18 
ON THE THEORY OF PERMUTANTS. 
[104 
of a particular set, and replace each of the terms so obtained by a simple permutant 
having for its characters the characters of the previously unpermuted set, the result 
is obviously the original compound permutant. Thus, in the above-mentioned case, 
where the first and second blanks form a set and the third and fourth blanks form 
a set, 
((F 1234 )) = (F 1234 ) - (F 1243 ), 
Or (( F a234 )) = ( F a234 ) — ( F 2J34), 
in the former of which equations the first and second blanks in each of the permutants 
on the second side form a set, and in the latter the third and fourth blanks in each 
of the permutants on the second side form a set, the remaining blanks being simply 
supernumerary and the characters in them unpermutable. It should be noted that 
the term quote, as previously defined, is only applicable to a system of blanks belonging 
to the same set, and it does not appear that anything would be gained by removing 
this restriction. 
The following rule for the expansion of a simple permutant (and which may be 
at once extended to compound permutants) is obvious. Write down all the distinct 
terms that can be obtained, on the supposition that the blanks group themselves in 
any manner into quotes, and replace each of the terms so obtained by a compound 
permutant having for a distinct set the blanks of each assumed quote; the result is 
the original simple permutant. Thus in the simple permutant (F 1234 ), supposing for 
the moment that the first and second blanks form a quote, and that the third and 
fourth blanks form a quote, this leads to the equation 
(F 1234 ) = + ((F,034)) + ((F 134 ,)) + ((F 1423 )) + ((F :j412 )) + ((F 4213 )) + (( 
where in each of the permutants on the second side the first and second blanks form 
a set, and also the third and fourth blanks. 
The blanks of a simple or compound permutant may of course, without either 
gain or loss of generality, be considered as having any particular arrangement in space, 
for instance, in the form of a rectangle : thus F 12 is neither more nor less general than 
34 
F 3234 . The idea of some such arrangement naturally presents itself as affording a means 
of showing in what manner the blanks are grouped into sets. But, considering the 
blanks as so arranged in a rectangular form, or in lines and columns, suppose in the 
first instance that this arrangement is independent of the grouping of the blanks into 
sets, or that the blanks of each set or of any of them are distributed at random in 
the different lines and columns. Assume that the form is such that a symbol 
a '£Y... 
is a function of symbols F a|3v ..., F^y..., &c. Or, passing over this general case, and 
the case (of intermediate generality) of the function being a symmetrical function, 
assume that 
r.ir- 
a'P'y'—