468 Proceedings of Royal Society of Edinburgh. [sess. 
CIRCULAR SUBSTITUTIONS. 
(35) Before proceeding further with the subject of “ inter- 
changes,” or “transpositions” as Cauchy called them, it will he 
convenient to take up that of “ circular substitutions ” and 
Cauchy’s rule of 1812. 
On referring back to § 3 it will be seen that the first thing need- 
ful in connection with the latter is to write down the substitution 
which is required for the transformation of the given permutation 
into the standard permutation, and then decompose or factorise it 
into substitutions which are circular. The best mode of obtaining 
the factors will be readily understood by observing the applica- 
tion of it to a particular case, say the case of the permutation 
8 7 1 5 4 6 3 2. 
The substitution needed to change this into the standard per- 
mutation being 
■pi 2 3 4 5 6 7 8\ 
\8 7 1 5 4 6 3 2/ 
we begin with the first element of the lower line, viz., 8, and learn 
of course that it has to he changed into 1 ; proceeding then to 1 
in the lower line we find that it has to be changed into 3 ; simi- 
larly that 3 has to be changed into 7, 7 into 2, and 2 into the 
element with which we started. This process gives us the partial 
substitution 
which from its nature is called “ circular ” or “ cyclic.” After this 
there remains the substitution 
M 5 6\ 
\5 4 6/, 
which being dealt with in similar fashion is separated into 
fi 0 and ©. 
so that we have finally 
71 2 3 4 5 6 7 8\ _ pi 3 7 2 8\74 5\ 
\8 7 1 5 4 6 3 2/ “ \8 1 3 7 2/ \5 4/ \6/ 
