[104 
104] 
ON THE THEORY OF PERMUTANTS. 
17 
STTS. 
As an instance of a simple permutant, we may take 
( '123) = l 7 )23 4" 1^231 + 1^312 F132 F<213 F 321 J 
and if in particular F 123 = clA 2 c 3 , then 
(F 123 ) = + aAci + a Ac* — ciAc* - aA^s 
It follows at once that a simple permutant remains unaltered, to the sign pres according 
to the rule of signs, by any permutations of the characters entering into the per 
mutant. For instance, 
(v m ) = (V m ) = (V m ) = - (F 132 ) = - (F 213 ) = - (F 321 ). 
Consequently also when two or more of the characters are identical, the permutant 
vanishes, thus 
FLi3 = 0. 
The form of the symbol may be such that the symbol remains unaltered, to the sign 
pres according to the rule of signs, for any permutations of the characters in certain 
(1852), pp. 40—51.] 
particular blanks. Such a system of blanks may be termed a quote. Thus, if the first 
and second blanks are a quote, 
are to be filled up by 
! of which are so filled 
ks of a symbol in the 
derived. Suppose the 
presented in the first 
Vj234 ... And it will be 
;o which reference will 
3, 4,... instead of as 
. . When the form is 
F 123 = — F 213 , F 132 =— F 312 , F 231 = f 321 , 
and consequently 
(F 123 ) = 2 (F 133 + F 231 + F 312 ); 
and if the blanks constitute one single quote, 
(F 123 ...) = A r F 123 ..., 
where N = 1.2.3 ... n, n being the number of characters. An important case, which 
will be noticed in the sequel, is that in which the whole series of blanks divide 
themselves into quotes, each of them containing the same number of blanks. Thus, 
if the first and second blanks, and the third and fourth blanks, form quotes respectively, 
^ (F t234 ) = F 1234 + F 1342 + F[ 423 + 1 3412 + Fj 213 + F 2314 . 
nitive arrangement and 
imber of inversions or 
e arrangements derived 
3ns or interchanges of 
e termed “ the rule of 
y possible arrangement 
jement, may be termed 
>ns which will presently 
Y enclosing the symbol 
aore elliptical than the 
bant, or the characters 
It is easy now to pass to the general definition of a “ Permutant.” We have only 
to consider the blanks as forming, not as heretofore a single set, but any number of 
distinct sets, and to consider the characters in each set of blanks as permutable 
inter se and not otherwise, giving to the symbol the sign compounded of the signs 
corresponding to the arrangements of the characters in the different sets of blanks. 
Thus, if the first and second blanks form a set, and the third and fourth blanks form 
a set, 
((F 1234 )) = F 1234 — F 2I34 — F 1243 + F 2143 . 
The word ‘ set ’ will be used throughout in the above technical sense. The particular 
mode in which the blanks are divided into sets may be indicated either in words or 
by some superadded notation. It is clear that the theory of permutants depends 
ultimately on that of simple permutants; for if in a compound permutant we first 
write down all the terms which can be obtained, leaving unpermuted the characters 
c. ii. 3