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

PERSYMMETRIC DETERMINANTS (LIGOWSKI, 1861) 311
LIGOWSKI, W. (1861).
[Nacltrag zu der Abhandlung "Ueber die Inhaltsberechnung der
Kbrper." Archiv d. Math. u. Phys., xxxvi. pp. 181-185.]
Ligowski, while on the same track as Schelibach (1836), reaches
the same set of equations as Joachimsthal, and solves them in
Sylvester's way. What is new and interesting is his evaluation
of the special determinants to which Sylvester's method leads. The
first of these is the determinant got by leaving out the last column
of the array
1 1 1 1 1
2 3 n+1
1 1 1 1 1
2 3 4.... n-i n+ 2
1 1 1 1 1
3 4 5 n+2 n+3
1 1 1 1 1
n n+1 n+2 2n-1 2n
This he has the insight to recognize as a case of Cauchy's double
alternant
1 1 1.
ai-f% a1-/32 al nl
2-/32 A2/32 02 a2-fn
1 1 1
a.O a,,-/2 n-A
namely, the case where the a's are
n+-i, n+2, n[-3., 2n,
and the 3's
and where therefore the general value

a(a,) a...) /n) ]TI (ar - 31) (a1,. - 82).I2. I=. )
r=l