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

BIGRADIENTS (TRUDI, 1862)

a 1) C- (1..(a b c d.
t U V' X
t UV X.

11 am,- bt
I(tV -— ct

(('0-Ct (Ix -d(tI
u1x-cit Ixd
+ bV - cm x- u

a u = jX lau-bt (tv-ct ax-dtflj
where each array on the left is got from the one that precedes it by
deleting the first row, the first column, and the last row; and each
array on the right by merely deleting the last row. It is noted that
the leading determinant of the condensed array is axisymmetric.
Lastly, it is pointed out that cases where mn.> nt present almost
no additional difficulty, as they are readily brought under the foregoing. Thus, if the case be
(a, b, (tdie,)I
we have only to take a b c d e f
0 0 0 t u1 v

for our generating
being
~a bc d ef
a becd e f'

array and proceed exactly as before, the results

I -

I.. t.t u. tu v
t U V.

t u
u v
V

V
V
A
I
I

Iau
1av
au
av

at
at au +bt

au
av~bu

au+bt av~bu-vct bv+cu

av+bu bv+cu
bv cv

V + du-let
dy -ft

av
bvy
eV +a'
dy -ftI
ev -fu
V.dy -ft
eu -fu

U
t U V

U
av+bu
by

V
by +cu
cv

cv+du-et
dy-ft

a b
t
I t U

t U V i I1
U V..11
V.. j

t ~U V
I U V
U V
av +bu bv +cu cu j-du -et dy -ft