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

PERSYMMETRIC DETERMINANTS (SYLVESTER, 1862) 317

Ux, UX+1~ Ux+2~~~ U~,$nl
U+1 2x+2 US ~ +3-n I = constant
ux+nn-I Uxfn Ux+n+l. X+2nis satisfied by the same integrals as the equation
uX-plUX+i +P2uX2- -... + ( -l)n-1'Pn1u~1+n- ( -i)U~,+, = 0.'
An analogous result is obtained in regard to the differential
equation.dy d2y
Y dx dx2
dy d2y d3y
dx dx2 dx3
dn-'y dn y d1-'y
dxn-' dxn dxn+l
where the determinant is seen to

dxn-1
dny
* dx = constant,
d2n-2y
be a special form of Wronskian.

HANKEL, H. (1862).
[Ueber die Transformation von Reihen in Kettenbrilche. Zeitschrift f. Math. u. Phys., vii. pp. 338-343.1
This is merely a fresh presentation of results already known:
for example, from Brioschi (1854) and Sylvester (1850). (Hist..
ii. pp. 343-346, 333-335.)
SYLVESTER, J. J. (1863).
[Sequel to the theorems relating to 'canonic roots,'.... Philos.
Magazine, (4) xxv. pp. 453-460; or Collected Math. Pajpers,
ii. pp. 331-337.]
A simple problem here solved is the elimination of x from two
such equations as
1 X2 x3 =0, 1 2 x =0,
0X X
a b c d a b c
b c d e b c d
c d e f