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 ,