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

CONTINUANTS (NACHREINER, 1872)
The third section opens with the identity

405

ao 1
bi al
bb
I 2

1...
a2...... b,, a.

ao 1
1 Al
1

1
A2.b b,

A..1 1.... 1 Af,

1 b, b2
where A, al A ab
bl' 2 a-2 b A 13b
where A1 ma- 2 A3 a2
br-lb?'- -6..
and generally, At. a- bibr.-br..4
This is reached by a rather curious process, of which only the first
two steps need be indicated.* They are, in the case of the fifth
order,

a.
bi

1
al
b2

a2 1
b a3 1
b4 a4

ao
b '

1
al
b2

1..
a2 I
bs a, 1
1 a
b4

= b3b4

ao 1
ba, 1
b2 a2 1
b3
3

cin- 1

_U
b, E, 1
b2 a2 1
*.1

I
a4
b4

1

a~b, I

*The transformation is probably most readily effected by dividing the 2nd,
3rd, 4th,. rows by
b1, b2, blb, bb4., blb b., b2bjb,, blbsb,b7.
respectively, and multiplying the 3rd, 4th, 5th.. columns by the same, the
most instructive order of performing these operations being that in which each
division is immediately followed by the corresponding multiplication. As the
number of the divisions is necessarily one more than the number of multiplications
the reason for the outside factor bb,,2b,4.... on the left member of the
identity is also thus made apparent.