24 ON THE THEORY OF PERMUTANTS. [l 04 
and, observing that the 1, 2 form a permutable system as do also the 3, 4, the 
second and third terms vanish, while the first and fourth terms are equivalent to 
each other; we may therefore write 
t 
' X 
1' 
= 
' X 
r 
y 
2 
y 
4 
X 
3 
X 
3 
,y 
4, 
,y 
2 , 
where on the first side of the equation the bar has been introduced into the second 
column, in order to show that throughout the equation the 1, 2 and the 3, 4 are 
to be considered as forming distinct sets. 
Consider in like manner the expression 
r x 1' 
V 7 
s 6 
x 8 
V 2 
* 9 
x 4 
V 5 
3, 
where in the first column the sets are distinguished by the horizontal bars and in 
the second column the characters 1, 2, 3 and 4, 5, 6 and 7, 8, 9 are to be 
considered as belonging to distinct sets. The same reasoning as in the former case 
will show that this is a multiple of 
x 1 N 
y 2 
z 3 
x 4 
y & 
z 6 
x 7 
y 8 
and to find the numerical multiplier it is only necessary to inquire in how man) 
ways, in the expression first written down, the characters of the first column can be