220 
Proceedings of the Poyal Society of Edinburgh. [Sess. 
5. The determinant factors constituting any one of the binary products 
are most readily indicated by the numbers which their columns bear in the 
array ; for example, the binary products mentioned in the preceding 
paragraph are then 
| 1 2 34 | | 5678 | , | 1 2 3 5 j [ 4 6 7 8 J , ... 
We thus quite safely dispense with the use of letters altogether. Further, 
when the number of columns is double the number of rows, there is an 
additional advantage in denoting the second n columns by the accented 
numbers 1', 2', . . . , n, our examples then becoming 
| 1 2 3 4 | | 1'2'3'4'|, | 123 1' | | 4 2'3'4' | , . . . 
6. The full number of binary products of the determinants of an 
n-6y-2n array is 
n ) 2 + (Gw, n-l ) 2 + * * * +(C„ )0 ) 2 }. 
This is seen at once on proceeding to classify the products according to the 
number of columns taken from the first n columns of the array to form the 
first factor of the product; for example, 
n columns taken, 1 product ; 
n - 1 columns taken, (C Wi n _f) 2 products ; 
and so on. 
Another expression for the same is Czn-^n-i* 
7. When n is odd we can perform the division by 2 in the expression 
for the number of products, the result being 
f‘2 , p2 1 
v ^'n, n T ^n, n— ] ' 
+ Ck 
n, |(n+l) ; 
but when n is even it is 
Cfi, n + C 2 | + * * * + i n , 
there being the difficulty of the J in the final term. This, however, is only 
apparent, for putting 2m for n we have 
2m(2m — 1) • • • (m + 1 ) 
< ’ = 
1 • 2 
m 
and . * . 
_ 2(2m — 1) 
1 . 2 • 
(m + 1) 
(m — 1 ) 
= 2G 
2 m- 1 , m — 1 j 
ur — \2 
2\ KJ 2m,mJ ‘ J \ v -'2m— 1, m— 1/ 5 
* 
the half of a square being supplanted by two squares, as exemplified already 
in § 4. 
The number of squares is thus \(yi + 1 ) when n is odd, and hn-\-2 when 
n is even. 
