720 
ME. A. CAYLEY ON THE SYIMMETEIC FLNCTIONS 
2cf =%2«2 +%i2i, 
2ag=^22a;2+^2i^i, 
2h'h=y\x^y^-\~ylx,y,. 
Proceeding next to the powers and products of the third order a^ a"b, &c., the total 
number of linear relations between the symmetric functions of the thii’d degree in 
respect to each set of roots exceeds by unity the number of the symmetric functions of 
the form in question ; in fact the expressions for abc, aP, bg^, ch^, fgh contain, not five, 
but only four symmetric functions of the roots ; for we have 
ohc—x^y-^z^.x^y-iZ^ 
4aP = {x,y\x^zl+x^yyc^ z]) + 2x,y^z,x^^z^, 
4bg" = ( y,z\y^^ + y^ zly.oi^,) + 2x{y,z,x^^z^, 
4ch® = ( z^x\ z^yl + z^\z, yX) + 2x,y,z,x,yi^^ 
8fgh = {x,y\x^z% + x^ylx, zX ) ' 
+ ( y 1 + y% Ay A) !> + ^x,y,z,x.^^z^ 
+{z,x\zA2+z^xlz,y\) 
and consequently the quantities a, b, c, f. g, h, are not independent, but are connected 
by the equation 
abc — af ® — bg^ — ch® fi- 2fgh = 0, 
an equation, which is in fact verified by the foregoing values of a, &c. in terms of the 
coefiicients of the given system. 
The expressions for the symmetric functions of the third degree considered as functions 
of a, b, c, f, g, h, are consequently not absolutely determinate, but they may be modified 
by the addition of the term A(abc — aP — bg^— -ch^+2fgh), where X is an indetermmate 
numerical coefficient. 
The simplest expressions are those obtained by disregarding the preceding equation 
for fgh, and the entire system then becomes- — 
q 3 _ /y»3/y,3 
t/v jt/v 25 
b® =y\y% 
e :=z\z% 
b'c —y\z{y\z^, 
c^a =zlxizlx2, 
affi ^xXy.xly^, 
be" —yAy^A, 
ca" =ZiXlz^'l, 
ab" =xAiX 2 yl, 
abc—x^yiZiX^^^i^ 
