456 
ON THE SYMMETRIC FUNCTIONS OF THE ROOTS 
[149 
then the two linear equations give 
x : y : z = ^-yy : yf - a£ : ay-/3g; 
and substituting in the quadratic equation, we have for the resultant of 
equations, 
(a, b, c, f g, h][f%- yy, y£ - af, ay - /3£) 2 = 0, 
which may be represented by 
(a, b, c, f, g, h$f, y, O 2 = 0 > 
where the coefficients are given by means of the Table. 
a b c f g h 
a = 
+ y 2 +/3 2 
-2/3y 
b = 
+ y 2 + a 2 
— 2ya 
c = 
+ /3 2 + a 2 
— 2 a/3 
f = 
-Pr 
— a 2 + a/3 + ya 
§ = 
— ya 
+ a/3 -/3 2 + /3y 
h= 
— a/3 
+ ya + /3y — y 2 
(a 
(V) 
(a 
2 m 
2 m 
viz. a = by- + c/3 2 — 2f/3y, &c. 
But if the roots of the given system are 
(a?i, Vi, Zi), O2, y 2 , z*), 
then the resultant of the three equations will be 
Oi, Vi, V, £) • (®2, 2/2, z&Z, V, (f) = 0; 
and comparing the two expressions, we have 
t> = 
C = ¿pg, , 
2f = y x z 2 + y 2 z x , 
2g = ZiX 2 + z< x x x , 
2h = x x y z + # 2 y 1( 
which are the expressions for the six fundamental symmetric functions, or 
functions of the first degree in each set, of the roots of the given system. 
the three 
symmetric 
By forming the powers and products of the second order a 2 , ab, &c., we obtain 
linear relations between the symmetric functions of the second degree in respect to 
each set of roots. The number of equations is precisely equal to that of the