460 
ON THE SYMMETRIC FUNCTIONS OF THE ROOTS 
[149 
where the cubic function written at full length is 
ax 3 + by 3 + cz 3 + 3fy*z + 3gz 2 x + 3hx 2 y + 3iyz 2 4- 3jzx 2 + 3kxy 2 + Qlxyz. 
Joining to the system the linear equation 
(£ v, y, *) = o, 
the linear equations give 
x \ y \ z = /3^-yy : 7: ay - /3%, 
and the resultant is 
(a, b, c, f g, h, i, j, Jc, Z#/3£- yy, y£- a v - /3f) 3 = 0, 
which may be represented by 
(a, b, c, f, g, h, i, j, k, l$f y, £) 3 = 0, 
where the coefficients a, b, &c. are given by means of the Table :— 
a 
b 
c 
/ 
9 
h 
i 
j 
k 
l 
a = 
+ f 
~3/?y 2 
+ 3/3 2 y 
b = 
-y 3 
+ a 3 
- 3ya 2 
4- 3y 2 a 
7* 
c = 
+ /3 3 
-a 3 
— 3a/3 2 
+ 3a 2 /3 
C 3 
f = 
+ /3y 2 
— a 2 /3 
+ a 2 /3 
-y 2 a 
— a 3 
— 2a/3y 
4- 2ya 2 
3^ 
§ — 
+ ya 2 
+ a/3 2 
+ ^ 2 y 
— 2a/3y 
+ 2a/3 2 
3£ 2 I 
h = 
+ y 2 a 
-/S 2 y 
— 2a/3y 
-7 3 
+w 
3^7? 
i = 
-Fy 
+ a 3 
+ /3 3 
+ 2a/3y 
+ a/3 2 
— ya 2 
— 2a 2 /3 
3^ 2 
j = 
-y 2 a 
4- 2a/3y 
— a/3 2 
+ /V 
-2/8 2 y 
3CI 2 
k= 
-a 2 /> 
+ 2a/3y 
+ y 3 
+ ya 2 
-/3y 2 
— 2y 2 a 
3^ 
1 = 
— ya 2 
-a/3 2 
~/V 
4- a 2 /3 
+ ^ 2 y 
4- y 2 a 
viz. a = by 3 — c/3 3 — 3f(3y 2 + 3i/3 2 y, &c. 
But if the roots of the given system are 
(«1» Vu *i), ( x 2, y„ z 9 ), Os, y s , z 3 ), 
then the resultant of the three equations may also be represented by 
Oi> yi, ^51, y, £)•(**, ya» *»££, y, 0-(^3,2/3, ^11, t?, 0; 
and comparing with the former expression, we find : 
cL ■— 0C-]0C<2pCfo 
b = 2/12/22/3, 
c = Z-^Z^Z^