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XXXI. Memoir on the Resultant of a System of two Equations. 
By Aethue Cayley, Esq., F.R.S. 
Eeceived December 18, 1856, — Eead January 8, 1857. 
The Eesultant of two equations such as 
(«, h, . yf=0, 
(q>, q, . yy=0, 
is, it is well kno^vn, a function homogeneous in regard to the coefficients of each equa- 
tion separately, viz. of the degree n in regard to the coefficients [a, h, . .) of the first 
equation, and of the degree m in regard to the coefficients (p, q, . .) of the second equa- 
tion; and it is natural to develope the resultant in the form kAF-{-Jc'MV «&c., where 
A, A', &c. are the combinations (powers and products) of the degree n in the coefficients 
{a, h, . .), P, P', &c. are the combinations of the degree m in the coefficients {p, q, . .), 
and Jc, Jd, &c. are mere numerical coefficients. The object of the present memoir is to 
show how this may be conveniently effected, either by the method of symmetric func- 
tions, or from the known expression of the Eesultant in the form of a determinant, and 
to exhibit the developed expressions for the resultant of two equations, the degrees of 
which do not exceed 4. With respect to the first method, the formula in its best form, 
or nearly so, is given in the ‘ Algebra’ of Meyee Hiesch, and the application of it is 
veiy easy when the necessary tables are calculated : as to this, see my “ Memoir on the 
SjTnmetric Frmctions of the Eoots of an Equation*.” But when the expression for the 
Eesultant of two equations is to be calculated without the assistance of such tables, it is 
I think by far the most simple process to develope the determinant according to the 
second of the two methods. 
Consider first the method of symmetric functions, and to fix the ideas, let the two 
equations be 
{a, b, c, d'Jx, yY=^, 
(p, q, r yf=0. 
Then writing 
(a, b, c, (VJl, zf = a(l — oiz)(l—l3z){l — yz), 
so that 
-l = a+l3+r =(1), 
+l =a/3+ay+^y=(l"), 
-^ = u(3y =(V), 
* Philosophical Transactions, 1857, pp. 489-497. 
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