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

130 HISTORY OF THE THEORY OF DETERMINANTS

where u21 =u2, by multiplying together
a1 - a2 a11 ' 12 -a2 a1
71 -72, '21 '22. -72 71
in row-by-row fashion; and then the generalisation of this
identity in two different directions.
The first generalisation consists in the proposition that the
two-line axisymmetric determinant

'/11 b'12
".. -'t21 22
'aNl Cln2
a1 act2,, ~n a a U11 'La12.. ' 62 a 2 'U21 q'22...* U an 'nI 'Un2,.. Ct 0 71 72

*.. ',ln 71 '11 '12.. 1ln.. 't2 72 al21 'a 22 *. q2,,n... 'nn 7n Uanl l 2,. * *nn... 72n 0 71 72 7n

an
0

where Z,,a = xa,,, contains

%11 U12
%21?622
all 'l 2l
'antl ', n2...?/6111.. U.
** *' ' znn

as a factor, and that the cofactor is an integral homogeneous
function of the a's and likewise of the y's. The case where n
is equal to 3 is treated as follows. The determinant

611
%21
'-31
71

'12 U13 a1
9'22 'a23 a2
Z'32 '33 a3
72 73 3

having uK^ = c,^,, is introduced and denoted by B, with the
result that the two-line determinant in question is representable by

DB D B DB
-a1 + -/a2+ Da3
77l ~7Y2 +73

DB,B aB
al '3a2u, 33
DB DB 3B
'alYl + a72 + a73

and its predicated factor by DB/Dl3. It is then pointed out that