130 PROCEEDINGS OP THE AMERICAN ACADEMY. 



where t is a variable parameter, a, b, c, . . . are linear functions of the 

 coordinates that are not expressible in terms of any v linear functions of 

 the coordinates where v ^ n, and m is an integer which is not less than n, 

 the number of ways of the space. Two consecutive i^„_/s intersect in 

 the F„_2, 



ar-'+(m - 1) 5 r-2 + l^?-iII|!!Liil)c r-« + . . . . + e = o, 



Jr-i+ (m - 1) cr-2+ . . . + et -\-f— 0. 



The elimination of the parameter from these equations gives the equa- 

 tion of 'S'^.j. The result is the discriminant of the original equation 

 placed equal to zero ; the order of S^_.^ is then 2 {ni — 1).* 



Three consecutive F,^_^% intersect in the i^„_3, 



a r-2 + (m — 2) i <'"-» +.... = 0, 



5r-2 + (m — 2) c r-'' + . . . . + e =r 0, 



cr-2+ -{. et +/= 0. 



The equations of /S'„_2 are found by eliminating the parameter from these 

 equations. The result is a restricted system equivalent to two inde- 

 pendent equations ; the order of the system, i. e., the order of »S'„_2 is 

 3 (m— 2).t 



Similarly k consecutive -^„_i's intersect in the F^, given by the k 

 equations, 



b r-*+i ■{- {m — h + \) e T"* +.... = 



-\-et+f=Q. 



The elimination of the parameter from these equations gives a 

 restricted system equivalent to ^ — 1 independent equations, the equa- 

 tions of 'S'„_;.+,. The order of *S'„_i.+j is seen to be {h -\- \) (m — k). 



Lastly the intersection of n consecutive i^„_i's is the point, Fq, given 

 by the equations, 



a r-"+'' + {m — n-\-\)b T"" +.... = 



b r-"+i + (m — n+\) c r-" +.... = 



-\.et+f=0. 



* Salmon, Higher Algebra, art. 105. 



t This is the condition that the three equations have a common root; Salmon, 

 Higher Algebra, art. 277. 



