THE EEY. GEOEHE SALMON ON QHATEENAET CHBICS. 
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Each of these breaks up into factors : choosing any set of these factors, the resultant of 
the four factors conies out easily in the form of the determinant 
d-j d , \/ d 
\/ d, \/ b -\- \/ d, \/ d 
\/ d, \/ d , \/ 6 -}- \/ d 
that is to say, 
\/bcd-\-y/ cda-\-\/ dah-\-\/ abc. 
We are then to multiply together the eight results obtained by giving all possible 
variations of signs to the radicals, and the product will evidently be the same as the 
result of clearing of radicals the equation 
s/bcd-\-\/ cda-\-s/ dab-\-\/ abc ^^ ; 
that is to say, 
{ Vedr + chW + ddcdl / + - 2abcd{ab -^ac^ad-\-hc-\-cd-\-dh)Y~^ 
or P=:64Sh 
Evidently then, by precisely the same process, the discriminant of the surface 
cix^ -\-bf-\-cz^-{- did + cv^ 
is the result of clearing of radicals the equation 
\/bcde-\-\/ cdea-\-\/ deab-\-\/ eabc-\-\/ abcd=0; 
and on performing the operation the result is found to be 
{A^-64Br=16384{D+2AC} ; 
where A, B, C, D are the fundamental invariants described above. 
It is evident, in like manner, that if it be requii’ed to find the result of elimination 
between three ternary quadrics, since we can write these functions in the form 
a x^-\-bf -^cz^ -\-dii\ 
c(! 5 ■ 
cd'x"^ + V'lf + d'z^ + d"id, 
we can reduce these to the form 
ax'^ = = hid \ 
that therefore, as before, the eliminant can be expressed as the result of clearing of 
radicals an equation of the form 
y'ba-\-\/ 
and therefore, as Mr. Sylvester first pointed out, the eliminant is of the form 
P=64S. 
But in this case S is not a perfect cube, as it turns out in the problem of finding the 
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