104] 
ON THE THEORY OF PERMUTANTS. 
25 
permuted so that x, y, z may go with 1, 2, 3 and with 4, 5, 6 and with 7, 8, 9. 
The order of the x, y, z in the second triad may be considered as arbitrary; but 
once assumed, it determines the place of one of the letters in the first triad; for 
instance, x3 and z9 determine y*l. The first triad must therefore contain xl and z6 
or x6 and zl. Suppose the former, then the third triad must contain z3, but the 
remaining two combinations may be either x4, y5, or x5, y4. Similarly, if the first 
triad contained xQ, zl, there would be two forms of the third triad, or a given 
form of the second triad gives four different forms. There are therefore in all 
24 forms, or 
f x 
1' 
' X 
V 
y 
2 
y 7 
z 3 
z 6 
x 4 
x 8 
y 
5 
y 
2 
z 6 
z 9 
X 7 
x 4 
y 
8 
y 5 
9, 
3; 
where the bars in the second column on the first side show that throughout the 
equation 1, 2, 3 and 4, 5, 6 and 7, 8, 9 are to be considered as forming distinct 
sets. The above proof is in reality perfectly general, and it seems hardly necessary 
to render it so in terms. 
To perceive the significance of the above equation it should be noticed that the 
first side is a product of determinants, viz. 
r X 1 
t 
f x 5' 
t 
[x 71 
y 2 
y 6 
OO 
5^ 
v. Z 3 , 
7, 
.z 9, 
and if the second side be partially expanded by permuting the characters of the 
second column, each of the terms so obtained is in like manner a product of deter 
minants, so that 
24 
' x 
V 
t 
' x 
4' 
t 
' x 
7 ' 
t = 
'x 
1' 
t 
f x 
8' 
t 
' x 
4 > 
y 
2 
y 
5 
y 
8 
y 
7 
y 
2 
y 
5 
V Z 
3, 
,z 
6, 
.z 
9 > 
.z 
6, 
»z 
9, 
.z 
3, 
the permutations on the second side being the permutations inter se of 1, 2, 3, of 
4, 5, 6, and of 7, 8, 9. 
It is obvious that the preceding theorem is not confined to intermutants of two 
columns. 
C. II. 
4