440 
[148 
148. 
MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO 
EQUATIONS. 
[From, the Philosophical Transactions of the Royal Society of London, vol. cxlvii. for the 
year 1857, pp. 703—715. Received December 18, 1856,—Read January 8, 1857.] 
The Resultant of two equations such as 
(a, h, ..Afx, y) m = 0, 
(p, q, ...J«, y) n = 0, 
is, it is well known, a function homogeneous in regard to the coefficients of each 
equation separately, viz. of the degree n in regard to the coefficients (a, b, ...) of 
the first equation, and of the degree m in regard to the coefficients (p, q, ...) of 
the second equation ; and it is natural to develope the resultant in the form 
kAP + k'A'P’ + &c., where A, A', &c. are the combinations (powers and products) of 
the degree n in the coefficients (a, b, ...), P, P', &c. are the combinations of the 
degree m in the coefficients (p, q, ...), and k, k', &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 functions, or from the known expression of the 
Resultant 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 Meyer Hirsch, [for proper title see p. 417], and the application of it is very 
easy when the necessary tables are calculated : as to this, see my “ Memoir on the 
Symmetric Functions of the Roots of an Equation’^ 1 ). But when the expression for the 
Resultant 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. 
1 Philosophical Transactions, 1857, pp. 489—497, [147].