20 
ON THE THEORY OF PERMUTANTS. 
[104 
the same symbol—in the case of an even number of columns constantly with the 
positive sign, but in the case of an odd number of columns with the positive or 
negative sign, according to the rule of signs. For the inversion or interchange of two 
entire lines is equivalent to as many inversions or interchanges of two characters as 
there are characters in a line, that is, as there are columns, and consequently intro 
duces a sign compounded of as many negative signs as there are columns. Hence 
Theorem. A commutant of an even number of columns may be calculated by 
considering the characters of any one column (no matter which) as supernumerary 
impermutable characters, and multiplying the result by the number of permutations of 
as many things as there are lines in the commutant. 
The mark -f* added to a commutant of an even number of columns will be employed 
to show that the numerical multiplier is to be omitted. The same mark placed over 
any one of the columns of the commutant will show that the characters of that 
particular column are considered as non-permutable. 
A determinant is consequently represented indifferently by the notations 
t + 
Hi' 
t 
'11' 
? 
'11' 
22 
22 
22 
^ nn j 
^ nn , 
and a commutant of an odd number of columns vanishes identically. 
By considering, however, a commutant of an odd number of columns, having the 
characters of some one column non-permutable, we obtain what will in the sequel be 
spoken of as commutants of an odd number of columns. This non-permutability will be 
denoted, as before, by means of the mark -f* placed over the column in question, and 
it is to be noticed that it is not, as in the case of a commutant of an even number 
of columns, indifferent over which of the columns the mark in question is placed ; and 
consequently there would be no meaning in simply adding the mark f to a com 
mutant of an odd number of columns. 
A commutant is said to be symmetrical when the symbols F a)3v „. are such as to 
remain unaltered by any permutations inter se of the characters a, /3, y ... A com 
mutant is said to be skew when each symbol V a/3y is such as to be altered in sign 
only according to the rule of signs for any permutations inter se of the characters 
a, /3, y, this of course implies that the symbol F a(3y ... vanishes when any two of 
the characters a, /3, y... are identical. The commutant is said to be demi-skew when 
V aif¡ty ... is altered in sign only, according to the rule of signs for any permutation 
inter se of non-identical characters or, /3, y,... 
An intermutant is represented by a notation similar to that of a commutant. The 
sets are to be distinguished, whenever it is possible to do so, by placing in contiguity 
the symbols of the same set, and separating them by a stroke or bar from the symbols