An Upper Limit for the Value of a Determinant. 
331 
the axisymmetric determinant in question is 
123 
123 
123 
123 
3 
12 
23 
1 
3 
13 
12 
3 
3 
23 
13 
2 
2 
23 
12 
1 
2 
12 
13 
3 
2 
13 
23 
2 
23 
1 
1 
12 
28 
3 
3 
13 
23 
2 
2 
23. 
This notation is very useful in that, if we wish to test whether the pro- 
duct of two different rows vanishes (as every such product ought), we 
have only to count the number of different digits in each of the four 
sections of the two rows ; thus, in the case of the rows 
. 123 . 123 
23 2 2 23 
there are 2 corresponding digits different in the first section, 2 in the 
second, 1 in the third, and 1 in the fourth — that is to say, 6 altogether, 
w^hich give —1 on performing multiplication, thus making the product 
6-6. Similarly in the case of the last two rows the like number is 
0 + 2 + 2 + 2. 
16. On the other hand, Hadamard's 20-line determinant appears to be 
essentially unsymmetric. As the result of a fresh investigation, in which 
axisymmetry was steadily kept in view, the following has been reached — 
12345 
12345 
12345 
12345 
45 
145 
135 
24 
45 
245 
234 
15 
34 
123 
345 
45 
35 
123 
123 
12 
3 
1245 
1245 
3 
345 
34 
24 
234 
345 
35 
15 
135 
