364 
Proceedings of Royal Society of Edinburgh. [sess. 
abode f 
-g 
abode 
-g -f 
a b o d 
JJL 
1 
1 
1 
a b o 
ft 
-g -f -e -d 
a b 
0 
1 
1 
Qi 
1 
1 
1 
a 
rC> 
1 
0 
1 
1 
<» 
1 
'Kv, 
1 
I 
— (a b + c — d + e — g -f- g') 
abode 
-g 
a b o d 
-g -f 
a b c 
JLL 
TT 
-g -f 
a b 
i 
1 
1 
1 
a 
1 
1 
X 
1 
ct> 
1 
1 
cs 
The square factor of the ehminant, left after the removal of 
(a + b + c + d + . . .) (a-b + c- d-y . . .), may thus he written — 
a 
b 
a 
c 
b 
a 
d 
c 
b 
a 
e 
d 
c 
b 
a 
-g 
-g -f 
-g -f -e 
-g -f ~ e -d 
g -f -e -d -c 
4. This squared factor possesses the property of remaining 
unaltered in substance on having its even-numbered elements, 
b, d , /, changed in sign. For, on making this alteration, the 
result is 
a - b o 
-d 
e 
-g 
a — b 
c 
-d 
-g 
f 
a 
-b 
c 
J_L 
+T 
-g 
f 
- e 
a 
-b 
-g 
f 
- e 
d 
a 
-g f 
- e 
d 
- c 
and, as this cannot he affected in substance by changing the signs 
of the even-numbered rows and columns, we have it 
abode 
~9 
abed 
-g -f 
a b c 
JLL 
tt 
1 
1 
1 
a b 
-g -f -e -d 
a 
l 
1 
1 
X 
i 
Css 
I 
as stated. 
