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

322 HISTORY OF THE THEORY OF DETERMINANTS
have it again in terms of the 4's. In doing this Hesse pays
attention only to the term in 12'.., making the assertion
that the coefficient of t1^2. i' a XeXe... x^ is either the
same as the coefficient of XX2... x i2n 12 *... 1 or differs
from the latter coefficient by a merely arithmetical multiplier.
From this it follows that the coefficient of 1...2 in any
term A2e... e. iSX a merely arithmetical multiple
of Ae2. e,; and, if the multiplier in question be denoted by
Oee2... en, there results from the equatement of coefficients
1- ~ L elee2... en-Ae e2.. en,'
Next, let us suppose in addition that our substitution transforms an n-ary quadric
fi(x, 2, x..., Xn) into g2 2 + g.. + + ge2
a step which, as we know, introduces the quantities whose reality
is in question. In regard to them Hesse first recalls Jacobi's
proof (1833) that they are such that
g2C2 + g2$2 +... + ~g g3 + g2 2 +... + g.n.
lS 1 2 52 it ~ ~ I5 1 2 2
are also expressible as homogeneous quadric functions of the x's,
and that the coefficients of these quadrics are rational integral
functions of the coefficients of the original quadric fl. It is
seen to be not inappropriate therefore to use
fP(cxX2,..., x,) for gqe + g22 +. + 12
and to denote the partial differential-quotient of fp(xl, X2,,,n)
with respect to x, by f(,(x), thus giving
(), = -a 1ga 1+ aagf, +...+ (%
V? (7) klgl g1l + ak2912 J **apn
The next step is the deduction of an important result from the
consideration of the determinant
Xc1 X c.. Xn
/fl'(1l) /fl'(X2) f* (Xl)
1f'() f (x2) I... f() or a say.
Yfn-l(Xl) /2fn-l(2) * 2 f-i( q)