in elimination between Quadratic Functions. 215 



we need only find the four systems in their lowest terms of 

 w.y.z, which satisfy the first two equations, and multiply 

 the four linear functions obtained by substituting these values 

 of x, y, % in the fourth : the product will contain the resultant 

 of the system affected with some numerical factor. In the 

 present case, the four systems of #, y, z are 



x=0 y=0 # = 1 



y=0 £ = x=l 



Z = # = j/=l 



x> = (a — b){a — c) y=(b — a)(b—c) z = (c — a)(c — b), 



and accordingly the product of 



fa\ + my 1 -f ?m l 



lx 2 + my 2 + m q 



lx 3 + my 3 + ?iz 3 



lx 4 + jny 4 + nz 4 

 becomes 



lmn{a—ba—cl-{-b — a.b—cm-hc—ac — b.n},. 



agreeing with the result obtained by my theorem, — a special 

 numerical factor 4. arising from the peculiar form of the equa- 

 tions, having disappeared from the resultant. 



A geometrical demonstration may be given of the theorem 

 which is instructive in itself, and will suggest a remarkable 

 extension of it to functions containing more than three letters, 



{K\J+[ji,y+(la>+my + n8)t} = 0, 



xyzt 



which is a quadratic equation in X : p, may easily be shown to 

 imply that the conic AU + |u,V is touched by the straight line 



lx + mz + nz = 0. 

 And we thus see that in general two conies, 

 AU + jw,V = 0, 

 passing through the intersections of two given conies, 



U = V = 0, 

 may be drawn to meet a given line. If, however, the given 

 line passes through any of the four points of intersection, in 

 such case only one conic can be drawn to touch it ; accordingly 



I 1 1 1 {*.U + i*.V + (lx + my + nz)t} 



must be zero when I, m, n are so taken as to satisfy this con- 

 dition, i. e, if 



lx x -f my x + nz { = 0, 



