1906-7.] Dr Muir on Axisymmetric Determinants. 
K(U) = 
A H G L 
H B F M 
G F C N 
L M N P 
Ppo(U) = 
• £ v £ 
a A H G 
/3 H B F 
y G F C 
3 L M N 
137 
L 
M 
N 
P 
and ¥ pp (U) is what is obtained from V po (U) on changing a, (3, y, S into 
*]> £ w respectively : but freed of all fresh notation it is nothing more nor 
less than 
f 
rj 
£ 
(0 
( 
A + a 2 
H + a/3 
G + ay 
L + a8 
r l 
H + /?a 
B + /1 2 
F + /3y 
M + /28 
t 
G + ya 
F + y/1 
C + y? 
N+y8 
(O 
L + Sa 
M + 8/3 
N+3y 
P + 8 2 
• £ v K w 
£ A H G L 
v H B F M 
C G F C N 
to L M N P 
A H G L 
H B F M 
G F C N 
L M N P 
A 
+ a 2 
H + a/3 
G + ay 
L + aS 
H 
+ /3a 
B + /3 2 
M + /38 
G 
+ ya 
F + y/3 
c +> 2 
N + y8 
L 
+ 8a 
M + S/3 
N + 3y 
P + 3 2 
V 
■£ 
(0 
a 
A 
H 
G 
L 
P 
H 
n 
F 
M 
y 
G 
F 
0 
N 
8 
L 
M 
N 
P 
Nothing is said about the mode of proving it. 4 
If we note that the first determinant can be written in the form 
- 1 . 
and the fourth in the form 
a $ 
l v 
A H 
H B 
G F 
7 
c 
G 
F 
C 
M N 
M N 
5 
L 
M 
N 
P 
light dawns at once, for the last three determinants of the identity are then seen to be 
principal minors of the first, and the identity itself to be a case of Jacobi’s theorem 
regarding a minor of the adjugate. 
