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

56 HISTORY OF THE THEORY OF DETERMINANTS

HAMBURGER, M. (1875).
[Zur Theorie der Integration eines Systems von n linearen partiellen Differentialgleichungen... Crelle's Journ., lxxxi. pp.
243-280.]
Incidentally (pp. 265-266) Hamburger states that the number of
independent relations of the type
IfgI. I hk - f II glk + If/t I.ghl =
or (f, g, h, k) = 0, say,
between 2-line minors of a 2-by-n array is ~(n-2)(n-3), that is
to say, the number of different sets of four columns in which two
fixed columns always appear; and what he claims to prove is that
if 1121 be not 0 and for all values of g from 3 to n[lg( and 12g[
be not simultaneously 0, then all the relations in question are
deducible from those of the type (1, 2, h, k)= 0. The simple
facts are, that from the equations
l121-hkl - 1A11-[2k1 + i-kl H 12hl = 0
121 ikgl - lkl'12g l + lg 12kl =
I121 Jghl - 1lgl'j2hj + hlhjj2gl = 0o
by using the multipliers 1lgl, Ilhj, Ilk|, and adding, there is
obtained
1121 (ghk) = 0;
and having thus got
(lghk) = 0, (gfh) = O, (Ighk) = 0,
there is similarly deducible
lgI~ (fghk) = 0.
Again, instead of (lghk) = 0, we might equally easily have got
(2ghk) = 0,
and thence
2gi ~ (fghk) = O.
Provided therefore Ilgl and 12gl be not simultaneously 0, it
follows that
(fghk) = 0.