Notes on Semicirculants. 



155 



we can make it appear that 



g -a f - b e-c 

 h-b g -c f-d 

 a-c h-d g -e 



also is a factor, and that the vanishing co-factor of the 5th order 

 may have w + e, w - e removed from it, leaving only 



g -a f - b e-c 



h-b g - c f-d 



t g -a f-b e-c 



which differs from the previous three-line factor in having its third 

 row identical with its first. 



By any of the methods here indicated it is easy to establish the 

 general theorem : Every semicirculant of order 4m vanishes. (H-) 



4. In the case of the other even orders, say the order 4m + 2, 

 the result is different. As before we readily see that w + e and w - e 

 are factors : their co-factor, however, does not now vanish. Thus 

 taking the six-line determinant 



a 



b 



c 



d 



e 



f 



f 



a 



b 



c 



d 



e 



e 



f 



a 



b 



c 



d 



/ 



e 



d 



c 



b 



a 



a 



f 



e 



d 



c 



b 



b 



a 



f 



e 



d 



c 



and performing the operations 



col x + col 3 + col 5 , col 6 + col 4 + col 2 , 

 row 6 - row 2 , row 5 - row 3 , row 4 - row 2 , row 3 - row x , 



we find it 



b c d e 



abed 



f - b a-c b -d c -e 



e-a d-b . b -d 



e - a d-b 



f - b e-c 



e-a d-b 

 f - b e-c 



f - b c-e 

 e-a b - d 



(a + b + c+ ...) (a -b + c- d+ ...) 





(o e 







£ (1) 



> 



•••) 



e - a 

 d-b 



f-b 



e-c 



