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

HESSIANS (GORDAN, 1875: NOETHER, 1876)

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forms; in the second Noether indicates a simpler way by which he
and Gordan had been able to include all quaternaries also, and
announces that Gordan had convinced himself of the impossibility
of further extension. In the third the two writers give a careful
and exhaustive exposition of their combined labours.
We may therefore now sum up by saying that the proposition
holds only for binary, ternary and quaternary quantics, and that
if we prefer to include also n-ary quadrics we have to bear in mind
that the Hessian is then practically identical with the discriminant.
Sylvester's dictum of 1853 has thus come to be justified.
GRAVELAAR, N. L. W. A. (1877).
[Eene stelling uit de theorie der lineare substitution. Nieuw
Archief v. Wi.7k., iii. pp. 193-202.
Accepting without question Hesse's pair of theorems of 1851, and
apparently being unaware of any subsequent investigations on the
subject, Gravelaar enunciates and seeks to establish generalisations
of both, the extension taking the direction of a greater reduction
than 1 in the number of variables. The direct theorem, it will be
remembered, had already received a slight extension of this kind
(see Hist., ii. p. 398 at bottom). In the case of the disputed converse theorem he satisfies himself of its truth by concluding too
readily, from the axisymmetry of the adjugate of the given vanishing
Hessian, that all the primary minors of the Hessian have a common
factor of the highest possible degree in the variables, and that
consequently all the cofactors are constants.
BARANIECKI, M. A. (1878-9).
[TEORYA WYZNACZNIKOW. xxiii+595 pp. Pary.l]
To Hessians Baraniecki devotes as many as twenty-two pages
(pp. 509-530), giving a full and well-informed exposition.
In the course of this he does not go beyond the binary quantic, in
which case he points out that we then have
U11-+- 12x2 = (~ — 1)1i,
'121x1 -+ u22X2 = (M - 1 ) U2

M.D. III.

2A