Dr Muir on Prohlem of Sylvester’s in Elimination. 305 
that any equation of the one set, say, differs from an equation 
in the other set, (eg) say, in only one term. Consequently, by sub- 
traction there results 
(2 A'E'C' - AA'2 - CC'2 - EB'2 -f ABQ)zx = 0 , 
whence A = 0 . 
Here, for the first time, A does not appear in the second poiuer. 
9. Leaving now the subject of the modes of obtaining the elimi- 
nant, let us see how the given problem comes to be connected with 
the finding of the discriminant of a quadric. 
Taking the quadric 
Ax" -f Bif 4- Cz? -f 2A'yz -t- 2^'zx -t- 2Cxy , 
let us suppose it resolvable into real factors, viz., 
a-^x -h B{y + and a^x + + y ^. ; 
then the coefficients of the three expressions are connected by the 
six relations 
^iy2'h ^27 i ~ > 
^l/?2 — L , yi®2 d" y2®’l “ j 
yiy2=C, ai^2 + a2^] = 2C'. 
Substituting in the last three the values of a. 2 ,p 2 j 72 obtainable 
from the first three, we have 
CP,^+By,^==2A'P,yA 
Ayi2+ Cai2=-2E'yiai > 
Ba,^ + Ap,^ = 2G\p,) ^ 
which are exactly the three equations of Sylvester’s problem. 
VOL. XX. 10/11/94 
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