1889 — 90 .] Dr T. Muir on Self -conjugate Permutations. 
13 
7. In a zero-axial determinant of the nth order, U n = 0 when n is 
odd , and = 1 . 3 . 5 .... (n - 1) when n is even. 
Since here 11 = 0, the unique product just referred to vanishes. 
Consequently the difference-equation in this case becomes 
U n = (rc-l)U n _ 2 , 
Further, in such a determinant = 0. Therefore 
0=u 3 =u 5 =, . . . 
Again U 2 = 1, therefore 
U 4 = 3 . 1, TJ 6 = 5 . 3 . 1 , .... 
8. Returning now to the general determinant, and expanding it 
in terms of zero-axial determinants, we obtain 
15 
35 
45 
55 ( 
34 35 
. 45 
54 . 
15 
25 
11 
12 
13 
14 
21 
22 
23 
24 
31 
32 
33 
34 
41 
42 
43 
44 
51 
52 
53 
54 
45 
i n a 
22^33 
54 
• 
+ 
11 m '22 
43 
53 
+ 21 a 
11 
. 23 
32 . 
42 43 
52 53 
24 
34 
54 
25 
35 
45 
12 
21 
31 
41 
51 
13 
23 
14 
24 
32 . 
42 43 
52 53 
34 35 
. 45 
54 . 
where under the first 5 is included C 5>3 terms, under the second 
C 5i2 , and so on. And since the number of self- conjugate terms on 
the one side must be the same as the number on the other, there at 
once results with the help of the theorem of the preceding 
paragraph, 
U n = 1 + C m> „_2 • 1 + C w n _ 3 . 0 + C Wi „_ 4 .3.1+ C n<n _ . 0 
+ C Wi „_ 6 .5.3. 1 + . . . 
= 1 + 1 . C M( 2 + 1 . 3C W(4 +1.3. 5C„ >6 + . . . 
as before found. 
