222 Proceedings of the Poyal Society of Edinburgh. [Sess. 
distinguishing the columns ; so that, for example, the product in the (2, 4) th 
place is 
2 12 3' 
4'1 3 4 |. 
The determinants of the arrays can thus be unambiguously specified by 
their diagonal terms, and this is not at all seriously interfered with by the 
further convention that the product in the (r, s) th place shall take the factor 
(-i y+ s . 
10. Denoting the elements of the three arrays by double-suffixed b’s, c s, 
d’s respectively, and the unique product 
| 1 2 3 4 | | 1'2'3'4'j by a n , 
we complete our scheme of notation, which may be epitomised thus : The 
35 binary products of the determinants of a 4<-by-S array are the 
elements of the determinants 
| | 1 2 3 4 | 
| 1'2'3'4'| 1 
* 
3 
| 1 2 1'2'| 
[ 3'4'3 4 | 
• | 1 3 1'3'| 
| 2'4'2 4 | • 
| 1 4 1'4'| | 
2'3'1 2 
| 1 2 3'4'| | 
1'2'3 4 | 
• | 1 3 2'4'| 
| 1'3'2 4 | • 
| 1 4 2'3'j | 
l'4'l 2 
1'2 3 4 | • 
| 2 1'3'4'j 
| 2'1 3 4 | - 
| 3 1'2'4'| 
| 3 1 2 4 | . 
| 4 1'2'3' 
and are thus effectively denoted by 
a \\ ’ I ^13 I ’ 
13 I 5 
d 
14 
11. As regards the brevity of the notation, nothing requires to be said. 
It may be well, however, to show by a few examples how helpful it is in 
suggestiveness, and especially what notable orderliness it introduces into 
the working formulae. 
12. In the first place we may instance the results of the application of 
Sylvester’s theorem to | 1 2 3 4 1 | f'2'3'4' | . When it is single columns 
that are interchanged the results take the very simple form 
d\\ 4~ d\2 4* ^13 + d 
d 01 9 + d g + d, 0A = 
14 
24 
d'3\ 4 - d^ + d 33 + d M — 
^41 4- d 4 . 2 4- d A3 + d 44 = J 
V a 
li 
— ^n 4* d^i 4- d^-y + d 4 y 
1 1 yo 4" dgg I d q.) — d 
= d 13 + d 23 + d 33 4- d i3 
= ^14 4” ^24 4“ d 3 y 4~ d 4/ y . 
In other words, we have eight expressions for a n , namely, the sum of any 
one of the eight lines of the determinant. 
