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

92 HISTORY OF THE THEORY OF DETERMINANTS

CAUCHY, A. L. (1 8 56, Feb.).
[Suir une formule tre's-simple et tre's-gene'ral.... Comnptes rendus...Acctd. des SciL (Paris), xl1ii. pp. 366-874; or (Euvres
comipletes (1), xii. pp. 30-2-311.]
The main theorem concerns the set of equations
r,1X1 + a,-2X2 +.. + a, ThrJ,=
where the number m of unknowns is greater than the number n
of equations', and the other theorem' deals with the case of this
where the u's all vanish. The weapons employed in the discussion
are the 'prodnit symbolique' of the u's and 'clefs alge'briques,' or
' clefs anastrophiques' as they are now called. The paper may be
compared with Sylvester's of the year 1839 on 'the derivation of
coexistence.'
HERMITE, C. (1856).
[Sur la theories des polynomies homogenes du second degre'. Note
vi. (pp. 154-190) of Programme.... d'Arith~me'ique, d'Alge'bre et de G~oe'6ntrie Analyfique,.. par MM. Gerono et
Roguet: 4e e6d. 215 pp. Paris.]
This note of thirty-seven pages, which is said to be 'd'apres.
M. Hermite, consists, of live sections, the -first of which (pp..
154-157) deals expressly and the others incidentally with d&'
terminants. The latter sections concern the invariance of the
discriminant (called 'the inyariant '), orthogonal transformation,,
etc., and are simply but suggestively written.
HEGER, I. (1856, July).
[Ueber die Aufidsung cines Systemnes von mehreren unbestimmten
Gleichungen des ersten Grades in ganzen Zahlen. Denkschr.
d. kc. Akad. di. W~iss. (Wien): mat h.-n ahtrw. C1., xiv. (2),.
pp. 1-1 22.]
Although in this lengthy paper the vanishing of 'the determinants. of a 2-by-n array is -repeatedly under consideration
(e.g. ~ 24, p. 8'7), nothing new on the subject presents itself.