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

SKEW DETERMINANTS (SPOTTISWOODE, 1851)

265

it is not improved in the second edition by alteration into
"quadratic skew," the fact being that the system is not skew
at all, but is symmetric with respect to the principal diagonal,
or, in later phraseology, is axisymmetric.
The treatment of the next theorem taken up is happier than
the foregoing, and is after the outset no less fresh. Taking an
even-ordered skew determinant with zeros in the principal
diagonal he develops it according to products of an element
of the first row and an element of the first column, the result
being written in the form
* 12... In = (12)2 * 34... 3n + 2(12)(13) 34 35... 32 +...
21 * 2n 43 e. 45... 42
nln2 * n23.4.. ~. n4n5....n2
where, be it observed, the second typical term on the right
has been altered from

- 2(12)(1':) 32 34
42 *
n2 2n4... 3... 4n

by the translation of the first column to the last place. The
determinant in this typical term is then further transformed
into the square root of the product of two determinants like
that in the term preceding it, the steps of the reasoning being —

32 34...
42...
n22 n4...

3Xx 2 23 24.
4o _43.
* 7n3 n4. 2. 2 32 34... 3n. 4n 42... 4n
*.. ~ 2 n4... *

*
43
~ n
~3

24... 2n * 34...
*... 4n 42 *...,4... 2 n4..

3
4n
*?

the deletion of 23 and 32 in the last step being warranted by the
fact that their cofactors are determinants similar to the original