28 
Proceedings of the Royal Society of Edinburgh. [Sess. 
author in 1897.* Starting with a particular case of the general theorem 
(C), where m = 5 and k = 3, and using a similar method, we have on 
multiplying 
1 2 3 
4 5 
4 5 
M= 
% 
— 4 
--5 
-34 
1-35 
1-45 
2 3 4 
2 3 5 
2 4 5 j 
|3 45 
\ 
1 2 4 
1 2 5 
1 3 4 
13 5 
1 4 5 
2 3 4 
2 35 
2 4 5 | 
345 
) 
3 5 
i 3 4 
2 5 
2 4 
2 3 j 
1 5 
1 4 
13 
1 2 
\ 
3 5 
3 4 
2 5 
2 4 
2 3 
1 5 | 
1 4 
13 
1 2 
) 
where the positions indicated by the dashes in M may be filled with any 
numbers from the set 12 ... n, as long as (1) no two numbers in the 
same element are alike, for in that case every element in that row of M 
would be zero; (2) the row numbers in the fth row of M and the com- 
plementary with respect to m of the column numbers in the fth column 
of M have no numbers in common, for if they had then the corresponding 
element in the principal diagonal of the product would be zero, and 
therefore the product zero; the product 
M.N = A 10 . 
45 
— 3 4 51 
--345 
-23451 
-2 3 45 
-2345i 
1 2345 
i 1 2 3 4 5 
12345’ 
1 2 3 45 
12345’ 
1 23 45 
12345’ 
1 2 345 
N = A 6 . 
1 2 3 4 5 
1 2 3 4 5 
45 
- - 3 4 5 
--345 
- 2 3 4 5 
- 2 3 4 5 
- 2 3 4 5 
1 2 3 4 5 
1 2 3 4 5 
'12345 
1 2 3 4 5 
1 2 3 4 5 
1 2 3 4 5 
For the product has every element on one side of the principal diagonal 
zero, and therefore it equals the product of the elements along the principal 
diagonal. 
By Sylvester’s theorem 
and dividing out the common factor from both sides we have 
M = A 4 
where of course the numbers in the places indicated by the dashes are the 
same with which we started in M. 
It will be observed that the 4 in the row numbers of the second row 
of M, the 5 in the third row, the 3 and 4 in the fourth, the 3 and 5 in 
the fifth, etc., are what make all the elements on one side of the principal 
diagonal of the product vanish and the minor break up into factors. This 
is true independent of the numbers (under the restrictions named) in the 
places indicated by the dashes. 
* Metzler, “Compound Determinants,” American Journal of Mathematics , vol. xx. 
No. 3. 
