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

318 HISTORY OF THE THEORY OF DETERMINANTS
when n is even it is already of the latter form. All values of z2
thus obtainable must be negative, and consequently all the values
of z save the value 0 must be imaginary and must occur in
pairs whose sum is zero. But as
A x-1
Z (l+x)A x+l
and
l+z
it is clear that for every pair of values of z that differ only in
sign there must be a pair of values of x that are reciprocals.
The theorem reached by Brioschi we may thus enunciate for
ourselves as follows:-The roots of the equation
11 -- 12. '. 0) n
021 C22 X... W C2n -
W)nl W(n2 ~ ~ ~ (nn- X
where Iwo wo22 * * * ol is Cayley's orthogonant, are arrangeable
in pairs of reciprocal imaginaries, save when n is odd, in which
case there is the single real root 1.
When instead of the cw's we take the coefficients of the
substitution which transforms a general quadric into itself, the
words "reciprocal imaginaries" need to be changed into "reciprocals." This generalisation Brioschi published a month or
two sooner (see Annali di Sci. mat. efis., v. pp. 201-206).
BRUNO, F. FAA DI (1854, September).
[Note sur un theoreme de M. Brioschi. Journ. (de Liouville) de
Math., xix. p. 304.]
On multiplying both sides of Brioschi's equation (1854,
August) by 11 co22 * * * c,,, land dividing by (- x)' an equation is
obtained which differs from the original simply in having x-1
for x. The portion of the theorem which concerns "reciprocity"
Bruno thus readily establishes.