or THE BOOTS OE CEETAIN SYSTEMS OE TWO EQUATIONS. 
7iy 
But if the roots of the given system are 
(<^15 y \1 (.^2, ^2? ^2)5 
then the resultant of the three equations will be 
and comparing the two expressions, we have 
b =3/1 3/2, 
0 = 2,22, 
2f =3/122+3/2^1, 
2g=2,^2+22a^i, 
2h=^i?/2+ ■T23/1, 
which are the expressions for the six fundamental symmetric functions, or symmetric 
functions of the first degree in each set, of the roots of the given system. 
By forming the powers and products of the second order a^, ah, &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 symmetric functions of 
the form in question, and the solution of the linear equations gives — 
P 2 _|, /yj2 
d" ■ tv Jw 2 5 
b" =y\yl 
be =3/, 2,3/222, 
Ca = 2i.T,22^2, 
ab=^, 3/1^23/2, 
ii^—2hc=y\zl-\-ylz\, 
4g^ — 2 ca = z\x\ + 22^.’,, 
4h* — 2 ab = ^,3/2+ /r23/,, 
2af — ^,3/1 ^2'^2 ~l” 
2bg=?/, 2,^23/2+^13/1^222, 
2ch = z,x,y^z^-\-y,z,z.jJc^, 
4gh— 2af =^3/222+/P23 /i 2 :i, 
4hf — 2bg=y?22^2+3/22i-^i, 
4fg — 2ch = z\x^y^-\- zlx,y,, 
2M=y\tj^z^-\-yly,z,, 
2cg = 2,22^2+ 
2ah = /rf^2 y^ + ,3/1 , 
5 A 2 
