9 
1889 - 90 .] Dr T; Muir on Self-conjugate Permutations . 
1. Where no element is changed in position. 
2. Where two are changed. 
3. Where four .... 
4. Where six 
In the first class there is manifestly only 1, viz., the primitive per- 
mutation. In the second class there are as many as there are 
different pairs of elements, viz., C nA for example, 
2, 1, 3, 4, 5, .... , n, 3, 2, 1, 4, 5, ... , n. 
In the case of the third class we have to find how many pairs of 
pairs are possible, each having the four elements involved all 
different. For the first of the two pairs we have, as has just beeiq 
seen, C M>2 to choose from; for the second there are only C„_ 2>2 to 
choose from, because there are two fewer elements to. make the 
pairs of : consequently the number required is |C W>2 C W _ 2)2 , the 
J being due to the fact that the order in which the two pairs are 
taken is immaterial. In the case of the fourth class, we have to 
ascertain how many triads of pairs are possible, each having the six 
elements involved all different; and the result is similarly found 
to be-ij Cn i2 C w _ 2i2 C n _ 4t2 . The remaining classes manifestly follow 
the same law, consequently the proposition is established. 
3. If XJ n stand for the number of self-conjugate permutations of the 
elements 1, 2, 3, . . . , n, then 
U„ = U „_ 1 + (»-l)U„_ 2 . 
If, having examined the case of the elements 1, 2, 3, ...,(«- 1) 
we bring our wth element n to join them, it is clear that we have 
two possibilities to consider, viz. (1) the element n remaining in its 
place, (2) the said element suffering interchange with one of the 
others. Now, when it remains in its own proper position, the 
number of self-conjugate permutations is unaltered from the previous 
case, that is to say, is U M _!; and when it is taken with one of the 
n- 1 other elements to form a pair for interchange, there will arise, 
by reason of the remaining n- 2 elements, U„_ 2 self-conjugate 
permutations, that is to say, in all (n- 1) U n _ 2 . We have, conse- 
quently, as was to be proved, 
U n = U w _ 1 + ( W -l)U n _ 2 . 
