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

ALTERNANTS (GARBIERI, 1874)

153

A simple mode of verification consists in comparing the cofactor
of any one of the w's on the one side with its cofactor on the other.
A similar expansion is obtained for the determinant got by substituting the w's in any other column of I aa...al than the
first.
BELTRAMI, E. (1874).
[I DETERMINANTI, con...., per G. Garbieri. pp. 169-171.]
Beltrami carries us a step further than the so-called theorem
noted by Baltzer (1864). Instead of multiplying I ao00oa1 * an-,, n-i
by c aa... a1-1 I he (or Garbieri) multiplies
aoo ao,...ao, a, a
2 nt, al,... a,, 1 a2 a2... a
by
22.............
an,,-1, an-1,1 * * * a,-l,.... U
CCn 1-..* CO co,
cm C.... Co I... 0,
where the c's are defined by the identical equation
(x — a- )... (x-an) COXn + CfX"-1 + C+n-2..;+
and denoting
a00 + aao 02 +ao+2 +. +ao^ by fi('),
ao + al + 2a +... +an by f2(),
an-l,0+a-_ll — +a,,_l,2.. +. +a,,4' by f,,(),
he obtains for his product
fi(al) l(a2).... fi(an) fi(x)
f2(al) f2(a2).... f.(a,) f2(x)
/n(al) fn(a2). fn(an) fan(x)
0 0.... 0 CX +C-n +. C.
As, however, the multiplier is ~ (a,a, a,.. a,, x), and therefore
= (x- a)(x-an-.,)... (x-a2)(x-a,) *. (a,, a...., x)
= (x"+cx"-l+... + c,) *. '(a,, a2,. *., an),