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

272 HISTORY OF THE THEORY OF DETERMINANTS

The rest of the paper deals with inverse mantrices, and with
the application of them to the problem afterwards known as the
utqtomoorphic tcransformaCtion of a qcadrric.
BRIOSCHI, F. (1854).
[LA TEORICA DEI DETERMINANTI, E LE SUE PRINCIPALI APPLICAZIONI. viii+116 pp. Pavia.]
In this, the second text-book, the same importance is given to
skew determinants as in Spottiswoode, the first part of the eighth
section (pp. 55-72) being devoted to them under the heading " Dei
determinanti gobbi," which Schellbach translates by Qberschlagene.
The arrangement and treatment of the matter, however, are much
more logical, zero-axial skew determinants being taken first,
then the functions connected with these, namely, Pfaffians, then
skew determinants which are not zero-axial, and lastly the use
of skew determinants in the consideration of the problem of
orthogonal transformation.
The precedence given to determinants which are "gobbi simmetrici " over those which are " puramente gobbi" is explained at
the outset by reference to Cayley's theorem regarding the expressibility of the latter in terms of the former, the quite general
theorem from which Cayley's immediately follows being carefully
enunciated thus:
"Indicando con Po il determinante nel quale si pongano equali a
zero gli elementi principali; e con (",Pi)o un determinante minore
principale delle' mn-esimo ordine del determinante P nel quale siensi
annullati gli elementi principali si ha:P = Po + Era,.,.(lPi,)o -+ i... (2Pi)o +.. + 1a22... a M."
The proof given of Jacobi's theorem regarding the value of an
odd-ordered skew determinant with zeros in the principal
diagonal is essentially the same as Cayley's proof (1847), but
fuller and clearer. The proof of the corresponding theorem
for a determinant of even order resembles Spottiswoode's, the
difference lying mainly in the use of the notation of differential-quotients in specifying the minors of the determinant.