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

AXISYMMETRIC DETERMINANTS (HESSE, 1853) 131
the former is the differential-quotient of the product of the two
determinants

aB
~y,
DB
Da3

aB
aY2
aB
Da2

aB
aB
Da3

(11 C2
Y1I Y2

Ca3
Y3

I ml M92 9T73 1, 91 n2 n3,
called M and N, taken with respect to m1 + 2nm2 +mn,3; and
that consequently it is equal to

DM DN DM
Dvnl Dn1 + vn2

aN
Dn2

DM DN
am3 Dan3

(m)

Since, however, we have
DB DB DB DB
3Y1 11 + t2h12 + d- t13 + al1 = 0,
and other similar identities, it follows that

MB
I 3B.e.
M.- ie.
3B
DB
DUal
aB
-= -a2
DB
- a3
and therefore?B al
M a- B
a3

Da1 a
9Th21 9
aB
aB
_B?B
- -/3 72

3B
'72
B
'a2
n2

3B
aY3
DB
Da3
W13

U11
i21
i31

tI12 in13
in22 u23
in32 in33
a613i3
q33m3

Th19121 + i12792 + I
U21m1 + i122092 +'
m310z91 + Ui329T2 +

Y7i l111l + 6- 129T2 + i139n3
Y2 l21ml + 22m92 + U23iT3
73 U31M1 + in32m2 + U33m3

3B
= 3~. eP say.

The expression (m) then becomes
3B f DP 3N + P N + P 3N }
d al that remin1s is the 2 Dn f the b d
and all that remains is the evaluation of the bracketed factor