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

412 HISTORY OF THE THEORY OF DETERMINANTS
In illustrating he uses the fourth order, that is to say, where
the initial expression is
x y z u x u z y
y z u x y x U z
or -
z u x y x u
u x y z u z y x
or
-x4 + y4 - z4 + 14 - 4y2xz - 4xU2Z + 4x2yu + 4yuz2 + 2X2z2 - 2m2y2.
BAEHR, G. F. (1860).
[Solution de la question 432. Nouv. Annales de Math., xix.
pp. 170-174.]
After dealing as we have seen with the circulant whose
elements are in equidifferent progression, Baehr proceeds to the
circulant whose elements are in equirational progression, namely,
C'(a, ar, ar2,.., aCr-l ).
This he first changes into
*?C'(1, n r2,. ~, rn-l)
and then into
(-_)( )-). a * C(n-I, r- 2,..., r 1).
On the determinant thus reached the operations
r row1 - row2, row2 - ro,...
are performed, with the result that its value is found to be
(-l)n-l.:. (1 ),-l
and thence the value, of the original circulant to be
( (n - l). nl.
It is worth noting that instead of the last set of operations we
might substitute with advantage the set
rown - r row, w - r~own-2,...;
also, that Baehr's circulant is a special case of that referred to
under Cremona's third lemma.