22 
ON THE THEORY OF PERMUTANTS. 
[104 
each permutable column 0 corresponding to a 12 (*) and a non-permutable column 0 
1 1 
changing UU into U'°U'\ Similarly 
'0 0 
1 
2, 
t 
becomes (12.13. 23) • UUU, 
0 0^ becomes 12.13.23 U'°U A U'*, 
1 1 
12 2 y 
^0 0 
becomes (12.13.14.23.24.34)“ UUUU, &c. 
The analogy would be closer if in the memoir on Linear Transformations, just as 
&e., for 
Î2 is used to signify 
fi. Vi 
, 123 had been used to signify 
£i 2 , IjiVu Vi 
Vi 
%sVi> Vi 
ÇsVs, Vs 
then f0 0^ would have corresponded to 123 .UUU, [0 0 ) to 123 Z7’ 0 Z7 ,1 Z7’ 2 ; and this 
11 11 
V 2 2 J U 2; 
would not only have been an addition of some importance to the theory, but would 
in some instances have facilitated the calculation of hyperdeterminants. The preceding 
remarks show that the intermutant 
fo 0 0) 
111 
0 0 0 
.1 1 1; 
(where the first and fourth blanks in the last column are to be considered as belonging 
to the same set) is in the hyperdeterminant notation (12.34) 2 . (14.23) UUUU. 
1 Viz. 0 corresponds to 12 because 0 and 1 are the characters occupying the first and second blanks of a column. 
1 
If 0 and 1 had been the characters occupying the second and third blanks in a column, the symbol would have been 
28 and so on. It will be remembered, that the symbolic numbers 1, 2 in the hyperdeterminant notation are 
merely introduced to distinguish from each other functions which are made identical after certain differentiations 
are performed.