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

360 HISTORY OF THE THEORY OF DETERMINANTS
of which a generalisation of A, namely,
a2h2 + b2g2 + c2f2- ~ d2e2 + 4adfg + 4bceh or Q say
- 2ahbg - 2ahcf - 2ahde - 2bgcf - 2bgde - 2cfde J
is an invariant; expresses Q as a two-line determinant
ah - bg - f + de - 2(eh-fg)
- 2(cad-be) ah.- bg - cf + de
makes the substitution
Xl = (ah-bg-cf + cde) - 2(eh-fg) C2
ayi - =2 - (ad-bc)4 + (ah-bg -cf +de);'
and, as the multiplier connecting the new Q and the old is now
the second power of the modulus, he obtains what was wanted.The substitutes found turn out to be
I 3Q 1 3Q 1 DQ
23a' 2 ab 2 Dc
but no explanation of this is vouchsafed.
Eisenstein's case is the degeneration reached by putting
a, b, c, d, e, f, g, h
= a, b, b, c, b, c, c, d.
Had the two other sets of variables been at the same time
transformed with the same determinant for modulus, we should
have had the new Q equal to Q7.
BOOLE, G. (1844, June).
[Notes on linear transformation. Cambridge Math. Journ., iv.
pp. 167-171.]
The third note being devoted to the proof of his well-known
theorem of 1841 regarding what afterwards came to be known
* This short paper of Cayley's teems with misprints, both in the original and
in the Collected M1ath. Papers. -