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XXXII. On the Symmetric Functions of the Roots of certain Systems of two Equations. 
By Aethue Cayley, Esq., F.R.S. 
Eeceived December 18, 1856, — Eead January 8, 1857. 
Suppose in general that (p=0, '4^=0, See. denote a system of (n — 1) equations between 
the n variables {x, y, z , . .), where the functions (p, See. are quantics (i. e. rational and 
integral homogeneous functions) of the variables. Any values (Xi, y,, Zi, . .) 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 q)?’es, i. e. only the ratios inter se 
and not the absolute magnitudes of the roots of a set are determinate. 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 
(Xi, yi, 01 , . . .) and (x^, y^, 02 ? • •) are interchanged (that is, when x., and x.^, y^ and y.^. Sec. 
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 (.'Tj, yj, {x^,y 2 ), then the 
functions y„ yiya are each of them of the form in question ; but the first and 
third of these functions, although homogeneous of the fii’st degree in regard to each 
entire set, are not homogeneous 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 investigation of such expression for a symmetric function of the roots 
is indicated in Professor Schlapli’s Memoir, “ Ueber die Resultante eines Systemes 
mehrerer algebraischer 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 <p = 0 be the linear equation or one of the linear equations of 
the system, the resultant will be of the form ?>i?> 2 --? where <p^, (p. 2 , Sec. are what the 
function (p becomes upon substituting therein the different sets (x^, y„ 0i..), (^ 2 ? y 2 ? "a--) 
of the remaining (n — I) equations -4^ = 0, %=0, &c. ; comparing such expression with 
the given value of the resultant, we have expressed in terms of the coefficients of the 
functions Xi &c-? certain symmetric functions which may be called the fundamental 
symmetric functions of the roots of the system >' = 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 •4z = 0, >^=0, Sec. maybe expressed in terms of the coefficients, and then com- 
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