149] 
ON THE SYMMETRIC FUNCTIONS OF THE ROOTS, &C. 
455 
Gleichungen,” Vienna Transactions, t. IV. (1852). The process is as follows:—Suppose 
that we know the resultant of a system of equations, one or more of them being 
linear; then if <£ = 0 be the linear equation or one of the linear equations of the 
system, the resultant will be of the form <^$2 ... , where (p 1 , <£ 2 , &c. are what the 
function </> becomes upon substituting therein the different sets (x 1} y 1} z 1 ...), {x 2 , y 2 , z 2 ...) 
of the remaining (n — 1) equations yjr = 0, % = 0, &c.; comparing such expression with 
the given value of the resultant, we have expressed in terms of the coefficients of the 
functions &c., certain symmetric functions which may be called the fundamental 
symmetric functions of the roots of the system yjr = 0, % = 0, &c.; these are in fact 
the symmetric functions of the first degree in respect to each set of roots. By the 
aid of these fundamental symmetric functions, the other symmetric functions of the 
roots of the system yjr = 0, % = 0, &c. may be expressed in terms of the coefficients, 
and then combining with these equations a non-linear equation <£> = 0, the resultant 
of the system 0 = 0, ^ = 0, % = 0, &c. will be what the function ... becomes, upon 
substituting therein for the different symmetric functions of the roots of the system 
-\}r = 0, %=0, &c. the expressions for these functions in terms of the coefficients. We 
thus pass from the resultant of a system </> = 0, = 0, % = 0, &c., to that of a system 
0 = 0, t/t = 0, % = 0, &c., in which the linear function <p is replaced by the non-linear 
function O. By what has preceded, the symmetric functions of the roots of a system 
of (n — 1) equations depend on the resultant of the system obtained by combining the 
(n— 1) equations with an arbitrary linear equation; and moreover, the resultant of any 
system of n equations depends ultimately upon the resultant of a system of the same 
number of equations, all except one being linear; but in this case the linear equations 
determine the ratios of the variables or (disregarding a common factor) the values of 
the variables, and by substituting these values in the remaining equation we have the 
resultant of the system. The process leads, therefore, to the expressions for the 
symmetric functions of the roots of any system of (n— 1) equations, and also to the 
expression for the resultant of any system of n equations. Professor Schlafli discusses 
in the general case the problem of showing how the expressions for the fundamental 
symmetric functions lead to those of the other symmetric functions, but it is not 
necessary to speak further of this portion of his investigations. The object of the 
present Memoir is to apply the process to two particular cases, viz. I propose to 
obtain thereby the expressions for the simplest symmetric functions (after the funda 
mental ones) of the following systems of two ternary equations; that is, first, a linear 
equation and a quadric equation; and secondly, a linear equation and a cubic 
equation. 
First, consider the two equations 
(a, b, c, f g, h^x, y, zf = 0, 
(a, ¡3, yjx, y, z) = Q, 
and join to these the arbitrary linear equation