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

164 HISTORY OF THE THEORY OF DETERMINANTS

We then have from Garbieri
I a 2y5 6 I =)
t(a, 3, -, 6) =

1

a/3y6 -Ya/33y fa/3 -la
a/3y -E2a/3y la/3 -. 3y -2a/3y
from which, manifestly, there comes on the right
- -2a/3y -2a 1
a3yS 2a/3 -la
-2a/3y la/3;

~ 1.
1.
-Ea I
Ea/3 -Ea I

and this is Naegelsbach's result.
Garbieri's next theorem concerns the double alternant formable
from the bivarial function
i, k=0, 1, *..., n+h
(aikx Y i=0,19,. n+h
and two sets of variables
a1, a2,..., a, and 1 /32,... 3,,.
Calling the function F(x, y), Garbieri affirms that
I F(a131) ~ F(a2,2)... F ( (,/n)
~(a,,,2, a)., a () * (/ )13 * ' )

= ( —I)h+1

a00
aoo
a,,0
an+h,O
Cn

aoI
an
6n-l
n+h,:
Cn-1
* f

a... a *.ann+1... C,.... * +h,n a n+h,n,+l ''... C C...... C2 C1
Ch Ch-

ao,n+h
al,,n+h
~. ~
ar,nq+h
an+h,n+h
Co

Cn C.....
' I
'n-l * * * -
Co C1 2 ' '
~.. o
~....
~....