412 
Proceedings of the Royal Society 
nents on the left ; the second Permanent on the right is obtained in 
4 - 4 - 4 - + _ 4 - 4 * 
the same way from \a 1 b 2 c%\ , \x 1 z 2 y 8 \; the third from | a 1 b 2 c 2> |, 
+ + 
1 I > an d so on throughout the remaining possible permuta- 
tions of the rows of the second Permanent on the left. 
This theorem may be established as follows : — Taking the prin- 
cipal term of the first Permanent on the right along with the cor- 
responding terms of the other five Permanents, we see that the sum 
must be 
+ 4 * 4 " 4 " 
that is, the product of | x x y 2 z z | and the principal term of | a 1 b 2 c i |. 
In like manner, taking any other term of the first Permanent on the 
right along with the corresponding terms of the other five Perma- 
4 - + 
nents, we must obtain the product of | x x y 2 z z | and the term of 
4 - 4 - 
I a i^ 2 c 3 I corresponding to the said six terms. Hence, when this 
process is completed, we must have in all 
4 * 4 - 4 - 4 - 
I tt 1^2 C 3 I * I X dJ$3 I * 
6. The product of two Determinants of the n th order is expressible 
as an aggregate of n l Permanents of the same order. Thus — 
$1 ^2 
\ \ h 
C 1 C 2 C 3 
+ 
4 - 4 
+ 
x x 
x 2 
X 3 
apc-^ 
a l x l 
(^2*^2 
y± 
y 2 
y* 
= 
h \V\ 
^$2 
hVs 
— 
\h 
i ) 2 Z 2 
^3 Z 3 
h 
*2 
*3 
c l z l 
C 2 Z 2 
C 3 Z 3 
<Wi 
c 2^2 
C 3^3 
+ 4 
-- 
«2^2 
«3^3 
<hV i 
^2^2 
«3^3 
— 
b Y x x 
b 2 x 2 
b?x 5 
4 - 
Vi 
t J 2 Z 2 
h 3 Z 3 
c i z i 
C 2 Z 2 
C 3 Z 3 
c 2 X 2 
C 3 X 3 
+ 
* 4 - 
+ 
a i z i 
a 2 z 2 
a '3 Z 3 
a 
a 2 Z 2 
a 3 Z 3 
+ 
b 1 x 1 
b 2 X 2 
bgx z 
- 
\V\ 
b 02 
c dh 
C 2^2 
c &3 
c 1 x 1 
C 2 X 2 
C 3 X 3 
The mode of formation of the Permanents here is exactly the same 
a 3 in § 5 ; the sign preceding any one of them is + or - according 
