8 Proceedings of Royal Society of Edinburgh. [sess. 
are conjugate, because 3 is in the 1st place of (A) and 1 is in the 
3rd place of B, and so on in every case. A permutation may be 
conjugate with itself. Thus the permutation 
6 3 2 4 5 1 
which has 6 in the 1st place, has also 1 in the 6tli place, and so 
on. A self-conjugate permutation may consequently be defined 
independently, as one in which each element is either in its original 
position or has taken part in one interchange. 
Conjugate permutations are identical with those which Jacobi* 
calls reciprocal , his definition being that two permutations are so 
called when the performance of the one upon the other gives rise 
to the primitive permutation. For example, 3 5 2 1 4 and 4 3 15 2 
are reciprocal permutations, because 
(35214) (43152) = 12345. 
In regard to self-conjugate permutations, there is an interesting 
unsettled question which Rothe raised, viz., as to the number of 
them corresponding to any particular number of elements. As a 
partial solution he gave the difference-equation which it satisfies, 
but nothing further. No proof even of the equation was given, and 
it is the only result so left in his paper. Our present purpose is to 
furnish a full solution of the problem. 
2. The number of self-conjugate permutations of the elements 
1, 2, 3, ... , nzs 
1 4- C w>2 + J . C Mi 2 C m _ 2( 2 + i • 3 • C Wi2 C w _2 i2 C n _4 >2 + . . . . 
where as usual C w>r stands for n{n - 1) . . . (n - r + 1)/1 . 2 . 3 . . . . r. 
This is best established as it was first obtained, viz., by classify- 
ing the instances of self-conjugateness, and then making a census of 
the classes. A basis of classification exists in the varying number 
of elements retaining their primitive positions in the permutations. 
Taking this basis, we see at once that we have to consider in order 
the classes 
* “De formatione et proprietatibus determinantium,” Crelle’s Journ., xxii. 
p. 287. 
