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

-70 HISTORY OF THE THEORY OF DETERMINANTS

For example, the value of
a b
c cl
ef
g h
will be found by taking the (1.2)3 arrangements, as below,
ab ab ab ab ab ab ab ab
ed dc cd dc cd dc cd dc
ef ef fe fe ef ef fe fe
gh gh gh gh hg hg hg hg,
the signs of cd, ef, gh being supposed -t, those of dc, fe, hg will be
each -. Consequently the sum of the terms will be expressed by
aceg. bdfh - adeg. bcfl - acfg. bdelh + adfg. bceh
- aceh. bdfg + adeh e bcfg + acfrh bdeg - adfh. bceg."
It will be observed that the example here given is the quadratic
function which Cayley would have denoted by
000
TT
\ 111 1/1
and which, on the supposition that generally VO, B,,, = Va+/3+vy+
and in particular that V0 = c, V1 = b, V2 = c,.... would represent
cee - bd - bd + c2 - bd + c2 + c2 - db,
i.e. ae - 4bd + 3c2.
In his applications of the theory of commutants to that of
'forms,' Sylvester uses for the first time differential operators as
umbrae, speaking, for example, of the commutant
Dx 3y. _ D',
a D '
DX2~~ Dy. Dz.
this being possible from the fact that the coefficients of a
'form' nu are, save for a constant factor, representable as differential-quotients of u. This we may illustrate for ourselves by
the result