156 Transactions of the South African Philosophical Society. 



5. The ten-line and other cases may be similarly dealt with. 

 There is, however, a much more instructive method ; for, if the last 

 m + 1 columns of such a determinant be moved in a body so as to 

 occupy the first m + 1 places, the determinant will be found to be 

 centrosymmetric, and therefore resolvable into two determinants 

 of the order 2m. + 1, from one of which the factor a + b + c + d + ... 

 can be removed, and from the other the factor a-b + c -d + ..., 

 leaving co-factors which are identical. Thus, shifting as stated the 

 last three columns of the semicirculant of the 10th order, and using 

 1, 2, 3, ... for elements, we have it 



89*1234567 

 789*123456 



8 + 7 

 7 + 6 

 6 + 5 

 5 + 4 

 4 + 3 



9 + 6 

 8 + 5 

 7 + 4 

 6 + 3 

 5 + 2 



* + 5 

 9 + 4 

 8 + 3 

 7 + 2 

 6 + 1 



8 9*12 

 7 8 9*1 

 7 8 9* 

 * 9 8 7 

 1*98 

 2 

 3 



3 4 5 



2 3 4 

 12 3 



6 5 4 



7 6 5 

 1*9876 

 2 1*987 



4 3 2 1*98 



1 + 4 



* + 3 

 9 + 2 

 8 + 1 

 7 + * 



2 + 3 



1 + 2 

 * + l 

 9 + * 

 8 + 9 



8-7 9-6 * 



7-6 8-5 9 



6-5 7-4 8 



5-4 6-3 7 



1-4 2-3 

 *-3 1-2 

 9-2 *-l 

 8-1 9-* 



4-3 5-2 6-1 1-t 8-9 



The first of these factors, when there is performed on it the 



operations 



colj + col 2 + col 3 + 



row 5 — row 4 , row 4 — row, 

 is seen to be 



'3' 



= (1 + 2 + ... + *) 



(1 + 2 + ... + *) 



8+5-9-6 



7 + 4_8-5 

 6+3-7-4 

 5+2-6-3 



9+4-*-5 

 8+3-9-4 

 7+2-8-3 

 6+1-7-2 



$+3-1-4 1-3 



9+2-*-3 *-2 



8+1-9-2 9-1 



7+*-8-l 8-* 



8-6 9-5 *-4 1-3 



7-5 8-4 9-3 *-2 



6-4 7-3 8-2 9-1 



5_3 6-2 7-1 8-* 



and similarly the second factor is found 



= (1-2 + 3-...-*) 8-6 9-5 *-4 1-3 



7_5 8-4 9-3 *-2 



6-4 7-3 8-2 9-1 



5_3 6-2 7-1 8-* 



