30 Transactions of the South African Philosophical Society. 



— 3c* ..., and whose main diagonal is a, a - b - c, a - 2b — 2c, ..., is 



-equal to 



(a-n-l.b)(a-n-2.b-c)(a-n-3.b-2c) (a-n- l.c). 



By way of illustrative proof let us take the case where n = 5, viz., 



a b 



- ic a-b -c 2b 



-3c a -2b- 2c 36 



-2c a -3b -3c . 46 



- c a - 46 - 4c 



Performing the operation 



row x + row 2 + row 3 + row 4 + row 5 , 



removing the factor a — 4c and lowering the order of the determinant 

 we have 



(a - 4c) 



a -6 + 3c 26 + 4c 4c 4c 



-3c ft -2b -2c 3b 



-2c a -3b -3c 46 



- c ft - 36 - 4c 



Performing on this determinant the operation 



row! + 2 row 2 + 3 row 3 + 4 row 4 



we obtain 



(a - 6 - 3c) 



12 3 4 



-3c ft -26 -2c 36 



-2c ft -36 -3c 46 



- c ft - 46 - 4c 



the determinant factor of which reduces to 



ft -26 + 4c 36 + 9c 12c 



- 2c ft - 36 - 3c 46 



- c ft - 46 - 4c . 



On this we now perform the operation 



row, + 3 row 2 + 6 row 3 



obtaining 



(ft - 26 - 2c) 



13 6 



- 2c ft - 36 - 3c 46 



- c - ft - 46 - 4c 



