220 Proceedings of Royal Society of Edinburgh. [sess. 
mr 
mr 
On the Eliminant of a Set of Ternary Quadrics. 
By Thomas Muir, LL.D. 
(Read December 7, 1896.) 
(1) In the Cambridge Math. Journal, ii. p. 233, Sylvester 
showed how to eliminate x, y, z from the set of equations j 
Ax 2 + ayz + bzx + exy = 0 
My 2 + lyz + mzx + nxy — 0 
R z 2 -\-jpyz + qzx + rxy = 0 
His method consisted in deriving other three equations involving 
the variables x 2 , y 2 , z 2 , yz, zx, xy, and then eliminating these six 
variables from the six equations. The result obtained was * 
A 
M 
. . R 
. ra - c(M + p) ma - &(R + l) 
n( A + q) . ma - l ( R + b) 
q ( A + n) ra - j?(M + e) 
a b c 
l m n 
p q r 
- A(R + 1) - A(M + p) 
-M(R + 6) X - M(A + q) 
- R(M + c) - R(A + n) & 
where d> = a(n + q) - 6(M +^>) - c(l + R) , 
X = m(p + c) - ?z(R + b) - l(g + A) , 
xp = r (b + 1 ) - p( A + n) - q(c + M) ; 
— a result which, on account of its complexity, it is impossible to 
rest satisfied with. 
The manifest fact that when A = M = R = 0, the eliminant takes 
the form 
a b c 
l 7ii n 
p q r , 
and the further fact, that when a = m — r — 0, the equations become 
linear, with the eliminant 
A c b 
n M l 
q p R 
* There are several troublesome misprints in the original. 
