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

288 HISTORY OF THE THEORY OF DETERMINANTS

HESSE, 0. (1862): HENRICI, 0. (1864).
[Zerlegung der Bedingung fur die Gleichheit der Hauptaxen eines
auf einer Oberflaiche zweiter Ordnung liegenden Kegelschnittes
in die Summe von Quadraten. Crelle's Journ., Ix. pp. 305 -312; or Werke, pp. 497-506.]
[Bemerkung zu "Hesse, Zerlegung der Bedingung....." Crelle's
Journ., lxiv. pp. 187-192.]
The equations of the surface and the plane being respectively
ax2 +by2 + cz2 + 2yz + 2mzx +2nxy = 0,
and Ax+By+Cz+D = 0,
where A2+B2+C2 = 1, and consequently
a-\ n mn A
n b-X I B
m, I c-. C 0
A B C
being the equation for determining the axes of the section, Hesse
expresses the discriminant, (X1- 2)2, of this equation in four
different ways as a sum of squares, namely, as a sum of 10, of 7,
of 6, and of 5 squares. To this Henrici contributes an expression
in the form of the sum of two squares.
The subject is continued by C. Souillart, G. Bauer, C. F. Geiser,
E. Sourander, and C. Souillart.*
Geiser makes an extension to the case where the determinant
is of the (n + l)t) order and the equation is of the (n- l)th degree:
he also recurs in another paper to the original allied problem of
Kummer.t
SYLVESTER, J. J. (1867).
[Thoughts on inverse orthogonal matrices, simultaneous signsuccessions,.... Philos. Magazine, xxxiv. pp. 461-475;
or Collected Math. Papers, ii. pp. 615-628.]
Sylvester, having been led to study determinants whose elements
are inversely proportional to the corresponding elements of the
* Crelle's Journ., lxv. pp. 320-334: lxxi. pp. 46-52: Annali di Mat., (2) viii.
pp. 113-120; Crelle's Journ., lxxxv. pp. 339-344: lxxxvii. pp. 220-221.
t Crelle's Journ., lxxxii. pp. 47-53.