1906-7.] Dr Muir on Minors of a Product-Determinant. 85 
we can substitute the determinant 
- . tq a 2 a 3 a 4 
m i A 9i K A 
n \ A 9% A ^2 
r i fs 9s A ] h 
s i A 94 K A 1 
and, in the second place, the bipartite which is the equivalent of the minor 
in question having the square array 
i Fi G 2 1 |Fj H 2 |; j H x K 2 1 
I -^1 ^3 I I -^1 3 i I H-i K 3 | 
I f 3 g 4 ! 1^3 H 4 | |H S K 4 |, 
where every element contains | f\ g 2 h 3 1 as a factor, may be simplified 
by the removal of this factor throughout. The following interesting 
result is thus obtained : 
«1 
a 2 
a 3 
a t 
a i 
a 2 
a s 
a 4 
m 2 
A 
9i 
\ 
*1 
m s 
fi 
9i 
l h 
K 
n 2 
/ 2 
92 
n s 
A 
92 
h 2 
h 
r 2 
/ 3 
9s 
^3 
h 
r s 
fs 
9s 
S 2 
/ 4 
94 
h. 
K 
H 
A 
94 
K 
K 
C J 
c 2 
C 3 
C 4 
c i 
C 2 
C 3 
C 4 
m s 
fl 
9i 
K 
K 
m s 
A 
9i 
\ 
K 
n 2 
A 
92 
h 
K 
n s 
/ 2 
92 
h, 
r 2 
fs 
9s 
h 
h 
r s 
fs 
9s 
h 
*3 
S 2 
A 
94 
\ 
K 
“V s 3 
A 
94 
\ 
K 
1 «i ( 2 1 
“l % 1 • • - 
1 a S C 4 \ 
\h 3 k 4 \ 
— 1 9 s ^4 1 • • • 
l/s&l 
1 m i n 3 
\f l 2 k 4 \ 
\g 2 K \ . . . 
• ■ -1/2 04 1 
\ m 2 r 3 
1 A A 1 
~ 1 9\ k'2 1 • • • 
• • l/l 02 1 
1 r 2 S S 
This includes a notable theorem of Hesse’s regarding axisymmetric 
determinants, to which he devotes six pages (pp. 246-251) of his well- 
known paper “ Ueber Determinanten und ihre Anwendung in der Geometrie, 
. . . . ( Crelle’s Journ., xlix. pp. 243-264), the last sentence of which is 
“ Auf dem angegebenen Wege lasst sich auch, unter der Voraussetzung dass 
Wick = Um sei, die allgemeine Gleichung : 
