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

280 HISTORY OF THE THEORY OF DETERMINANTS

arrives at formulae in which functions of both the forms {
appear; for example,
[231'{1}'-[13]'{2}'+[12]'. {3}',
where [23]', { 1}'.... stand for the complementaries of [23].
{1}.
We may note that Tanner might have expressed the odd-ordered
functions
[123], [12345].
also as Pfaffians, namely,
Yi Y2 Y 3 22!Y Y2 (3 Y4 Y5
12 13 112 13 14 15
23 23 24 25
34 35
ROBERTS, S. (1879).
[Note on certain determinants connected with algebraical expressions
having the same terms as their component factors. Messengerof Math., viii. pp. 138-140.]
Starting with Lagrange's identity
(x2 + ax 12 ~bx22 + abx32) (e2 + aX2 + b$92 + abe32)
= P2 +aP12 +bP92 ~abP,"2
where
P = 4 +axl,+bx2e2+ abx.~,
P1 = -4e,+ x1e -bXc4+ bx3e9,
P2 = -42+ +ax1e3+ x2 - axe,
P3 = -X43 - XI12 + x4C1+ X3e,
and considering the consequences of
e2+a412 + b$22+ abe.2
becoming 0, Roberts deduces the result
e ae1 be2 a b Cj
- -bC3 be = (~2+42+b$2+ab2 2)2.
$2 ae, ae
C3 $2 C