1904-5.] Dr Muir on General Determinants. 935 
where 
5 ^2 ’ * • • J S 2 ’ * • • ’ ’} i 1% > • • • 5 In J 
denote any permutation whatever, the same or different, of the 
series of integers 
1,2, ... ,re, 
and where ± r denotes the sign + or — according as the number 
of inversions in , r 2 , . . . , r n is even or odd. Using a f over the 
column of p’s to indicate that these are unpermutable, he shows 
that 
f 
A 1 
f A 1 
Pi 0-1 . . . 
Pi (Ti . . . 

- = 1*2 • ■ • • m - 
- 
^ Pn • • • - 
. Pn °"n • • • . 
when the number of columns, m, is even ; and vanishes when 
m is odd. In the former case it is clear that the placing of 
the f over any other column would have had the same result, and 
therefore that it is better to mark this indifference by placing it 
over the A. The other theorems, including a multiplication 
theorem, need not be given, our object being simply to show the 
position occupied by determinants among the new functions ; and 
this we can now do by quoting one sentence, viz., “ an ordinary 
determinant is represented by 
t 
t 
1 A „ 1 
A., 
“i& 
1 1 
.... 
- or 
.... j 
v a w fin - 
„ n n ) 
the latter form being obviously equally general with the former 
one.” 
In his paper of 1845 a further generalisation was made, the 
functions then reached being called ‘hyperdeterminants,’ and a 
hyperdeterminant defined as an expression representable as a 
homogeneous p th -degree function, H^, of certain of the deter- 
minants of a rectangular array, each of whose elements is 
