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

76 HISTORY OF THE THEORY OF DETERMINANTS

viz.aa+b /3 ay+b8 aa ay b 2 2 a b
ca+d/3 cy+dd, ca cy d 2 2 c d
/3., a..
>, d...
In regard to the 2's which occur in the last form his remark is:"Any quantities might be substituted instead of 2...., as such terms
do not influence the result: this figure is probably, however, the proper
quantity arising from the application of the rule, because.... the
value of the determinant represented by a matrix of no places is not
zero but unity."
In the case where the two given determinants are of the third
order, say

a b c
a' b' c
aC" b" c"
he gives only the second and
forms, namelyaa +b13 aa'
a'a + b'/3 ((a'
a"a + b"/ a"a'
7

a i3 y
and a' f' 7'
third of the four ( ) possible
third of the four (g +1) possible

+ b3'
+ b'/'

aa" + b/3"
C'a" + b'/3"
a a- +7
ct"a + b/3"
7,

C
C
C/

and
aa aa aca" b c
a'a 'a' a/'a b' c
ac'a a"/a' / "a/ b" c
/3 /'..
7 y 7..,
pointing out by way of demonstration that the former of these
is arrived at by transforming the given determinants into
a b c. a /3. y
a' b c'. 'and B.
a" b" c". a' / 3..... 1.. 1.
and applying the ordinary rule of multiplication, and similarly
that the latter is got by multiplying