 rows and / + m — 2 columns. This restricted 

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

 (I + m — 2) (/ + m -3) (1+ m — 4). 



The equations of 2„_ r 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 -\- m — (r — 1) columns. This is a restricted system equiva- 



* Salmon, Higher Algebra, Art. 275. 



tlbid., Art. 285. 



