391 
1905-6.] Dr Muir on the Theory of Circulants. 
of all the rows of A, and every other row of A' is got by subtracting 
the corresponding row of A from the row preceding it in A, then 
A' = ( — l) n_1 wA.* 
Strange to say almost all the examples given in illustration of this 
theorem (of § 13) are of the special form distinguished at a later 
date by the name “ circulant,” and consequently fall now to he 
considered. He says (§17) : — 
“ Alin de sortir de ces generalites, considerons les equations 
-x 1 + x 2 + x 8 + . . 
. + x n = u 1 , 
x 1 - x 2 + x B + . . 
. + x n = u 2 , 
X i + X 2 + X 3 + • • 
• X n 'U'n ) J 
Pour obtenir le determinant A , je remplace d’abord les equations 
donnees par les suivantes : 
(n - 2)x 1 + (n - 2)x 2 + ... + (n - 2)x n = u 1 + u 2 + ... + u n , > 
- 2x 1 + 2x 2 = u 1 -u 2 , 
— 2x 2 + 2x 8 = u 2 — u 8 , > 
-2x n _ 1 + 2x n = u n _ 1 -u n . ) 
D’apres ce qui precede, le determinant A' du nouveau systeme 
sera ( - l)” - hi A. Mais, d’un autre cote, en comparant A' au 
determinant A" du systeme 
x 1 + x 2 + . . . 
+ X 3 ~ 
-*1 
d~ x 2 = 
X 2 
+ X S ~ 
X n—1 
d* X n = 
on a A' = (n — 2)2 n_1 A". Enfin, d’apres le n° 13, et en observant 
que les quantites - A 2 , A 2 - A 3 , . . . . ont ici change de 
signe 
A " = n . 
* This result is reached in a way different from Catalan’s by performing on 
A' the operation 
rowj + (?t-l)row 2 + (%-2)row 3 + . . . + rown , 
separating out the factor n, and then showing that the resulting determinant 
is ( — l) n—1 A . 
