458 on 
and consequently the 
by the equation 
THE SYMMETRIC FUNCTIONS OF 
quantities a, b, c, f, g, h, are not 
THE ROOTS [149 
independent, but are connected 
abc — af 2 — bg 2 — ch 2 + 2fgh = 0, 
an equation, which is in fact verified by the foregoing values of a, &c. in terms of 
the coefficients 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 X (abc — af 2 — bg 2 — ch 2 + 2fgh), where X is an 
indeterminate numerical coefficient. 
The simplest expressions are those obtained by disregarding the preceding equation 
for fgh, and the entire system then becomes : 
a s —— /yi 3/yi 3 
tUq } 
b 3 = 2/i %*f 
C 3 = Z X Z 2 , 
b 2 c = y?z x y?z 2 , 
c 2 a = z x x x zgx 2 , 
a 2 b = x x 2 y x xfy 2 , 
be 2 = y x z-?y&i, 
ca 2 = z x x?Z'ix£, 
ab _ = x x y x x 2 y 2 , 
abc = ah&iz 1 x 3 y&, 
2a 2 f = 
rp 2^# r? rp 2 
yi^2^2 
+ X 2 y 2 Z x X x , 
2b 2 g = 
y\Z x x 2 y? 
+ yiz&iyi, 
2c 2 h = 
Z \ aC iy-2 Z 2 
+ z *n 2 y x z x 2 , 
2a 2 g = 
rp 3 nr rp 2 
_ I - rp 3^ rp 2 
i *¿'2 ¿2^1 y 
2b 2 h = 
Vx^yi 
+ yiœ x y x \ 
2c 2 f = 
z \y‘i z ‘i 
+ Z 2 3 yi Z l, 
2a 2 h = 
rp 3 a* 2 ,p 
try^ zj 2 iX/2 
+ X 2 3 Z x 2 X x , 
2b 2 f = 
y x z x?y 2 + yix 2 y x , 
2c 2 g = 
Z iy* Z -2 
+ Z 2 3 yi% , 
2bcf = 
yi z iV‘2 z i 
+ ?j2 Z SJl Z l , 
2cag = 
z x X x z ZC 2 
+ Z 2 X$Z x X x , 
2abh = 
Xiyi^y? 
+ xiy&iyi, 
2bcg = y x z 2 x 2 y 2 z 2 + y^x x y x z x , 
2cah = z x x x -x 2 y 2 z 2 + z^x x y x z x , 
2abf = x x y x 2 x 2 y 2 z 2 + x 2 yix x y x z x ,