[104 
as constantly with the 
i with the positive or 
or interchange of two 
3 of two characters as 
md consequently intro- 
columns. Hence 
may be calculated by 
ich) as supernumerary 
iber of permutations of 
lumns will be employed 
same mark placed over 
he characters of that 
e notations 
104] 
ON THE THEORY OF PERMUTANTS. 
21 
iy. 
af columns, having the 
will in the sequel be 
)n-permutability will be 
olumn in question, and 
ant of an even number 
question is placed; and 
mark f to a com- 
V*,... are such as to 
•s a, ¡3, 7 ... A com- 
to be altered in sign 
se of the characters 
shes when any two of 
to be demi-skew when 
for any permutation 
of a commutant. The 
T placing in contiguity 
bar from the symbols 
of the adjacent sets. If, however, the symbols of the same set cannot be placed con 
tiguously, we may distinguish the symbols of a set by annexing to them some auxiliary 
character by way of suffix or otherwise, these auxiliary symbols being omitted in the 
final result. Thus 
n 1 la 
2 2 2 b 
3 3 5a 
v 4 3 66, 
would show that 1, 2 of the first column and the 3, 4 of the same column, the 1, 2 
and the upper 3 of the second column, and the lower 3 of the same column, the 1, 5 
of the third column, and the 2, 6 of the same column, form so many distinct sets,— 
the intermutant containing therefore 
(2.2.6.1.2.2 = ) 96 terms. 
A commutant of an even number of columns may be considered as an intermutant 
such that the characters of some one (no matter which) of its columns form each of 
them by itself a distinct set, and in like manner a commutant of an odd number of 
columns may be considered as an intermutant such that the characters of some one 
determinate column form each of them by itself a distinct set. 
The distinction of symmetrical, skew and demi-skew applies obviously as well to 
intermutants as to commutants. The theory of skew intermutants and skew commutants 
has a connexion with that of Pfaffians. 
Suppose V al3 y = F a+(3+v ... (which implies the symmetry of the intermutant or com 
mutant) and write for shortness V 0 — a, V x = b, V 2 = c, &c. Then 
0 0 0 0 
.1 1 1 1J 
.1 1. 
= 2 (ac — b~), 
= 2 (ae — 4 bd + 3c 2 ), 
= (ac — 6 2 ), &c. 
The functions on the second side are evidently hyperdeterminants such as are 
discussed in my memoir on Linear Transformations, and there is no difficulty in 
forming directly from the intermutant or commutant on the first side of the equation 
the symbol of derivation (in the sense of the memoir on Linear Transformations) from 
which the hyperdeterminant is obtained. Thus 
12 4 . UU, 
is 12 U'*U'\ 
'0 
0' 
is 12 2 . UU, 
"0 
0 
0 
0" 
.1 
1_ 
_1 
1 
1 
1_ 
“0 
t 
0“ 
is 12 U^U’ 1 , 
i° 
0 
0 
t 
0" 
_1 
1- 
Ll 
1 
1 
1_