726 ME. A. CAYLEY ON THE BOOTS OF CEETAIN SYSTI3IS OF TWO EQUATIONS. 
3(6il — 3fg — ck)=S 3/1^3, 
3(6jl — 3gh— ai)==S 
3(6kl — 3hf — bj)=S 
6(— fj— gk— M+4P)=S 
As an instance of the application of the formulae, let it be required to eliminate the 
variables from the three equations, 
(a, h, c,f, g, A, i,j, k, IJx, y, zf=^, 
(«', c\f\ g\ A' ^f= 0’ 
(«, /3, y Jx, j/, 2) =0. 
This may be done in two different ways ; first, representing the roots of the linear equa- 
tion and the quadric equation by y^, z^), (x^, y^^ 22)5 Hi® resultant will be 
which is equal to 
(«, ..X^I, yx, z,y.(a, ..Jx^, y^, z.,f, 
(2^ ^1^2 + &C., 
where the symmetric functions x\xl, &c. are given by the formula &c., where 
the coefficients of the quadratic equation being (a', h\ d, f\ g\ A') I have -written a' 
instead of a. Next, if the roots of the linear equation and the cubic equation are repre- 
sented by {Xi, 2i), (^25 y 2 -> ^2)5 (^35 ^3; then the resultant -will be 
(«', ..X^„ yx Z,)\{a!, .X^2, ^2, z.,f{a!, ..X^I’ 3 , ^3, ^3)', 
which is equal to 
a!^ x\xlxy^-^Scc,., 
the symmetric functions x\xlx% &c. being given by the formulse a-=j^a’i?’§, &c. The 
expression for the Resultant is in each case of the right degree, viz. of the degrees 6, 3. 
2, in the coefficients of the linear, the quadric, and the cubic equations respective! v ; 
the two expressions, therefore, can only differ by a numerical factor, which might be 
determined without difficulty. The third expression for the resultant, -viz. 
(a, yX^i’ 2^1’ yX^'25 ^ 2 ? z.j)- ■ .( 05 , /3, yX'^^C’ ys^ "g)’ 
(where (^„ y„ z^), . . (xg, yg^ 20) are the roots of the cubic and quadi-atic equations) com- 
pared -with the foregoing value, leads to expressions for the fundamental symmetric 
functions of the cubic and quadratic equations, and thence to expressions for the other 
symmetric functions of these two equations; but it would be difficult to obtain the 
actually developed values even of the fundamental symmetric functions. I hope to 
return to the subject, and consider in a general point of view the question of the forma- 
tion of the expressions for the other symmetric functions by means of the expressions 
for the fundamental symmetric functions. 
