474 Proceedings of Royal Society of Edinburgh. [sess. 
corresponding to it in the case of the conjugate permutation an 
item of substitution /3- into-X ; for, X in the former being the element 
and (3 its place, there must by definition be in the latter an element 
/3 in the X th place. Consequently if 
/ filP * ' * 3 ^-\ 
VX.A/X, . . . , £/ 
be a circular substitution for the case of the original permutation, 
then 
p X\ 
\X, . . . , <9, /*,/?/ 
must be a circular substitution for the case of the conjugate permu- 
tation. 
(45) Included in this is the fact that conjugate permutations have 
the same number of circular substitutions , and therefore have the 
same sign. Also, from § 41 it follows that conjugate permutations 
require the same number of interchanges. Indeed, whichever of 
the five rules of signs we employ, the number of things to be 
counted in the case of any permutation is the same as in the case 
of the conjugate permutation. 
(46) Since in a self-conjugate permutation the circular substitu- 
tion must remain unaltered by reversal, it follows that the circular 
substitutions of a self -conjugate permutation must be either mono- 
mial or binomial. 
This enables us to determine the total number of self-conjugate 
permutations of n elements. For, confining ourselves to circular 
substitutions of these two kinds, we have only to count the 
number of possible cases with no binomial circular substitutions, 
those with only one, those with only two, and so on. The result is 
1+C + 1 r n + _JL C C C + . . . . 
x T 2 T 2,2 ^ 1.2.3 n ,2 n - 2,2 n - 4,2 
or 
1 + bC n2 + 1-3-C^ + 1 3.5.C Bi6 + . . . . 
as is already known.* 
* See Muir, “On Self-Conjugate Permutations,” Proc. Roy. Soc. Edin., 
xvii. pp. 7-13. 
