344 Proceedings of the Royal Society of Edinburgh. [Sess. 
of every order, even the 0^^, being taken, and the sign prefixed to any 
product being + or — according as the first factor of the product is of 
even or odd order, then the sum of the whole is the determinant got from 
I ^in i "yyiGbking all the elements of the fixed array zero. 
(5) The character of the theorem being as thus insisted on, it remains 
to be ascertained whether the knowledge gained may not be utilised as a 
step towards simplification or generalisation. 
Returning to the example of § 3, let us consider the twelve products 
in the aggregate 
2/*^! I I • 
The sum of the first four of the twelve can be expressed as a determinant, 
namely, 
1 
«2 
a. 
h 
h 
h 
h 
h 
^1 
^2 
<■3 
^4 
^8 
h 
A 2 
h 
Ag 
h, 
Ag 
and the sum of the second four and the sum of the third four as 
«2 
ag 
a^ 
«5 
ag 
and 
^2 
ag 
a^ 
«5 
% 
«7 
ag 
K 
^3 
K 
• 
h 
h 
h 
h 
h 
h 
<•1 
^2 
^3 
‘-•4 
^5 
^8 
<^2 
. ^'3 
«4 
dg 
d. 
d.Q 
d. 
d. 
d^ 
d. 
d^ 
d^ 
dg 
h 
A 2 
hg 
h 
h 
hg 
h, 
hg 
A 2 
hg 
K 
h 
K 
h. 
K 
Similarly, the sum of the eighteen products 
+ ,2/ i I I 
can be expressed as the sum of three determinants like 
(I 2 ^3 ^4 
Cq Cg 
'1 2 
h^ h^ hg h ^ Ag Ag Aij, Ag , 
and the remaining four products as a single determinant of similar con- 
struction. Instead, therefore, of an aggregate of 35 items, we have an 
aggregate of 8, the theorem reached being : — If A be any n-line deter- 
minant with its first r rows conceived as separated into two arrays, 
namely, an r-by-s array (r^s) called P and an r-6^-(n-s) array called 
Q, and if be the sum of all the determinants got from A by zero-ising 
