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

460 HISTORY OF THE THEORY OF DETERMINANTS
minant the elements are arranged in the order of the columns
from which they are taken, Cayley points out that if the elements
be quaternions
7T~ ~ rr - rr= 0,
7r 7F~ r? 7
but
r = 7 jf- 0.
7r 7F
He is thus led to inquire what the non-zero value is in this latter
case and in other similar cases. Taking
'7r = x'+iy'+jZ'+kw',
7 = cv + iy" ~ jZ" ~ low",
7r = X +~t iy +SjZ /+ 70II
he says it is easy to show * that
7r w = -2 i j Ic
7r 7r 71 Z W
7:1 ZI W
7r 7r = =-2 3 i j 1<
7r 7r 7rX 71 Z W
7r 7F '7F X 7 Z
X 71Z/ W
*Probably the easiest way is to express the determinant as a sum of determinants with monomial elements. In the case of the third order the number of
such determinants is 64, of which 40 vanish, the sum remaining being
123 + 132~+ 213 + 231 + 312~+ 321
+124+142+....
+134+143+....
+234+243+....
where rst stands for the determinant whose columns are in order the rth, 8th, tth
columns of the array
x iy jZ klw
Xf iyl jz' kw'
xi/ iy' jeN 7hZO
and where therefore
123~132+... = xy'z"I(ij+ij+ij- ij-j+ij) = 2k I xy'",
and so on. The multiplication table of i, j, k, it may be recalled, is
(ii ij ik) -I k-j)
ji.j jie -k -I i
ki Iqj k/c j -i - I