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

AXISYMMETRIC DETERMINANTS ("H (1)," 1846) 115
and
2aa' ab' + ba' ca' + ac'
ab'+ ba' 2bb' bc'+cb' = 0.
ca' + ac' be'+ cb' 2cc'
It is seen that both determinants are axisymmetric, that the
second is expressible as the product of two vanishing determinants, and that the first is formable from the second by
subtracting aa' + bb' + cc' from each diagonal element,-a fact
which, taken along with the vanishing of the second, shows that
aa' + bb' + cc' is a factor of the first.
CAYLEY, A. (1847).
[Note sur les hyperdeterminants. Crelle's Journal, xxxiv.
pp. 148-152; or Collected Math. Papers, i. pp. 352-355.]
The second paragraph of this note concerns the expression
6abcd + 3b2C2 - ad2 - 4ac3 - 4b3d, or V say,
soon afterwards (1851) to be called the "discriminant" of the
binary cubic
ax3 + 3bx2y + 3cxy2 + y3;
and Cayley's proposition is that the determinant whose elements
are the second differential-quotients of V with respect to a, b,
c, d, namely, the axisymmetric determinant
- 2d2 6cd 6bd- 12c2 6bc-4ad
6cd 6c2-24bd 6ad+12bc 6ac-12b2
6bd-12c2 6ad+ 12bc 6b2- 24ac 6ab
6bc-4<ad 6ac- 12b2 ab - 2a2
is a numerical multiple of V2. As a matter of fact he says the
multiplier is 3; but this is because, instead of writing the determinant as here, he removes from it the factors 2, 6, 6, 2. A verificatory proof, unsatisfactory to himself, is given, the determinant
being as a preliminary again altered into a multiple (by 64) of
a2 ab 2b2- ac 3b c- 2ad
ab 4ac-lb2 2bc+ 'aCd 2c2-bd
2b- ac 2bc + -ad 4bd- c2 cd
3bc-2ad 2c2-bd cd d2