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

310 HISTORY OF THE THEORY OF DETERMINANTS

SYLVESTER, J. J. (1852, July).
[A demonstration of the theorem that every homogeneous
quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative
squares. Philos. Magazine (4), iv. pp. 138-142; or Collected
Math. Papers, i. pp. 378-381.]
The terms "orthogonal transformation" and " orthogonal substitution" date from the year 1852, the former appearing in a
paper of Sylvester's published in the February part of the
Cambridge cnd Dub. Math. Journ. (see vol. vii. p. 57), and the
latter in the title of the paper now reached. In the former
paper, too, the word "unimodular," as applied to a transformation, is first used (see p. 52), the meaning being that the
modulus-that is to say, the determinant of the coefficients of
transformation-is then unity.
As has been already noted * when dealing with axisymmetric
determinants, this opens with the proposition that when ar =asr,
all + a12... al a-x a12...
a21 a22 -+x... a2n a21 Ca22-. Ca2
nl an * * *.nn +X anl a.2... ti
q1- X2 q12... qln
q21 q22- 2 ~ * * 2n
qnl qn2 * * * nn -X2
where q.'S = (cala,2... car a,a2s... a~),
and where therefore
q11 q22 * *. q = Ialla22. ~nn 2
It is then pointed out that the last determinant multiplied by
(-1)n is expressible in the form
(X2)n - Q1(X2)n-1 + Q2 (X2)-2 -.....;
* On verifying this, see also the account of the related paper published in the
Nouv. Annales de Math. for November 1852.