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

ALTERNANTS (KOSTKA, 1875)

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and attractive way, which we may illustrate by the example already
used, namely, a0b2c5d6 |. Writing this in the form
a0 a1 a2 a3 a4 a5 a6
bO bl b2 b3 b4 b5 b6
C0o 1 C2 C3 C4 C5 c(
- dO d d2 d3 d4 d5 d6.... 1..,
he performs the operations
col7 -C colg +C2 col5 -C col4,
col6-C col5 col-+C 3 col3,
col1-C0 col4+C0 col-3-C colb,
which give the result
a0 al a2 a3
b0 b1 b2 b3
c0 c1 c2 c3
- d~ l d2 d3. 1.. — C. C4
1 -C 02 -C3
~.... i -C 02,
that is,
a0bl'c2d3. C' 4
C1 C2 03
1 CI C2
and thence, as before,
2 Cl 1
I a b2c5d6 - I
i aOblc2d3~ - 0
He also makes the first rediscovery of Schweins' multiplicationtheorem of 1825, and then takes the fresh step of using it to express
any symmetric function as a sum of determinants. Thus, since
Schweins' theorem gives
aObl2d3 (a2b2 +a2c2+a2d2+b2c2+b2d2+c2d2)
= alb2c3d4 j - ab2c3d5 1 + ablc4d5 I,