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

404 HISTORY OF THE THEORY OF DETERMINANTS

Lastly, he attacks the general circulant, or, as he calls it, "le
determinant du systeme
a a2 a3... a
a2 a3 a4... a,
a3 a4 a5... a2
a(n a 2 a2....t:_ 1.
The procedure, however, -is rather perverse, the theorem of ~ 13
being forced into service. This gives
= (- 1)n-l a,
where A' is the determinant of the system
s ca1 -a- a2 a2 3.......n-1
S 2 a- C 3 -a4.... an - a1
s 3 -a 4 a4- a5.... a1 -a2
s - a,- al-a2 (.. a_2-_....
82a.-.1..-a-2 (.tit-1.
after which A'/s is partitioned into determinants with monomial
elements, and certain more or less evident reductions made. The
result is "Le determinant du systeme propose s'obtiendra en
multipliant a1 + a2 +. + ac (i.e. s) par le determinant du
systeme
ai-a a2 - a3,.... a,,n, - an
a2-a3 a 3-a 4 *... an -a 1
aCn ---a a-c n ( —. ~ ~ a_3n-3 - an-2n
a theorem which afterwards came to be written in the form
C(a1, a2., an. ) = (a +a2+... +an)
P(aL-a2,,.., an _-a,, a.. a a_ n-a 2),
the symbol P(x, y, z, w, v) being used to stand for the "persymmetric" determinant
x y z
z w.