1896 - 97 .] 
Dr T. Muir on Ternary Quadrics. 
231 
the 4th column into the 6th place, and thereafter the 1st row 
into the 3rd place and the 4th row into the 6th place. 
(16) A consequence of this is that if the determinant he 
expressed in terms of the elements of the last row and their 
complementary minors, only two of the seven resulting expressions 
require to be fully expanded in order to obtain the final develop- 
ment. The first of the two is 
(<r + cr') x its complementary minor , 
and the second is any one of the remaining six, say 
(AM# + bnr) x its complementary minor. 
The former is invariant to all the substitutions ; the latter by 
means of the substitutions gives rise to the remaining five similar 
expressions. In other words, the eliminant may be written 
(AMR + amr + bup + clq) 
+ 22(AM# + bnr) 
A 
a 
• 
9 
9 
c 
• 
M 
m 
l 
9 
9 
r 
. 
R 
. 
a 
9 
e 
• 
b 
a 
A 
9 
n 
• 
. 
. 
m 
M 
9 
P 
• 
R 
• 
r 
a 
• 
9 
9 
c 
b 
M 
m 
l 
9 
e 
n 
9 
R 
• 
? 
0 
P 
9 
b 
a 
A 
9 
c 
9 
9 
9 
m 
M 
l 
V 
e 
R 
i 
r 
<1 
where, it may be noted, the first determinant equals 
(AMR + amr ) 2 - 2AMR2A.p.Z + 22 AM .pb.lq - 2AM .am.pq - bnp.clq. 
(17) Strange to say there is another form of the eliminant, which 
has all the properties of that of § 14, and differs from it only in 
the last row. This is due to the fact that from the original three 
equations there is another way of obtaining an equation in 
x 2 y, y 2 z, z 2 x, yz 2 , zx 2 , xy 2 , xyz . 
