224 
Proceedings of Boyal Society of Edinburgh. [sess. 
— that is to say, the determinant is a function which is symmetrical 
with respect to the interchange 
/ am r b n p\ 
I 
\A M E c l ql . 
(9) Of course an immediate deduction from this is, that if the 
said interchange he made in the original set of equations, the 
eliminant will not he altered ; in other words, that the eliminant 
of the set 
Ax 2 + ciyz + bzx + cxy = 01 
My 2 + lyz + mzx 4 - nxy — 0 
R 2 2 -\-pyz + qza + rxy = 0 ) 
is the same as the eliminant of the set 
ax 2 + A yz + czx + bxy = 0 
my 2 + nyz + M zx + Ixy = 0 
rz 2 + qyz + yzx + Rxy = 0 
This is readily established from a consideration merely of the two 
sets of equations. For, multiplying both sides of the original 
equations by yz , zx, xy respectively, we have 
a(yz) 2 +A (zx)(xy) + c(xy)(yz) + b(yz)(zx) = 0 
m(zx) 2 + n{zx){xy) + AL(xy)(yz) + l(yz)(zx) — 0 
r(xy) 2 + q{zx)(xy) + P (xy)(yz) + T&(yz)(zx) — 0 
and the original problem of elimination is thus changed into 
another perfectly similar problem in which the variables are yz, zx, 
xy instead of x, y, z, and in which a takes the place of A, and A 
of a, etc. Or, substituting h, ~ f 0 r x, y, z in the original set of 
equations, we obtain a set with necessarily the same eliminant, viz., 
A 
a 
b 
c 
= 0 
0 
+ 
— 
+ 
— 
+ 
— - 
XX 
yz 
zx 
xy 
M 
l 
m 
n 
= 0 
+ 
— 
+ 
— 
+ 
— = 
yA 
yz 
zx 
xy 
R 
p _ 
q 
r 
= 0 
0 
+ 
+ 
+ 
— = 
£4 
yz 
ZX 
xy 
and this set when cleared of fractions is the second set given 
above. 
