149] OF CERTAIN SYSTEMS OF TWO EQUATIONS. 457 
symmetric functions of the form in question, and the solution of the linear equations 
gives— 
a 2 __ /yi 2/yi 2 
tU] i/(y O j 
b 2 = y?yi, 
c 2 = z?z£, 
be = y x z x y 2 z 2 , 
ca = z x x x z 2 x 2 , 
ab = x x y x x 2 y 2 , 
4f 2 — 2bc = y?z£ + yiz-c, 
4g2 _ 2ca = z x x 2 + z 2 x x , 
4h 2 — 2ab = x 2 y 2 + x 2 y x , 
2af = x x y x z 2 x 2 + z x x x x 2 y 2 , 
2bg = y x z x x 2 y 2 + x x y x y 2 z 2 , 
2ch = z x x x y 2 z 2 + y x z x z 2 x 2 , 
4gh - 2af = x x y 2 z 2 + xiy x z x , 
4hf 2bg = y x z 2 x 2 -f- y?z x x x , 
4fg - 2ch = z x 2 x 2 y 2 + z 2 x x y x , 
2bf = y x y 2 z 2 + y 2 y x z x , 
2cg = z x 2 z 2 x 2 + z 2 %x x , 
2ah = x 2 x 2 y 2 + x 2 2 x x y x , 
2cf = z x y 2 z 2 + ziy x z x , 
2ag = x?z 2 x 2 + x 2 %x x , 
2bh = y x x 2 y 2 + y 2 2 x x y x . 
Proceeding next to the powers and products of the third order a 3 , a 2 b, &c., the 
total number of linear relations between the symmetric functions of the third degree 
in respect to each set of roots exceeds by unity the number of the symmetric functions 
of the form in question; in fact the expressions for abc, af 2 , bg 2 , ch 2 , fgh, contain, 
not five, but only four symmetric functions of the roots; for we have 
abc = x x y x z x . x 2 y 2 z 2 , 
4af 2 = (x x y x 2 x 2 z 2 2 + x 2 y 2 2 x x z x 2 ) + 2x 1 y 1 z x x 2 y 2 z 2 , 
4bg 2 = (y x z x 2 y 2 x 2 2 + y 2 zly x x x 2 ) + 2 x x y x z x x 2 y 2 z 2 , 
4ch 2 = (z x x x 2 z 2 y 2 2 + z 2 xfz x y x 2 ) + 2 x x y x z x x 2 y 2 z 2 , 
8fgh = (x x y x 2 x 2 z 2 2 + x 2 y 2 2 x x z x 2 ) j 
+ (yiZiy^i + y£z 2 y x X 2 ) 'r + %X x y x Z x X 2 y 2 Z 2 , 
4" (z x x 2 z 2 x 2 + z 2 x 2 z x y 2 ) j 
C. II. 
58