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

124 HISTORY OF THE THEORY OF DETERMINANTS

ZEIIFUSS, G. (18602).
[Zwei Satze iltber Determinanten. Zeitschrift f. Math. u. Phys.,
vni. pp. 436-439.]
Zehfuss' second theorem concerns a determinant whose elements
are situated symmetrically with respect to the centre of the square.
When the order-number is even, 6 say, it is to the effect that

a, a2 a. CL4 a. a6
bi b2 b3b4 b5b6
1 C al~a6 aL2+aL a3+a4 aa a2-a5
- b, ~b6 b2~b5 b3~b4 b -b6 b2 b5
C6 C5 4 C3 C2 C C
b.b5 b4 b b2 b1 el C6 2 +C C3 +C4 ~1 -C6 C2 - 5
a6 C5 4 a3 2 1
the result being reached by performing the operations

CL3 - a4
b3 -b4
c3 -C4

row1+row6, row2 +row5,~

row3 +row4

followed by o-ol 0. C2 C4C13
When the order-number is odd, 5 say, it is similarly seen that

C1 23 aL C4 CL
el 53 CS C2 el
b5 b4 b3 b2 b1
CL5 04 a.3 C2 al1

a1 ~
-b 1 +

b5 b~,+ aL 203 a- L L - L
b4 23 ba -ba ba -ba
C C~. 2 4
2 a

The latter case, however, is incorrectly stated by Zehfuss, the 2's
in the first determinant on the right being omitted and a power of
2 prefixed to the said determinant as a factor.

FERRERS, N. M. (1866).
[Ax ELEMENTARY TREATISE ON TRILINEAR COORDINATES. (Chap.
mi. pp. 59-72.) 2" edition. xiv-[182 pp. London.]
Ferrers' last exercise when correctly stated * depends on the result
* Another mode of correctin would make the underlying result
aQ2 (c +a)2 (a +b)2
(b+ C)2 b2 (a +b)2 = (a +b +c)3(-la3 - a2b).
(b+ C)2 (c +a)2 C2