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

320 HISTORY OF THE THEORY OF DETERMINANTS

thus reaching the elements
21L,, 21LS,.
A I A
of the desired substitutions without more trouble. On the other
hand, he fails to note that Cayley's 0's are so introduced as
to ensure from the outset the equality of x12+x22+...and
12 + 22+.., and thus he is led to prove propositions already
established (~15, 5).
Brioschi's equation of August 1854 being denoted (~15, 9)
by f(x)=0, he multiplies f(x) by f(-x), and obtains for
f(x). (-x)/xn a skew determinant having each diagonal element
equal to 1/x-x. This determinant being therefore expressible
as a sum of squares when n is even, and as 1/x-x times a sum
of squares when n is odd, the part of Brioschi's proposition
which asserts the unreality of the roots follows by a reductio ad
absumrdum.
SALMON, G. (1859).
[LEssoNs INTRODUCTORY TO THE MODERN HIGHER ALGEBRA,...
xii+147 pp., Dublin.]
In Salmon's treatment of the subject (~~ 118, 139, 142, 156-7,
163-4) only two points call for remark. In the first place,
"orthogonal transformation" with him is not as with his predecessors a transformation which merely changes
x2+y2+2+,.. into e2+22++2+...,
but one which at the same time changes
ax2 + by2 + cz2... + 2fyz + 2gzx + 2hxy +... into A2 + B2+C2 +...
In the second place, he has a fresh mode of arriving at the
equation for determining A, B, C,... Calling the four quadrics
just mentioned V, V', U, U', he forms the discriminant of U-XV,
and asserts that the coefficient of all the several powers of X in it
must be invariants, and that, therefore, if the said discriminant
be put equal to 0 and the equation so obtained be solved for X,