130 PROCEEDINGS OF 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 F^s intersect in 

 the F n _ 2 , 



ar -i + (m _ j) br -2+ O- 1 ) ' | m ~ 2) c r- 8 + . . . . + e = 0, 



- 1 



bf a ~ 1 + (m — 1) ct m ~ 2 + . . . + et +f= 0. 



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

 tion of S n _ v The result is the discriminant of the original equation 

 placed equal to zero ; the order of *S'„_ 1 is then 2 (m — 1).* 



Three consecutive F n _^s intersect in the F n _& 



a r~ 2 + (m — 2)b t m - 3 +.... = 0, 



bt m ~ 2 + (m — 2) c t m -' +.... + = 0, 



ct m ~ 2 + + et +/= 0. 



The equations of S n _ 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 n _ 2 is 

 3 (m- 2).f 



Similarly k consecutive F n _^s intersect in the F k , given by the k 



equations, 



a fn-*+i + (m — k+ 1) b t m ~ k +....= 



b r- &+1 + (m — h + 1) c r~* +.... = 



+ 0*+/=O. 



The elimination of the parameter from these equations gives a 

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

 tions of S n _ k+y The order of S n _ k+l is seen to be (k + 1) (m — k). 



Lastly the intersection of n consecutive F n _^s is the point, F , given 

 by the equations, 



a r-" +1 + (m — n + l)b t m ' n +.... = 



b r- n+1 + (m — ii+l)c t m ~ n +.... = 



+ 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. 



