154 Transactions of the South African Philosophical Society. 



where w denotes the sum of the odd-placed elements, and £ the sum 

 of the others. If on this the second set of operations 



•ow 8 — row 6 , 



row 7 - 



- row 5 , 



row 6 • 



- row 4 , 





raied, we have 













U) £ C 



d 



e 



/ 



9 



h 



£ 0) b 



c 



d 



e 



f 



9 



a — c 



b-d 



c — e 



d-f 



e-g 



/-* 



. . h-b 



a — c 



b-d 



c — e 



d-f 



e-g 



. . g-a 



h-b 



a — c 



b-d 



c — e 



d-f 



• ■ f~h 



g-a 



h-b 



a — c 



b-d 



c — e 



• • e-g 



/-* 



g-a 



h-b 



a — c 



b-d 



• • d-f 



e-g 



f-h 



g-a 



h-b 



a — c 



or 



(io + e) (u)-e) A 6 , 



where A 6 is a function of the eight differences 



a-c, b-d, c-e, d-f, e-g, f-h, g-a, h-b, 

 being in fact, save as to sign, the persymmetric determinant 



\d- 



e-g 



g-a 



a-c 



c-e 



e-g 



f f-h h-b b-d d-f f-h 



). do 



A similar result is of course obtainable in reference to all circulants 

 of even order. 



3. Taking now the given semicirculant, if we may so term it, and 

 proceeding in almost quite the same manner, we find it equal to 



/ 



9 



f 



h 



d-f c - g b-h 

 c-e b -f a-g 



u) £ c d e 



e io b c d 



a-c b-d c-e 



h-b a-c b-d 



f-h e - a d-b 



g-a f - b e-c 



h-b g -c f-d 



a-c h-d g - c 



and therefore equal to 0, since the co-factor of w 2 - e 2 here vanishes. 

 Of course the same final result might be reached without re- 

 moving the factor w 2 - £ 2 . Further, by performing in succession 

 the operations 



row 8 - row x , row 7 - row 2 , row 6 - row 3 , 

 col 7 -f col 5 , col 8 -f col 4 , coli + col 3 , 



