OF THE EOOTS OF CEETAIN SYSTEMS OF TWO EQUATIONS. 
723 
a b c f g h i j ^ 
^3 
3 ?c" 
3 e^ri 
6?r;? 
^iz. a=^7*— cf 3 ^— 3y'/3y^+37j8^7, &c. ; 
but if the roots of the given system are 
J/l? ^l)? ('^25 ^2? ^2)? (i^S! 3^3? ^3)? 
then the resultant of the three equations may also be represented by 
J/n O’ (^'2; j /25 . 5 ^ 3 ? 0 ’ 
and comparing with the former expression, we find — 
a = 
b= 
c = 
1 + 
+ y3 
+ 1 
- 3 / 37 '^ 
— 3ya® 
-3a/32 
+ 3/3+ 
+ 3y2a 
+ 3a2/3 
f = 
+ /37® 
+ 0.^13 
—y'^a 
— 2a,f3y 
+ 2ya® 
g= 
+ ya® 
+ /3+ 
-+ 
— 2cx,j3y 
+ 2a/32 
h = 
+«+ 
+ 7'“ 
-/3+ 
— 2a/3y 
— y3 
+ 2/3y2 
i = 
-(3-7 
+ a? 
+ 2a/3y 
+ ad2 
— ya® 
-2a®/3 
j = 
— y-a 
+ /3S 
— a /32 
+dy® 
-2+y 
k= 
-a^d 
+ 2a/3y 
-|-y3 
+ ya2 
-/3y2 
— 2y®a 
1 = 
— ya® 
—a /32 
— /3y2 
+ a2/3 
+ /32y 
+ 7®a 
a =XiX 2 ^' 3 , 
b 
C=Z,Z, Z3, 
3f =^ii/2 23 +M3 Zi ^ 2 , 
3g = Z2 .T3 + ^2 23 + ^3 ^2, 
Sh=x,X3y3+x^3^i +^’ 3 ^, ^ 2 , 
3 i =3/, 22^3 +^2 ^3 +3/321^2, 
3 j = 2 , 0 ^ 2^3 + 22 ^ 3 .^, 4 - 2:3 
3 k =0:, 3/23/3+^23/3^1 +^33/13/2, 
61 ^^0^1 3/2 ^3 “h 'I' 23/3^1 “h ^33/1 ^2“i~ ^13/3^2“}” '^23/1^3 ‘+^33/2^1’ 
But there is in the present case a relation independent of the quantities a, See., viz. we 
have («,(3,7 Xo;„ 3/„2,) = 0, («, ^, 7l^’2,3/2, 22)=0, («, ^, 71^3,3/3, ^3) = 0, and thence elimi- 
nating the coefficients (a, ( 3 , 7), we find 
V = ^13/2^3 + ^^23/3^ , + ^’33/1^2 — ^‘13/3^2 — 1 = 6. 
By forming the powers and products of the second degree a^, ab, &c., we obtain 55 
equations between the symmetric functions of the second degree in each set of roots. 
But we have V*= 0 =a symmetric function of the roots, and thus the entire number of 
linear relations is 56 , and this is in fact the number of the symmetric functions of tlie 
second degree in each set. I use for shortness the sign S to denote the sum of tlie 
