1889-90.] Dr T. Muir on Self-conjugate Permutations. 11 
It therefore follows that 
+ (n- l)U n _ 2 = 1 + 1 . C W(2 + 1 . 3C Mi4 + 1.3. 5C n>6 + . . . . 
= D W . 
5. To every problem regarding the permutations of 1,2,3, . . . , n, 
there corresponds a problem regarding the terms of a determinant. 
What problem on determinants, then, corresponds to the problem 
here dealt with ? The answer is most easily perceived if the 
elements of a self-conjugate permutation be considered the column- 
numbers of the determinant, and as such be suffixed to the row- 
numbers 1, 2, 3, . . . , n, taken always in this the natural order. 
For example, if we combine the row-numbers 1 2 3 4 5 6 with the 
self-conjugate permutation 3 6 1 4 5 2 of the column-numbers, we 
obtain 
1 3 2 6 3i4 4 5 5 6 2 , 
a term of the determinant 
I !> 
and at once see that it possesses the property of remaining un- 
altered when the row and column numbers of every factor of it are 
interchanged — that, in fact, such an interchange merely alters the 
order of its six factors, 1 3 , 2 6 , 3 1? . . . . That this is the characteristic 
property of such a combination of row-numbers in the natural 
order with column-numbers in any order constituting a self- 
conjugate permutation is evident on recurring to the definition 
with which we started. The same property may be expressed with 
regard to the vertical line notation for determinants, by saying that 
the elements of the determinant which compose the term are cut 
symmetrically by the main diagonal. Our problem is thus trans- 
formed into finding the number of terms of a determinant of the 
wth order, which are such that the n elements taken to form any one 
of them are in the square array distributed symmetrically about 
the main diagonal. Terms of this kind may fitly be called self- 
conjugate; and, generally, the word conjugate may be used in 
regard to the terms of a determinant exactly as in regard to per- 
mutations. We may even go further, and apply it to the elements, 
calling a TiSi a s<r conjugate; in which case, two terms could be 
defined as conjugate when the elements of the one were conjugate 
