1896 - 97 .] 
Dr Muir on a certain Eliminant. 
365 
From this and the relation between the linear factors, it imme- 
diately follows that the difference between the two original deter- 
minant factors lies in the fact that, if we view the first as a function 
of a , b, c, d, e, /, g, the second is the very same function of 
a, -b, c, - d , e, -/, g , — a fact which is easily proved directly, as 
follows : — 
The second determinant factor, as we have seen, is 
a b c d e f 
~9 
abode 
~9 ~f 
abed 
JUL 
~9 ~f ~e 
a b c 
tt 
~9 ~f ~ e ~d 
a b 
-g -f -e -d -e 
a 
- g -f -e - d -c - b 
Now, this will not be altered in substance by changing the signs of 
all the elements in the odd-numbered rows and thereafter in the 
odd-numbered columns, so that it must be equal to 
a -b c - d e — / 
9 
a -b c - d e 
9 ~f 
a -b c -d 
J_L 
9 ~f e 
a -b c 
tt 
g -f e -d 
a - b 
g -/ e -d e 
a 
g -f e -d e -b 
and this is simply the first determinant factor with 5, d , / replaced 
by - b , -d, -/. 
5. The factorisation of the eliminant may thus be suitably 
epitomised as follows : — 
b , c, d, e,f, eg) = \p(a, b, c, d, e,f, g) . -b,c,- d, e, -/, g), 
— (a + b + c + d + e +/ + g)-X ( a ? /> o) ) 
(a-b + c-d + e -/+ g\x(a, b, c, d, e,f } g) j , 
= (ci + b + c + . . .)(u — b + c — . . .)-X' j 
where and % are functions analogous to (p } viz. 
