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

312 HISTORY OF THE THEORY OF DETERMINANTS

becomes.(n)][(i-)12!3 (n)]
[1! n! (n-I-i)! (n +2)! (2n -i)!
1 1! 2!.. (n-i)
i~~~e. ~[1!1 2! 3! (n.i)!]n
n! (n-I-i)! (n +2)!...(2n-1)!
The other determinant* is that got from the same array by leaving
out any other column, say the (k + i)th, and is made dependent on
the double alternant also, the a's this time being
n+2, n%3, n+4,..., 2n+i,
the p3's
n+1, n, n-i,1.., n-k+2, n-k, n-k-i,, 1,
and the value of the determinant
[1! 2:3. (n -i1)13 n!(n +k)!,
(k!)2 (n-k)! (n+l1)! (n+2)!...(2n)!'
The quotient of the latter determinant by the former, namely,
(n!)2I (n +k)!
(k!)2 (n-k)! (2n)!
is, save for a sign-factor, the value of one of the unknown's corresponding to those which, in Sylvester's paper, were denoted by 'y X
2:XX2....The value had already been obtained (1832) by
Murphy without the use of determinants.
HANKEL, H. (1861).
[Ueber eine besondere Classe der symmetrischen Determinanten,
Dissert. 29 pp. Gb5ttingen.]
The special determinant with which ilankel is concerned is the
persymmetric, but he does not speak of it as having been previously
considered, and names it orthosymmetric. It and the special form
a1 al a2.-l...(_tn1 a2 a 2 a... a2a3... t
1 a3 C(3a4. 3U4... "(P')
1a~ anCa.0+1... an~a Ctn
*Apparently not seen to include the former.