III. " Memoir on the Symmetric Functions of the Roots of cer- 

 tain Systems of two Equations." By ARTHUR CAYLEY, 

 Esq., F.R.S. Received December 18, 1856. 



(Abstract.) 



The author defines the term roots as applied to a system of n 1 

 equations 0=0, ^=0, &c., where <j>, \l/, &c., arequantics (i. e. rational 

 and integral homogeneous functions) of the n variables (a?, y, 2, . .) 

 and the terms symmetric functions and fundamental symmetric func- 

 tions of the roots of such a system ; and he explains the process 

 given in Professor Schlafle's memoir, " TJeber die Resultante eines 

 Systemes mehrerer algebraischer Gleichungen," Vienna Transactions, 

 t. iv. (1852), whereby the determination of the symmetric functions 

 of any system of (n 1) equations, and of the resultant of any sy- 

 stem of n equations is made to depend upon the very simple question 

 of the determination of the resultant of a system of n equations, 

 all of them, except one, being linear. The object of the memoir is 

 then stated to be the application of the process to two particular 

 cases, viz. to obtaining the expressions for the simplest symmetric 

 functions, after the fundamental ones of the following systems of two 

 ternary equations, viz. first, a linear equation and a quadratic equa- 

 tion ; and secondly, a linear equation and a cubic equation ; and 

 the author accordingly obtains expressions, as regards the first system, 

 for the fundamental symmetric functions or symmetric functions of 

 the first degree in respect to each set of roots, and for the symmetric 

 functions of the second and third degrees respectively, and as regards 

 the second system, for the fundamental symmetric functions or sym- 

 metric functions of the first degree, and for the symmetric functions 

 of the second degree in respect to each set of roots. 



IV. " Memoir on the Resultant of a System of two Equations." 

 By ARTHUR CAYLEY, Esq., F.R.S. Received December 18, 



1856. 



(Abstract.) 



The resultant of two equations such as 



(a, b, . vgx, y) m = 

 (p, q, . .x, y)" = 



