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

DETERMINANTS IN GENERAL (BRUNO, 1852)

73

C(~l a.2 *.. Cn)... P. (..;.. b&n)
=2 s(^lf**^fl *6 E(~ 1... b q
= [I ( a a2. aa P b'+ bp+2 b ). (~ b b2. P +1 a+, +2. 2Lp+1 ~p+2 ~ O 01 02 ' + 1 p) l -2 j
using two other sets of letters like 01, 2,..., O. This change
in notation being allowed for, the new proof is in general
character exactly the same as the old; it is, however, more
concise and more clearly set forth. It starts with the fact
that any term arising from the expansion of the typical product
on the right-hand side may be written
a] a2... aP a+ +l... a2 bl' b V2...b b p bf+l bp+2... b0.
2 1P P+1 n 0O 02 Op 0p+1 Op+2 On
Then observing the'indices superieures' attached to the b's in this,
we are asked to consider two possible cases. In the first place, we
have to note that if no one of the VI's be identical with any one
of the O's, the term is a term of the expansion of the product on
the left-hand side, and that the number of such terms in the
expansion of each product on the right-hand side being
(1.2.3... n). (1.2.3... p) (1.2.3.... -p)
and the number of products
n(n-1)(n-2).... (n-p +l)
1.2.3... p
the total number of such terms is
(1.2.3... n)2,
which is exactly the total number on the left-hand side. In the
second place, if one of the Vt's be identical with one of the p's, say
%i = 01,+h,
it is pointed out that there must exist another term in which, in
place of having.... i......
Vi p01)+h
we shall have............ b......
pi h.. i
and that these two terms having necessarily different signs, must
cancel each other.