323 



is, it is well known, a function homogeneous in regard to the co- 

 efficients 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 MP + '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, W, &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 sym- 

 metric 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 ex- 

 ceed 4. With respect to the first method, the formula in its best form, 

 or nearly so, is given in the ' Algebra ' of Meyer Hirsch, and the appli- 

 cation 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." 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 determi- 

 nant according to the second of the two methods. 



V. " Memoir on the Symmetric Functions of the Roots of an 

 Equation." By ARTHUR CAYLEY, Esq., F.R.S. Received 

 December 18, 1856. 



(Abstract.) 



There are contained in a work, which is not, I think, so generally 

 known as it deserves to be, the ' Algebra ' of Meyer Hirsch, some very 

 useful tables of the symmetric functions up to the tenth degree of 

 the roots of an equation of any order. It seems desirable to join to 

 these a set of tables, giving reciprocally the expressions of the powers 

 and products of the coefficients in terms of the symmetric functions 

 of the roots. The present memoir contains the two sets of tables, 

 viz. the new tables distinguished by the letter (a), and the tables of 

 Meyer Hirsch distinguished by the letter (b) ; the memoir contains 



VOL. VIII. 2 C 



