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

SKEW DETERMINANTS (SPOTTISWOODE, 1851) 263
whose determinant is got from the determinant of the former
set by the change of rows into columns, and may therefore be
denominated by the same symbol A. Solving the two sets of
equations, we have

XiA = [11]m1 + [12>U2 +
x2A = [21]ul + [22]m2 +
x,,A = [nl],t 1~[n2]n2 +

and... + [1n]wu,
+ [rn]u,,... + [Er~n]u, *... + [n2]v,, I...+ [n2]V?'b

xIA -
x2A =
X",A =

[11]v1 + [21]V2 +
[12]vl + F22]v2 +
[Ln]vl + [2n]v2 +

where, be it remarked, it would have been much better if in
every case the coefficients of u,. and v,. had been interchanged,
for then [r-s] would have stood for the cofactor of (rs) in A.
From these by addition and subtraction and by utilising the fact
that u, + v, = 0 t two others are obtained, viz.,

2x1
2x2
2x,,
all

A = 0 + ([12] - [21])n2 +
A = ([21]-[12])u + 0 +
A = ([al] - [n] )m + (['n2]-[2n])u2 +
o= 2 [II1 + ([12] + L21])m2 +
o = ([21] + [12])ml + 2[22]U2 +
o = ([n1] + [1n])ul + ([n2] + [2n])zu,2 +

+ ([2n] -[n1])m,,
+
+- ([ln3 + [n2 )U.u
+ ([1m]+[n1])u.
+ 2 [nwn]m, I

Then follows the very curious sentence-curious, that is to say,
from a logical point of view" There is herein used the fact, first noted by Rothe in 1800, that the cofactor
of rs in any determinant is equal to the cofactor of sr in the conjugate determinant.
t Along with this fact Spottiswoode associates the statements that
L,+ +... ~ui,, = 0, v1~v 2 + -... = 0,
which are manifestly incorrect.