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

RECURRENTS (GLAISHER, 1876)

2:35

Similarly, having reached Hammond's first recurrent with the
concomitant result,
12.....
1.3 3.... =0,
1464....
2n-1
he obtains, by the like partitionment,

2 3-5... (2n+) 1
B-Add1 ~1
1
and

2....
3 3....
5 10 5
7 21 35....
f+1
6....;
15 15.
28 70....
* * * * 4 n%+i

(22 )
Bn = (2n+2)!

1 2
1 4
1 6
1 8

To these expressions for B. have also to be added

(2Xn)!
22n-1 (22n-1 )

1
3!
2
5!
3
7!

1
-1
2!
1 1
4! 2!
o o o

(2n)!
22n —1

-1
3!
2 1
5! 3!
3 1
7! 5!.

I....
1
3!....
~. *

n-il

the one originating in the quotient tan x/x and the other in x/tan x.
The latter, however, is readily obtained from Nagelsbach's first
expression of 1874 by dividing the rows in order by 3!, 5!, 7!,...
and multiplying the columns in order by 0!, 2!, 4!,.... We may
note also that a like process of transformation applied to the other
expression given under Nigelsbach leads to