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

PERSYMMETRIC DETERMINANTS (HANKEL, 1861) 315

where A = (I-q)(1-q2)...
and B = (1 - qY)(1 - qy+')2(1 - qY+2)2...
the expression obtained for the determinant is neither short nor
simple. Among the interesting specializations made there may
be noted
T-cZ -
(1)q=1, and I X*
(2) q 1,y= o, and.. q= 0;
(3) q 1, a= xoc, and.x qa= 0.
Section fifth concerns sets of linear equations of the type
a,1 +ar+lX2 +-. +a,+nlxn +ar~n = O<:0
whose determinants are the persymmetric forms dealt with in the
preceding sections. Section sixth is a continuation, the special
set of equations considered being
Clr Cara, ar2 n.. -2 a... a,.+,-, __
Y1. - x,..+.) $ 0,7rr~ 7,~Y'~i, 7r., ~ ~Yr'$-3n —2 Yr Yr+,n-1
The last section (pp. 25-29) is occupied with the transformation of
1 +a,x+a2x2+.
into a continued fraction of the form
1 p1~ p2X
1-1I -1 -..
The resolution obtained is accurate so far as it goes: it leaves,
however, a misleading impression in that the expression reached
for p,, is different from and more complicated than that for Pn+1
The details need not be given, as, on putting
b = 1, b = b3 =... = 0
in Heilermann's result of the year 1845, we obtain
ao+alx+a2x+
A O A1X A2x A0'&X AlA4X zA2Ax
_ z xzal/A0 xt~jz/0z~ xAob3/aza2 xziz~i4/z~,2A3
or -1 A -1 - 1 - 1