148] MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. 
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Consider first the method of symmetric functions, and to fix the ideas, let the 
two equations be 
(a, b, c, d\x, y) s = 0, 
(p, q, r \x, yf = 0. 
Then writing 
(a, b, c, d\ 1, zf = a (1 - az) (1 - ¡3z) (1 - yz), 
so that 
h ~ = a + P + 7 = (1) 
+ - = a/3 + ay + fiy = (l 2 ), 
CL 
d 
ci/3y 
a 
the Resultant is 
(p, q, r\a, l) 2 . (p, q, r\fi, l) 2 . (p, q, l) 2 , 
which is equal to 
r 3 + qr 2 (a + /3 + 7) +pr 2 (a 2 + /3 2 + <f) + pqr (a 2 /3 + a/3 2 + /3 2 7 + fdy 2 + 7a 2 + y 2 a) + &c.; 
or adopting the notation for symmetric functions used in the memoir above referred 
to, this is 
{ r 
{+ qr 2 (1) 
f+pr 2 (2) 
\+ q 2 r (l 2 ) 
(+ pqr (21) 
[+qs (l 3 ) 
C+p 2 r (2 2 ) 
\+pq 2 (21 2 ) 
[+p*q (2 2 i) 
[+p z (2 3 ) , 
the law of which is best seen by dividing by r 3 and then writing 
and similarly, 
C. II. 
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