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

386 HISTORY OF THE THEORY OF DETERMINANTS
for all values of r and s a; and consequently from (y)
ax =( -(m)-(Nxx1 + Nmx +
=-(m-1)Nsx {ju+2jx +..} +
= -(n- l)(mr- 2)Nxul,
whence
D2A -2u
axzv = -a(-1)(m- 2)N ~ —
This result we may formally enunciate as follows:-If the first
differential-quotients of a homogeneous rational integral
function all vanish, the elements of the Hessian of the function
are proportional to the elements of the Hessian of the Hessian.
TERQUEM, O. (1851, March).
[Note sur les determinants. Nouzv. Annales de Math., x. pp.
124-131.]
This is an elementary exposition of Hesse's determinant, with
simple illustrations from algebraic geometry, the property of
"covariance" being made prominent. A curious distinction is
made between what are called the "first" and "second"
determinants: for example,
4ac - b2
*The first two of these results, which follow from (a) and (P), there is no
pressing reason for mentioning: it would have been equally pertinent to note
that z=0. The third result Jacobi probably obtained (see Crelle's Journal, xv.
p. 304) by taking in every possible way n - 1 of the initiatory set of equations
and deducing
X1: X2 ' ~:' Xn = U11: U21: '* * *.:Unl
= U1: U,2:...: Un2,
= Ulz: U:.'U..: U.
This implies that U,.s: Urs' = Xs: Xs,
and US'r: Us/r' = x2.: Xr;
and from these, by reason of the equality of U,,, and Use. there is got by multiplication
Urs: UsIrt = XsX: XsXrf.