162 Transactions of the South African Philosophical Society. 



where a is the sum of the elements, and the M's are matrices 

 formed from the first m rows of the semicirculant, viz., M x con- 

 sisting of the m consecutive columns beginning with the first, M 2 

 of the m consecutive columns beginning with the second, and M 3 

 of the last m columns reversed. 



For example, in the semicirculant of the 9th order, it being the 

 second column which is centrosymmetric, the first column is moved 

 to the end, whereupon the first four rows stand thus — 



234567891 

 1234 5 6789 

 912345678 

 89123456 7, 



and the factors of the circulant are seen to be 



(1 + 2 + .. . + 9) 



2-1 3-9 4-8 5-7 



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



9_8 1-7 2-6 3-5 



8-7 9-6 1-5 2-4 



3_ i 4_9 5_8 6-7 

 2-9 3-8 4-7 5-6 

 1_8 2-7 3-6 4-5 

 9_7 i_6 2-5 3-4 



11. From this we naturally pass to the semicirculants of odd 

 order in which the two cyclical changes are effected in different 

 directions. It is not necessary, however, to continue to give full 

 details : the results alone will suffice. 



First of all there is the theorem corresponding to that of § 9, viz., 

 In every odd-ordered semicirculant with opposite cyclical movements 

 the middle column has the first element in the middle place, this being 

 led up to by, the (m-{-l)th., mth, (m — l)th, ..., 3r<i, 2nd elements and 

 followed by the same : and the first column has the first element in 

 the first place, this being followed by the (27?i + l)th, 2mth, ..., (m + 2)th 

 elements and these repeated. (VIII.) 



This suggests the opening set of operations 



row 2ni+I - row w+I , row 2m - row TO , row 2m _ t - row m _ n ..., 

 after which we are led as before to the required value, viz., 



(_)§«Km-i) ff> | M I -M 2 |.| M I -M 3 | (IX.) 



where a is the sum of the elements, and the M's are matrices formed 

 from the first m rows, viz., M z consisting of the m consecutive 

 columns beginning with »the first, M 2 of the m consecutive columns 

 beginning with the second, and M 3 of the last m columns in reversed 

 order. 



12. The foregoing investigations suggest others of a more general 



