149] OF CERTAIN SYSTEMS OF TWO EQUATIONS. 461 
3f = 
3g = Z x Z 2 X 3 + Z 2 Z^C X + Z 3 Z x X 2 , 
3h = x x x 2 y 3 + x 2 x 3 y x + x& x y 2 , 
3i = y x z 2 z 3 + y 2 z 3 z x + y & z x z 2 , 
3j = Z x XJX 3 + ZrfC 3 X x + Z 3 X x X2, 
3k = x x y 2 y 3 + x 2 y 3 y x +x 3 y x y 2 , 
61 = X x y 2 Z 3 + x 2 y 3 z x + X 3 y x z 2 + X x y 3 Z 2 + X 2 y x z 3 + X 3 y 2 Z x . 
But there is in the present case a relation independent of the quantities a, &c., viz. 
we have (a, /3, yjx u y x , z x ) = 0, (a, /3, y^x 2 , y 2 , z 2 ) = 0, (a, /3, yjx 3 , y 3 , z 3 ) = 0, and 
thence eliminating the coefficients (a, /3, 7), we find 
V = x x y 2 z 3 + x 2 y 3 z x + x 3 y x z 2 - x x y 3 z 2 - x 2 y x z 3 - x 3 y,z x = 0. 
By forming the powers and products of the second degree a 2 , ab, &c., we obtain 55 
equations between the symmetric functions of the second degree in each set of roots. 
But we have V 2 = 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 the second degree in each set. I use for shortness the sign S to denote the sum 
of the distinct terms obtained by permuting the different sets of roots, so that the 
equations for the fundamental symmetric functions are— 
a = x x x 2 x 3 , 
b= 2AM3, 
c = z x z 2 z 3 , 
3f = S y x y 3 z 3 , 
3g = S z x z<>x 3 , 
3h = S x x x 2 y 3 , 
3i = >Sf y x z 2 z 3 , 
3j = Sz x x 2 x3, 
3k = S x x y 2 y 3 , 
61 = Sx x y 2 z 3 ; 
then the complete system of expressions for the symmetric functions of the second 
order is as follows, viz. 
a 2 — nr* 2/y» 2no 2 
lA/y lAj2 tA/g j 
b 2 = y x y 3 y 3 , 
c 2 = Z X 2 Z 2 2 Z 3 2 , 
be = y x z x y 2 z 2 y 3 z 3 , 
ca = z 1 x 1 z 2 x 2 z 3 x 3 , 
ab = x x y x x 2 y 2 x 3 y 3 ,