272 
Transactions of the Royal Society of South Africa. 
right-hand side it and the 7-th column of B are simultaneously excluded. 
Thus, taking the case 
m, n, k, r = 4:, 3, 3, 3 
we have 
1 «2T ((^2 «44 1 • 
^.2 
^3 
^Z2 
^.3i 
K 
^22 
&23 
«22 
<^23 
f 1 «ii ^22 ^^44 1 • 
^.2 
^ZI 
^:2 
^22 
^23 
^2Z 
^22 
k\ 
«3i 
«32 
«'33 
«4i 
«42 
"•43' 
= I a„ fl^33 a^J • 
where neither on the one side nor on the other do the elements 631, h^_^^, b^^ 
(that is to say, the third row of B) occur. As a matter of fact there are 
only four elements of B found on both sides : and it is the introduction of 
the two extra elements &13, b^. which makes the identity possible. 
3. The following mode of establishing the theorem in the foregoing 
particular case throws additional light on the point raised. 
The right-hand member of the identity is clearly equal to 
«i4 • 
«2I 
«22 
(X23 
«3i 
^32 
<^33 
a,, . 
«4x 
«^42 
«43 
a,, . 
^X2 
^13 
. Z>„ b,. 
^22 
^23 
. b,, b,. 
which, again, by subtraction of the first two columns from the last two 
columns is equal to 
<^''l2 
«X3 
«X4 
<^2I 
«22 
«24 
^3, 
«33 
«34 
- ^3^ - (232 
«4i 
«43 
^44 
- ^41 - ^42 
^X2 
K 
^21 
^22 
K 
and the expansion of this by means of Laplace's theorem gives the left- 
hand member. 
