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

LESS COMMON SPECIAL FORMS (ROSANES, 1872) 483
The next step, naturally, is to express each element of the determinant
on the right in terms of the f's. For any element in the first row
there is no difficulty, the transformation just mentioned giving us
what we require; and two operations with d will give the substitutions for the corresponding elements in the other rows. For example,

I x1Y2 i3' = m(m-1)2 l f3 f4
6fl 8f3 ef4
82fl1 2f3 82.4
l Xy2 13852 = m(m-1) (134 '

= m(m-1)2 (012)
k,134'

i X y1322 = m(m 1)2{( ) +( )}
The result of the full substitution thus is
The result of the full substitution thus is

Ixly2 l12 = - 3( - 2)(3m - 7)2 {m(n - 1)2}3. 012)
\134)
(013)
1341
023 \ 014\
134 +\ 134

_/012
1241
(013)
1241
_ 023) /0141
\124 \124/

012)
123/
f0130
\123/
(~23+ (014
123/ \123/

where the determinant on the right is partitionable into two determinants, the first of which is equal to a principal minor of the adjugate
of 0123, and thus will be found equal to
/0123 2
J1 k1234/ '
and the second of which vanishes.* Thus far, therefore, we have
I 0123\2
Ixly2112 * ' = - 3m3( — 1)6(m —2)(3m —7)2.f,* 123/)
* Because, if we multiply
f2 f3 f4
af2 af3 af4
x U
rowwise by it, we shall obtain a determinant with two rows proportional. More
generally,
Ilalb3Ac41 -albd4I'l abe3I = 0,
where the left-hand member is our extensional of 16b3c4l \b2d4\' b2e3jl, and this
last is such that ba coli - b3 col2 + b4 col3 = 0.