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

PERSYMMETRIC DETERMINANTS (ROUCHE, 1858) 355
+1
which if we put i, for I J xrdx take the form
-1
ani o+ an-_i +....+ ali n- + n = 0
ani1 + an-i32.. + aii + + 1n+i = 0
(an'2 + al,-1_3 + +..+ Cqin+ + 3n+2 = 0
anin-1 + (an-,in4 + * *+ ai32n-2 + 3211-1 0=
Solving for a,, a2, a3,..., an and substituting in V, Rouchd of
course obtains
Vn = o0 3l 2 -* -n -n1 ',1 92 93....,, n in+1
i2 3 i4.... 4'n+1 in+2
1 x.2. xn...where a stands for the cofactor of Xn in the determinant, and
being independent of x may be left out of account; and, since
i, is l/(r+l) or 0 according as r is even or odd, it follows that
X, X2, X3,.... are proportional to
1 0 1 0. 1 0 0
O, ~ 3 o
lx, 0 0 0 5
1 x 2 0 -1 0
1 x 2 X.
respectively.
By reason of the peculiar distribution of the zero-elements
these determinants are changeable into
>3 1 x ~ i 1, 1 5 '
1x' 3 x1 1 x2 x 4
~ 5
1 x2 x