[104 
t 
i-permutable column 0 
1 
UUUU, &c. 
transformations, just as 
fi 2 > %iVi> Vi for 
& 2 , ^2, i?2 2 
£ 3 2 , ZsVs, Vz 
23 U'°U' l U' 2 \ and this 
the theory, but would 
inants. The preceding 
:onsidered as belonging 
V. 23) UUUU. 
d second blanks of a column. 
the symbol would have been 
.’perdeterminant notation are 
after certain differentiations 
104] ON THE THEORY OF PERMUTANTS. 23 
It will, I think, illustrate the general theory to perform the development of the 
last-mentioned intermutant. We have 
'0 
0 
(T 
= 
'0 
0 
0' 
- 
'0 
0 
0' 
- 
'0 
0 
11 
+ 
r° 
0 
1 j 
1 
1 
1 
1 
i 
i 
1 
i 
0 
1 
1 
1 
1 
i 
0 
0 
0 
0 
0 
0 
0~ 
0 
1 
0 
0 
1 
0 
0 
1 
,1 
1 
T 
a 
i 
i, 
a 
1 
1, 
A 
1 
0, 
a 
1 
oj 
t 
t 
t 
t 
2! 
0 0 0 
0 
0 
0 
- 
0 
0 
0 
"0 0 r 
t 
_1 1 1_ 
-1 1 1. 
-1 1 o_ 
_1 1 1_ 
= 2 {{ad — be) 2 — 4 (ac — b a ) (bd — c 2 )}, 
= 2 (a 2 d 2 4- 4ac 3 + 4<b 3 d - 3b 2 c 2 — Gabcd), 
the different steps of which may be easily verified. 
The following important theorem (which is, I believe, the same as a theorem of 
Mr Sylvester’s, published in the Philosophical Magazine) is perhaps best exhibited by 
means of a simple example. Consider the intermutant 
( x 1 > 
V_ 4 
x 3 
M 2, 
where in the first column the sets are distinguished as before by the horizontal bar, 
but in the second column the 1, 2 are to be considered as forming a set, and the 
3, 4 as forming a second set. Then, partially expanding, the intermutant is 
' X 
V 
- 
f y 
V 
- 
' X 
r 
+ 
f y 
1 ' 
y 
4 
X 
4 
y 
4 
X 
4 
X 
3 
X 
3 
y 
3 
y 
3 
y 
2, 
Ly 
2, 
1# 
2, 
K.X 
2 j 
or, since entire horizontal lines may obviously be permuted, 
+ + + + 
^ il-/«, n _ ^ 
' X 
r 
- 
'y 
r 
- 
' X 
r 
+ 
f y 
r 
y 
2 
y 
2 
X 
2 
X 
2 
X 
3 
X 
3 
y 
3 
y 
3 
,y 
4, 
4, 
,y 
4, 
A 
4 )