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

262 HISTORY OF THE THEORY OF DETERMINANTS

and
1 12 X13 1 12 13
-1i2 1 23 or -12 1 23
-X13-23 1 -13-23 ]
= 223 + \213 + \212 + 1
=[23]2 + [13]2 + [12]2 + 1.
SPOTTISWOODE, W. (1851, 1853).
[ELEMENTARY THEOREMS RELATING TO DETERMINANTS......
viii+63 pp. London, 1851. Second edition, as an article
in Crelle's Journ., li. pp. 209-271, 328-381.]
In this the earliest of modern text-books on Determinants, a
special section (~ ix. pp. 46-51; or ~ vi. pp. 260-266 in second
edition) is set apart with the heading " On Skew Determinants."
As a matter of fact, however, it is only the latter half of the
section which at present concerns us, as the other half deals in
reality with Cayley's determinant solution of the problem of
orthogonal transformation.
In a sense the mode of treatment is indirect, the general skew
determinant being viewed, not as a- separate entity, but in its
relation to a set of linear equations, the coefficients of which are
its elements. The set of equations is
(11) x, + (12)x2 +... + (ln))x = U1
(21)x1 + (22)X2 +... + (2n)x, =
(nl)x1 + (qn2)x2 +... + (nn)xn = %1,
where it has to be remembered that in every instance (rr)=0
and (rs) + (sr)= 0. The right-hand members of what he calls the
"derived" set are v,, v,..., vn; that is to say, there exists.simultaneously with the original the set
(1l)x1 + (21)x2 +.. + (nl)x,, = V
(12)x, + (22)x +.. + (2)n = V2
(l1n)x + (2n)x2 +... + (nn),, = v,