Notes on Semicirculants. 



157 



As the common four-line factor is the determinant of the difference 

 of the matrices 



8 9*1 



7 8 9* 



6 7 8 9 



5 6 7 8 



6 5 4 3 



5 4 3 2 



4 3 2 1 



3 2 1 £ 



we may say that the semicirculant of the ten elements 1, 2, 3, ..., t 

 is equal to 



(l + 2 + 3 + 4+... + $).(l : -2 + 3-4 + ...-Q 



8 9*1 6543 



7 8 9 £ 5 4 3 2 



6789 4321 



5 6 7 8 3 2 1 £ 



6. The matrices whose difference is the matrix of the squared 

 factor in the preceding are both per symmetric in form, and are both 

 obtainable from the first 2m rows of the given semicirculant, after 

 it has been made centrosymmetric, by leaving out the (2ra + l)th 

 and (4m + 2)th columns, and in the case of the second matrix 

 reversing the order of the columns. Thus in the case of the 

 fourteen-line semicirculant (i.e., where m = 3), if the first row be 



1, 2, 3, 4, 5, 6, 7, a, b, c, d, e, f, g 



we shift the last four elements to the front, and so alter the row into 



cl, e, f, g, 1, 2, 3, 4, 5, 6, 7, a, b, c ; 

 then we leave out the 7th and 14th, obtaining 



d, e, f, g, 1, 2 4, 5, 6, 7, a, b, 

 and thus finally are led to the squared factor 



d 



e 



/ 



9 



1 



2 



b 



a 



7 



6 



5 



4 



c 



d 



e 



f 



9 



1 



a 



7 



6 



5 



4 



3 



b 



c 



d 



e 



f 



9 



7 



6 



5 



4 



3 



2 



a 



b 



c 



cl 



e 



f 



" 6 



5 



4 



3 



2 



1 



7 



a 



b 



c 



d 



e 



5 



4 



3 



2 



1 



9 



6 



7 



a 



b 



c 



d 



4 



3 



2 



1 



ci 



f 



If, therefore, in addition to the usual notation for persymmetric 



determinants, viz., 



a b c 

 P (a, b, c, d, e) for bed 



c d e 



