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

AXISYMMETRIC ]DETERMINANTS (CALDAIREIRA, 1871) 103
The evaluation of the cofactor of an Caldarera effects with much
trouble (pp. 226-231) by means of his (or, rather, Chio's) condensation-theorem, not observing that a series of similar steps can be
made by simply diminishing each row by the row immediately
preceding: for example, when n =4, we have
1 2 3
D (a, 0) 1 3 1:3
4 1 4 i
a 1 4 10'
the theorem repeat edly used in performing the subtractions being
Cn'' c- Cmr = nlr1
The same process applied directly to D (a, 8) gives the value a" with
equal ease, showing at one and the same time that D (a, 8) =ID(a, 0),
and that the axisy mmetric cofactor of an in the latter is equal to 1.
SCIIULTZE, E. (1871).
[Uber die aus einer symmetrischen Determinaante An = 1 ~4 all1...an
gebildete iReihe An) A~1,-. O.,. Sch. Progr. 22 pp.
Berlin.].The title is misleading, the subject really being the linear transformation of a quadric, whose discriminant is An, into an aggregate
of multiples of squares. As, however, one of the modes of transformation referred to is that resuscitated by Brioschi in 1856,* but
originally due to Lagrange (1759 t), namely, that which changes
a13__xJ + a22c12 4-...+ annXn2 + 2a12XJ1X2 4 --into
'2Al2 A51 + 7 -*Brioschi, F., "1Sur les series qui donnent le nombre de racines re'elles des edquations alge'briques h une ou 'a plusieurs inconnues," Nouv. Annales de Math., xv. pp.
264-285; or Zeitschrift f. Math. u. Phys., ii. pp. 209-222; or Opere Mat., v. pp.
127-143.
-~Lagrange, J. L., "1Recherche sur la me'thode de maxim'is et minimis,"
Miscellt... Taurinensis, i. p. 18; or (~Euvres, i. pp. 3-20. See also Gauss (1823),
le~rke, iv. pp. 27-53, and Jacobi (1847), Crelle's Journ., Iiii. pp. 265-270.