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

172 HISTORY OF THE THEORY OF DETERMINANTS

first row from each row after the first, when it is found that the
factor
(C, - a2)(L-).*. (a, - a)
(l + b2)(a1 + b3)... (a,+ bn)
can be removed, and that the cofactor is a determinant similar to
the original but of the (? -l)th order, namely, the determinant
which is the cofactor of the element in the place (1, 1) of the
original. The final result thus obtained agrees with Cauchy's
save in having no sign-factor, the latter being only necessary
when the b's are all made negative.
BRIOSCHI, F. (1854, Oct.).
[Intorno ad alcune formole per la risoluzione delle equazioni
algebriche. Annali di Sci. mnat. e is., v. pp. 416-421; also
reprinted as half (~ 2) of the last note in the French translation of Brioschi's text-book; or Opere mat., i. pp. 157-161.]
All that occurs in this paper in connection with our subject is
the statement
So S1.... S_2 1
81 82.... Sn-1 X
A) s2 s8.... X2
)xi } S= 2.... S......
8n-2 8&_ 1....2.w-4 2
1 X.... n-2 1
where A is the determinant-form of the difference-product of
x1, x2,..., x. No explanation of the statement is given, nor
the mode of arriving at it. All is made clear, however, if we
note first that by x on the right hand is meant any x chosen at
will from the set x, x,..., xn: second that the differentialquotient on the left is intended to stand for the cofactor of the
( -l1)th power of that particular x in A, and therefore merely
denotes the difference-product of a certain n -I of the x's.
What the statement thus gives us is an alternative form for the
square of the difference-product of n-1 quantities.