411 
of Edinburgh, Session 1881-82. 
Either of these propositions may be viewed as a consequence of 
the definition above implied, or as an alternative form of the 
definition. 
4. Of course at the outset it is evident that from the theory of 
determinants there is suggested the possibility of analogous theorems 
regarding Permanents. As evidently, however, the number of such 
analogous theorems must be small : for no analogue can exist when 
the theorem in determinants depends upon the property of change 
of sign consequent upon the interchange of two rows. As an ex- 
ample of such analogues, we have evidently 
+ + + + + + + + + + 
I I = ^ii | + a 2 I J + cio\ bgiflj^. | + b^c 2 d^ | ? 
+ ++ + + + + + + ++ + 
= | ^1^2 I I C 3^4 I "1” | ^1^3 1 | c 2^4 I ’ * • 'H %^4 I I C 1^2 I > 
and so on throughout the whole range of Laplace’s expansion- 
theorem. 
The following, however, are properties of a different kind. 
5. The product of two Permanents of the n th order is expressible 
as the sum of n \ Permanents of the same order. Thus — 
+ 
+ " 
- - 
a 1 
a 2 
a 3 
rys ryt ryt 
fcvj 1^2 ^3 
&2%2 
a l x 1 
Cl2%2 ^ 3^3 
h 
^2 
h 
• 
Vi y% Vz 
= 
hvi hv-2 
+ 
\h 
b 2 z 2 b B z 3 
c i 
C 2 
C 3 
h h % 
c l z l 
% 
C 3'% 
hV-1 
C 2^2 C zVz i 
+ 
- + 
■f 
agji 
a 2 y 2 ayy 2 
c hVi 
a 2^2 a zVz 
+ 
b 1 x l 
b 2 x 2 
b$ x 3 
+ 
hh 
b 2 z 2 b 3 z B 
c ih 
C 2 Z 2 
c z z z 
c l x l 
Cffi2 ^ 3*^3 
¥ 
+ 
atf j 
a 2 z 2 
a zh 
<*i z i 
a 2 Z 2 a Z Z Z 
+ 
b-pc^ 
byx 2 
b^ x z 
\ + 
\V\ 
KVi hvz 
c iVi 
C 2^2 
c 1 x l 
C 2 X 2 CgXg 
Any element of the first Permanent on the right is obtained by 
multiplying together the corresponding elements in the two Perma- 
VOL. XI. 3 G 
