160 Transactions of the South African Philosophical Society. 



For example, when m — 4 we have 



9-0 



84-0 



126- 1 



0-0 



9- 1 



84- 36 



0-1 



0-36 



9-126 



= 2« 



8. If the two originating circulants be of odd order, we may 

 approximate to a semicirculant by taking one row more from the 

 one than from the other. In this case also the new determinant is 

 resolvable into factors, but now there is no factor repeated. Thus, 

 taking the five-line instance we have 



abc.de 



e a b c d 

 d e a b c 

 e d c b a 

 a e d c b 



a b c d e 



e a b c d 



d c a b c 



d-a c-b b -c a- d 



a b + e ' c + d 

 e a+d b+c 

 d c + c a + b 



— (a + b + c + d + e 



■b d 



d 



c 



b 

 b-c 

 c - d 



a - c b - d 

 e — d a - c 



c - d b 



e 



e 



d 



e 

 a - d 

 b - e 



b-c a- d 

 c - d b-c 



It must be noted, however, that the process here followed is not 

 applicable to the other instances without modifications, and that 

 consequently we are not readily led by it to the formulation of the 

 general result. The analogous difficulty in the case of even-ordered 

 determinants was got over, as we have seen, by a preparatory 

 process which brought about centrosymmetry : here this is im- 

 possible. The following theorem suggests a way out. 



9. In every odd-ordered scmicircidant there is one column %ohich 

 is centro symmetric [that is, reversible tvithout change), and there 

 is another, which, if deprived of its first element, has the same 

 property. (VI.) 



If the order-number be 4?7Z + 1, the middle row, being the 

 (2m + l)th row of the first originating circulant, is 



2m + 2, 2m + 3, ..., 4ra + l, 1, 2, ..., 2m + l ; 



and the (2m + 2)th, being the first row of the second originating 

 circulant, is 



4m + 1, 4??i, ..., 2;/i + 2, 2w+l, 2m, ..., 2, 1. 



Now, as we have here 2m consecutive integers arranged in 



