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

ORTHOGONANTS (CAUCHY, 1857)

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CAUCHY, A. L. (1857, Feb.).
[Sur les fonctions quadratiques et homogenes de plusieurs
variables. Conmptes rendus... A cad. des Sci. (Paris), xliv,
pp. 361-370, 416; or (Euvres completes (1), xii. pp. 421-432.
444-445.]
The second section of this bears the title "Sur l'equation
qui determine les maxima et minima d'une fonction reelle
quadratique et homogene de plusieurs variables dont les carres
donne pour somme l'unite," and at once recalls the important
memoir of 1829. The subject is the same, and any additional
result obtained is quite unimportant. Further, the mode of
treatment is not essentially different, the language and
notation of 'clefs anastrophiques' being for some obscure
reason substituted for those of 'sommes alternees.'
We have only to add, as being well worthy of note in
passing, that this was Cauchy's last contribution to the
literature of our subject, his first and greatest, and probably
the greatest of all, having been made so long before as fortyfive years. Three months after the last was presented to the
Academy he was dead.
BALTZER, R. (1857).
[THEORIE UND ANWENDUNGEN DER DETERMINANTEN, mit.....
vi+129 pp. Leipzig.]
Baltzer devotes a whole section (~15) of seventeen pages
(pp. 80-96) to the subject of "Die lineare, insbesondere die
orthogonale Substitutionen." The section, like its fellows, is
noteworthy, not for freshness of matter, but for good arrangement, clearness and compactness.
In treating of Cayley's orthogonant (~ 15, 6) he takes 1, not 1,
as the constant element of the basic determinant: and, when in
the course of the proof he obtains the two values for each of
Cayley's 0's, he does not equate them, but uses with each of them
Hermite's observation
xi+% = 210i,