1906-7.] Dr Muir on Axisymmetric Determinants. 
151 
9B , 0B 0B 
a l + a 2 + 
0 yi a y 2 a Ys 
0B . 0B . 0B 
— SB ^ 0B ^ aB 
a l + — a 2 + — - tt 3 — cq + — a 2 + — a, 
dcij ca^ c/a^ 
7 i 
ri + Frr 2 + 
0 y ; 
73 
y 3 
0B , 0B , 0B 
^-yi + a-y2+5— y 3 
0 aj 
0 a o 
0 a fi 
0B 
and its predicated factor by ^ . It is then pointed out that the former is 
the differential-quotient of the product of the two determinants 
0B 
0B 
0B 
0yi 
0 y 2 
0 y 3 
0B 
0B 
0B 
0aj 
0a 2 
0a 3 
m 2 
m 3 
a l a 2 a s 
7i y 2 yg 
% w 2 ? 7 3 
called M and N , taken with respect to m 1 ^ 1 + m 2 n 2 + m 3 w 3 ; and that 
consequently it is equal to 
0M 0JST 0M 0K 0M 0N 
0771] 0777 2 072 2 0?77 3 077 g 
(®) 
Since, however, we have 
0B 
, 0B 
V 11 Sr. 
and other similar identities, it follows that 
, 0 B , 0B n 
+ + a/ 3 ai = 0> 
M 
0B 
d/3 
i.e. 
0B 
0B 
0B 
u n 
U 12 
U \3 
0yi 
0 y 2 
0 y 3 
0B 
0B 
0B 
. 
U 21 
U 22 
U 23 
0cq 
0a 2 
0a 3 
7?7j 
m 2 
777 3 
U 31 
U 32 
u 33 
0B 
0B 
0/? ai 
8/3 Yl 
0B 
0B 
8/3 2 
~W 72 
SB 
0B 
0/3 3 
p Ys 
y 3 W 31 7?7 1 + W 32 777 2 + W 33 m 3 
and therefore 
a l 7l + w i2 m 2 + w 13 m 8 
(*2 y 2 ^ 21^1 ^ 22^2 ^ 23^3 
a 3 y 3 W 31 W 1 "h u 32 m 2 U 33 m 3 
The expression (gt) then becomes 
M = S -? 
0/3 
0B 
¥ 
P say. 
