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

BORDERED DETERMINANTS

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show that 'extended determinant' as here used would have been
suitably replaced a year later by 'bordered discriminant.'
In Cayley's work the type in question appears as early as 1846
(Hist., ii. p. 113), and probably he was ready to accept a name for
it if the need again arose; at any rate in 1854, when dealing with
skew determinants, he employed the expression 'un determinant
gauche borde' for the case where the border is a single row and
column.
With Hesse the type appears in 1853 (Hist., ii. p. 129); and in
the year following Brioschi published his theorem regarding what
the English mathematicians called the 'bordered Hessian,' and
thenceforward the usage tended to become more common. Certain
it is that the determinants themselves soon came to be of very
frequent occurrence, especially in geometrical investigations; good
examples are Painvin's articles on the 'Application de la nouvelle
analyse aux surfaces du second ordre' in the Nouv. Annales de
Math. for 1859, C. W. Baur's in the Zeitschriftf. Math. u. Phys. for
1860 and 1861, and Clebsch's papers in Crelle's Journ. for 1863
and 1864.
A bordered determinant is in this sense ar determinant got by
bordering another, there being some reason for keeping that other
prominently in view. Were this not the case it might be convenient to define it quite differently, namely, as a determinant of
the (n+r)t& order which has nothing but zero elements in the complementary of its first n-line minor. Theorems concerning such determinants may thus take two very different-looking forms. The
property, for instance, which has already appeared in connection
with the vanishing of one of the primary minors of a determinant
may with at least equal convenience be put as follows: If a null
determinant be bordered, the resulting determinant is divisible by a
linear homogeneous function of either set of bordering elements; thus,
if la1b2c3 = 0, then
x. $ y z- x y z I* a 2 ' lalb21
i a2 aCt3 a a2 a3 Y bi b2, b1 b2 b3 1 b2 3 ~ c1 c2
I, c c2 (3

= - (XC1+ yC2+zC3)(A+N +,B3+ MC3) + C3.
2E

M. D. III.