149] 
OF CERTAIN SYSTEMS OF TWO EQUATIONS. 
463 
9jk — 3af = S x 1 2 x 2 y 2 y 3 z 3 , 
9ki obg = 8 y^y2^2 z 3 x 3 , 
9ij — 3ch = 8 z 2 z 2 x 2 x 3 y 3 , 
9fi - 3bc = 8 y?y 2 z 2 zi, 
9gj — 3ca = 8 z 1 2 z 2 x. 2 x s 2 , 
9hk— 3ab = 8 x?x 2 y 2 y£, 
3 ( fj -f gk + hi - l 2 ) = 8 x^z^ysZs, 
3(2fj - gk - hi + l 2 ) = S x<y 2 x 2 y 2 zi, 
3 (2gk - hi - fj +1 2 ) = 8 y 1 z$ i zffif, 
3 (2hi - fj - gk +1 2 ) = 8 Zift.z& 2 yg, 
3 (6f 1 - 3ki - bg) =8 x^jgjgzg, 
3(6gl — 3ij — ch) =Sy 1 z 1 z?x 3 2 , 
3 (6hl — 3jk — af) =8 ZxXxXgyg, 
3 (6 il — 3fg — ck) = 8 z x Xxy 2 zg, 
3 (6jl - 3gh - ai ) =8 x x y x z?x£, 
3 (6kl — 3hf — bj ) =8 yxZ x x£y£, 
6 (— fj — gk — hi + 41 2 ) = 8xgygzg. 
As an instance of the application of the formulae, let it be required to eliminate 
the variables from the three equations, 
(a, b, c, f, g, h, i, j, k, l\x, y, zf = 0, 
(a', b', c', f, g', h! Jx, y, zf = 0, 
(a, /3, 7 \x, y, z) =0. 
This may be done in two different ways; first, representing the roots of the linear 
equation and the quadric equation by (x 1} y 1} Zj), (x 2 , y 2 , z 2 ), the resultant will be 
(«,...$>!, y l9 Zx) 3 .(a, ...Qx a, y 2 , z 2 f, 
which is equal to 
a 2 xgxg + &c., 
where the symmetric functions xgxg, &c. are given by the formulae a' 3 = xgxg, &c., 
in which, since the coefficients of the quadratic equation are {a', b', c', /', g', h'), 
I have written a' instead of a. Next, if the roots of the linear equation and the cubic 
equation are represented by {x 2 , y 1} z x ), (x 2 , y 2 , z 2 ), (x 3 , y 3 , z 3 ), then the resultant 
will be 
(a', ...Jxx, yx, Zxf.{a',..^x 2 , y 2 , z 2 f{a\ ...~§x 3 , y 3 , z 3 ) 2 ,