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

458 HISTORY OF THE THEORY OF DETERMINANTS

BORCHARDT, C. W. (1859, May).
[Ueber eine der Interpolation entsprechende Darstellung der
Eliminations-Resultante. Crelle's fJorn., lvii. pp. 111-121;
or ]Ifonatsb. d. Akcad. d. lWiss. (Berlin), pp. 376-388; also
abstract in Annani di Mat., ii. pp. 262-2(34.]
The representation in question is in terms of the values which
the two functions p(x) and -fr(x), both of the nthh degree, assume
for the values ac, a1, all, a., a,, of x. It emerges as a special
determinant of the form
Tj - (11) -(12) -12
-(21) 02-(22) -(2n)
-(nl) -(n2).... (nn)
where
(rO) + (i ) +.... + (rn) and (rs) = (sr),
a form which we readily recognise to be the axisymmetric case
of Sylvester's determinant of tile year 1855. To the consideration of it Borchardt, probably supposing it to be new, devotes
the last six pages of his paper.
Denoting it by (0, 1, 2,..., 'a), since it is a function of the
In (n + 1) quantities,
(01) (02).. (On)
(12)...(In)
(n - i, n),
he first shows with some prolixity that the cofactor of (01) in it
is (0+ 1, 2, 3,..., n), next that the cofactor of (O1)(02)...(Oi) is
(0+1+....+i, i+l, i+2.,
and finally that
(0, 1, 2,...,-n) C(01)(1, 2,..., n)
+ Y(01)(02)(1~2, 3,..., n)
+ E(01)(02)... (On)(1+2+. +, k lj) n
+ (01)(02).. (On).