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

116 HISTORY OF THE THEORY OF DETERMINANTS

or
a2 ab ac-3r ad~9q
ba b2 + 2r bc-q bd - 3p
ca - 3r cb-q e2~ ~2-p cd
da+9q db-3p de dc
where
p0 - Y~"(bd2), q = 7(be-ad), r = 2(ae b2),
the last change being probably due to the fact that it was
known that
V = 9(pr-q2)
and that verification would thus be easier.
CAYLEY, A. (1848).
[On geometrical reciprocity. Cambridge an-d Dub. Math. Jourrr.,
iii. pp. 173-179; or Collected Math. Papers, i. pp. 377-382.]
Incidentally Cayley gives the identity
2a a'+b a"// +c
a'" +b 2b' b" + c'
a a"+c b"~e' 2c"
4 a a' a"/ {ax 2+b'y2+c"z2 +(b"t+')yz+(c+ a")zx+(a'+b)xy
b b' b"/ - [cx(ab"f - a"b + a'c - ac') + y(b'c - be' + bUaV- b'a")
e C C" + z (c'a' - c'a" + cb" - clb 2
where
$= ax+a'y + a"z,;= bx + b'y + bz, ex= Cc+ cy +c'z.
No proof is adduced, and it is not noted that, when the
determinant Iab'c"I is axisymmetric, the expression in rectangular brackets vanishes, and the identity becomes in later
notation
a hg h b f a h g x
h b f g f c h bfy
~gfC gfc z