The theory of determinants in the historical order of development, by Sir Thomas Muir.

402 HISTORY OF THE THEORY OF DETERMINANTS
his own, which in his notation might have been written in the
form
de't. (A+A+A... -+A, A1-A2, A2-A3,..., A.,11-An)
=(-1)j-n1.n-det. (A,, A2,, * An),
and which is to the effect that If from a determinant A of the
nth order we form another A', such that the first row of A' is the
sum 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
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 conseqnently fall now to
be considered. He says (~ 17):
"Afin de sortir de ces g6neralites, considerons les equations
- 1+x2+x3+... +xn = U1, - X + +... X = U2J
X1 + X2 +,X3 + Xn - U = 17,
Pour obtenir le determinant A, je remplace d'abord les equations
donnees par les suivantes:
(it - 2)x, + (n - 2)X2 +. + (n2 - 2)x,, = U1 + U2, +. + Uns,
- 2x, + 2x2 = U1 - U2
- 2x2 + 2x3 = U 2 - U3
-2x,,v +~2x,, =Un-1 U
D'apres ce qui pr&eWde, le determinant A' du nouveau systeme sera
( - 1) )nA. Mais, d'un autre c6te, en comparant A' au determinant
A" du systeme
-X1 +x2 =
-x2 +X3 =
* This result is reached in a way different from Catalan's by performing on
A' the operation
row, + (n - 1) row2 ~ (n - 2)row3 +.. + row,,,
separating out the factor n, and then showing that the resulting determinant is
(-1)"n'A.