221 
1917-18.] Determinants of a 4-by-8 Array. 
8. In the next place it has to be noted that in devising a notation for 
the binary products it is desirable that the second factor of a product may 
be left out of consideration altogether, a result that will be obtained if the 
notation for it be readily recoverable from the notation for the first factor. 
Now, as soon as the columns of the first factor are known we know the 
columns that must go to form the second, so that all that remains to be 
attended to is a convention as to the order in which the columns are to be 
taken. The following is here adopted : (a) the first factor is to begin with 
the numbers taken from the first given determinant, (j8) the second factor 
is to begin with the remaining numbers of the second given determinant, 
(y) the numbers taken from each of the given determinants are always to 
be placed in ascending order : for example, if the products 
| 1 3'4 2 | | 3 1'4'2'|, | 1 3'3 4'| | 2 4 2'1'| 
turned up in the course of work, they would be written 
- | 1 2 4 3' | | 1'2'4'3 | , | 1 3 3'4' | | 1'2'2 4 | . 
9. When this convention is accepted it will be found that the products 
group themselves naturally into square arrays, the principle at the basis of 
the arrangement being that the products occupying a row of a square 
have in their first factors the same unaccented numbers, and those occupy- 
ing a column have the same accented numbers. For example, we have 
the square 
| 1 2 1'2' | 
| 3'4'3 4 | 
I 1 2 1'3'| 
j 2'4'3 4 | 
| 1 2 1'4' j 
| 2'3'3 4 | 
| 1 3 1'2'| | 
3'4'2 4 | 
I 1 3 1'3'| | 
2'4'2 4 | 
| 1 3 1'4' | 
| 2 / 3 / 2 4 | 
| 1 4 1'2'| 
| 3'4'2 3 | 
1 4 1'3'| | 
2'4'2 3 | 
| 1 4 1'4' 
| | 2'3'2 3 | 
where the rows are respectively distinguished by the presence of 
12 , 13 , 14 
in the first factors, and the columns by the presence of 
, 1'2', 1'3', 1'4' 
in the first factors. Similarly we have a square whose rows are dis- 
tinguished by 
12 , 13 , 14 
respectively, and columns by 
3'4', 2'4', 2'3' 
respectively. And lastly we have a four-line square with 
1, 2, 3, 4 
distinguishing the rows, and 
2'3'4', 1'3'4', 1'2'4', 1'2'3' 
