1896-97.] Dr Muir on Bordered Skew Determinant. 
351 
9. In the skew determinant which he borders, Cayley uses 
diagonal elements which are all different, and the same course has 
been followed here for the purpose of a perfectly satisfactory com- 
parison of the results. It is most important to observe, however, 
that there is no great gain in generality in doing this in preference 
to using diagonal elements which are all unity ; for, so long as the 
diagonal elements are not zero, any one of them, rn say, can be 
changed into 1 , without the determinant ceasing to be skew, by 
dividing the elements of the row and of the column in which it 
occurs by Jin ; thus — 
m 
a 
p 
1 
a 
Jmn 
P 
Jmr 
— a 
n 
7 
= m n r 
a 
Jinn 
1 
7 
Jnr 
- P 
-y 
v 
Jmr 
y 
Jnr 
1 
And as unit-axial skew determinants are those which occur in 
connection with orthogonal substitution, — the most important 
sphere for the application of skew determinants, — it is very desir- 
able to employ this seemingly special kind in the formal statement 
of theorems. 
Doing this we write, for the purpose of making clear the law of 
development, the first four cases of our theorem, as follows : — 
m Itj h. 2 
— k 1 1 
— k 2 — a i 1 
m + + h 2 k 2 ) + aj j m 7^ h. 2 
a. 
m 
\ 
ll 2 
h 
= m + (W + h 2 k 2 + hjrj 
h 
1 
a l 
a 2 
+ oL^I m h 2 
+ a 2 | in h x h 3 
h? 
— a i 
1 
Pi 
hi k% 
k 3 
h'z 
— a 2 
-Pi 
1 
a \ 
a 2 
+ Si I m hr 
k 9 k 
3 
Pi 
+ | \ Jl 2 h 3 
• 
Ji'ey ^ 
a l a 2 
a l a 2 
Pi 
Pi 
