46 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
followed by the operation of adding all the preceding rows to the (2n — l)st, 
we have a result from which[the factor (n— 1 . a-^ + n . <x 2 ) comes out at once, 
leaving a determinant whose elements along the secondary^ diagonal are 
(a 2 — (Xj), and all the elements to the left of it are zeros. 
It will be observed from (9) and Theorem YI that a circulant of odd 
order 2n — 1, having as elements (n— 1) a’s followed by n b’s, has the same 
value whether all the b’s follow all the as or alternate with them. 
7. In connection with Muir's paper of 1881, in which he shows that 
the cofactor of s in the circulant C of order 2n—l can be expressed in 
symmetric form, the use of these same properties of the roots of unity 
brings out the fact that this symmetric determinant may be factored into 
linear factors. The consideration of the case of a continuant of order 
2n — 1 = 9 and therefore n = 4 will serve to illustrate. 
The factors of the circulant other than s are 
a x + a 2 6 k + a 8 6 2k + a ^0 3k + a b 6 ik + a & 6 bk + + a 8 0 lk + a 9 0 8k = a k (k= 1,2,. . .8). 
The product of the first and eighth (a x and a 8 ) gives 
2a 4 2 + ^a-^a^O + 6 8 ) 4- 2 ^ 3(02 + 0 7 ) + + 0°) + 'Ia l a 5 (6 4: + 6 b ) 
or 
I. A 1 + A 2 (0 + 6 8 ) + A 3 (0 2 + 0 7 ) + A 4 (0 3 + 0«) + A 5 (0 4 + 0 6 ) = ai >8 , 
say, where 
0 
A& = 
Similarly, the other pairs, a 2 a 7 , a 3 a 6 , a 4 a 5 , give 
II. A, + A 2 (0 2 + 0 1 ) + A s (0 4 + 0 5 ) + A 4 ( 3 + 06) + Ag($ + 6 8 ) = a 2; 7 
III. A, + A 2 (0 3 + 0 Q ) + A 3 (0 3 + 0 6 ) + A 4 (2) + A 5 (0 3 + 06) = 03 1 6 
IV. A 4 + A 2 (0 4 + 0 5 ) + A 3 (0 + 0 8 ) + A 4 (0 3 + 6 6 ) + A 5 (0 2 + 0 7 ) = a 4 , 5 . 
These four relations may be written as follows : — 
(A, - A 2 ) + (A 2 - A 3 )(l + 0 + 0 8 ) + (A 3 - A 4 )(l + 0 + 6 8 + 6 2 + 6 7 ) 
+ ( A 4 - A 6 )( L + 0 + 0 8 + 0 2 + 0 7 + 0 3 + 0 6 ) - tti f 8 
(A 2 - A 3 )(l + 0 + 0 8 + 02 + #7) + (A 4 - A 4 ) - (A 2 - A 5 )( 1 A 6 + <9 8 ) 
+ (a 3 -a 5 xi + ^ + 0 8 +0 2 + 0 7 + ^ 4 + ^ 5 ) = «2, 7 
— (A 3 - A 4 )(l + 0 3 + 0 6 + 0 3 + 0 6 + 0 9 + 0 9 ) — (A 2 - A 5 )(l + 0 3 + 0 6 ) 
+ (A 4 — A 5 ) + (A 2 — A 4 )(l + 0 3 + 0 G + 0 3 + $ 6 ) = a 3 _ 6 
-(A 4 - A 5 )(l + 0 + 0 8 + 0 2 + 0 7 ) + (A 3 -A 5 )(l + 0 + 0 8 ) 
— ( A 2 — A 4 )(l + 0 + 6 8 4 - q* + (P + 0 3 + 0 G ) + (A 4 — A 3 ) = a 4 5 . 
Here we have not only the elements of the symmetric determinant 
but the multipliers for the columns which give the factors, and the factors 
themselves, which are real. 
The law of formation of these symmetric determinants is, perhaps, 
best seen by examining the elements along the principal and parallel 
