454 
[149 
149. 
ON THE SYMMETRIC FUNCTIONS OF THE ROOTS OF CERTAIN 
SYSTEMS OF TWO EQUATIONS. 
[From the Philosophical Transactions of the Royal Society of London, voi. cxlvii. for the 
year 1857, pp. 717—726. Received December 18, 1856,—Read January 8, 1857.] 
Suppose in general that </> = 0, yjr = 0, &c. denote a system of (n — 1) equations 
between the n variables (x, y, z, ...), where the functions </>, yfr, &c. are quantics (i.e. 
rational and integral homogeneous functions) of the variables. Any values (x ly y lt z x ,...) 
satisfying the equations, are said to constitute a set of roots of the system ; the roots 
of the same set are, it is clear, only determinate to a common factor pres, i.e. only 
the ratios inter se and not the absolute magnitudes of the roots of a set are deter 
minate. The number of sets, or the degree of the system, is equal to the product 
of the degrees of the component equations. Imagine a function of the roots which 
remains unaltered when any two sets {x x , y l , z lt ...) and (x 2 , y 2 , z. 2 , ...) are interchanged 
(that is, when x x and x 2 , y x and y 2 , &c. are simultaneously interchanged), and which is 
besides homogeneous of the same degree as regards each entire set of roots, although 
not of necessity homogeneous as regards the different roots of the same set ; thus, 
for example, if the sets are (x lt y x ), (x 2) y 2 ), then the functions x x x 2 , x x y 2 + x 2 y x , y x y 2 
are each of them of the form in question ; but the first and third of these functions, 
although homogeneous of the first degree in regard to each entire set, are not homo 
geneous as regards the two variables of each set. A function of the above-mentioned 
form may, for shortness, be termed a symmetric function of the roots ; such function 
(disregarding an arbitrary factor depending on the common factors which enter implicitly 
into the different sets of roots) will be a rational and integral function of the coefficients 
of the equations, i.e. any symmetric function of the roots may be considered as a 
rational and integral function of the coefficients. The general process for the investi 
gation of such expression for a symmetric function of the roots is indicated in Pro 
fessor Schlafli’s Memoir, “Ueber die Resultante eines Systemes mehrerer algebraischer