An introduction to the modern theory of equations, by Florian Cajori.

CHAPTER X

ON SUBSTITUTIONS
81. Notation. In the arrangement or permutation of four
letters, laaa,a,, let each letter be replaced by one of the others;
put, for instance, a4 for Ca, a3 for a,, ac for a3, and.c2 for (4, then
this operation, called a substitatiol, may be designated by the
notation
(a'2a?~a4',
\a4at3( lt2/
where each letter is replaced by the one beneath, or by the
notationL (alcc.4a3), where each letter is replaced by the one
imllediately following, the last letter, ac, being replaced by the
first, a. We shall use more frequently the second notation.
Observe that (/123\ (x13x),
\xoxX2/
and that \\24531} 24
12 4 5 3 5).
Just as the substitution (caa4aa,), effected upon the arrangement aa2c3a4, gives the new arrangement c4c3aa2, so when
effected upon c4a3acla, it gives a2ala4a3.
We shall agree that in a substitution a letter may be replaced
by itself, but that no two letters can be replaced by the same
letter. Accordingly
alaCa4a2/
is a substitution, but (cla,CaaC2a4) is not, because in the latter
ca and as are both replaced by as.
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