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

AXISYMMETRIC DETERMINANTS (SYLVESTER, 1853) 129

between the perimeter and area of a triangle, namely, the fact
that
(ah)2 (ac)2 1
(acb)2 ( i.e. ( (ab)4-2 (cab)2(bc)2
(ac). (bc)2. 1
1 1 1
is a divisor of the result of rationalising
12(bc)2 + </2(ca)2 + d2(ab)
where the radicands are the complementary minors of the
elements in the places 11, 22, 33 of the determinant, boldly
extends the proposition (without proof) to any triangular
number of arbitrary quantities, taking occasion also to point
out that when we leave geometry (ab), (ac),.... may be written
for (ab)2, (ac)2,....
HESSE, 0. (1853, April).
[Ueber Determinanten und ihre Anwendung in der Geometric,
insbesondere auf Curven vierter Ordnung. Crelle's Jouornal,
xlix. pp. 243-264; or Werke, pp. 319-343.]
The main subject of the first half of Hesse's paper (pp. 243-253)
is a property of axisymmetric determinants required for the
establishment of the geometrical results contained in the second
half. In the first three pages he considers the relations between
the minors of two general determinants A, B, and the minors of
their product C; or, as he unfortunately feels himself compelled
to put it, "wie die partiellen Differential-quotienten der Determinante C nach ihren Elementen c genommen durch die
partiellen Differential-quotienten der Factoren A und B nach
ihren Elementen genommen sich ausdrticken lassen." What
follows thereafter may be described as the establishment of the
simple identity
U11 a12 a 11 U12 71 11 U12 al 22
11 12 1 a a2
Ub21 ~22 a2 ( /:21 'a22 72 - _21 T22 a2
a1 a2 0 Y 20l Y2 7 2 1 / 21 22 71 72
M.D. II. 2
M.D. II. I