718 
iVIE. A. CAYLEY OIS THE SY3DIETEIC FI-A'CTIOXS 
billing with these equations a non-linear equation <1> = 0, the resultant of the s^'steni 
0 = 0, \p = 0 , &c. will be what the function OjOa... becomes, upon substituting 
therein for the different symmetric functions of the roots of the system %}/ = 0, x= 0 , &c. 
the expressions for these functions in terms of the coefficients. We thus pass from the 
resultant of a system <p = 0, '^^= 0 , %=0, &c., to that of a system 0=0, '^ = 0 . x=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 (« — 1) equations with an arbi- 
trary 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 substi- 
tuting 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—\) 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 par- 
ticular cases, viz. I propose to obtain thereby the expressions for the simplest s^Tumetiic 
functions (after the fundamental ones) of the follorving systems of two teniaiy equa- 
tions ; 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, h, c,f, g, lifw, y, zy= 0 , 
y, 2:)=0, 
and join to these the arbitrary linear equation 
(i, n, IXx, y, 2)=0, 
then the two linear equations give 
x\y\z=^l—Y^>Y^~^^'- ; 
and substituting in the quadratic equation, we have for the resultant of the three equations, 
(a, h, c,f, g, — 0|)"=O, 
which may be represented by 
(a, b, c, f, g, hjl, ri, 4)‘"=0, 
where the coefficients are given by means of the Table. 
a b c f g A 
a = 
+y- +/3- 
-2fty 
b= 
+ 7' +a^ 
— 2yoi. 
+ /3= 
— 2a,ft 
f = 
—fty 
— or +a.ft +ya 
(T — 
—ya 
+ a/3 —ft- +fty 
h = 
-a. ft 
+ ya +fty —y- 
(f) 
if) 
SC’jO 
2(??) 
viz. a=^7^+c,6'^ — 
