Notes on Semicirculants. 161 



ascending order, and placed under them the same integers arranged 

 in the reverse order, it follows that the two middle elements of the 

 upper group have below them the same two middle elements in the 

 opposite order : also as we have the first 2m + 1 integers arranged 

 in ascending order, and placed under them the same integers 

 arranged in the reverse order, it follows that the one middle element 

 stands over the same middle element. This amounts to saying that 

 the two rows must be partially represented thus — 



, 37?i + l, 377^ + 2, , m + 1, 



, 3m + 2, 3m + 1, , m + 1, 



Now by reason of the cyclical change from row to row, the 

 elements immediately above 3?7i + l in the first of these two rows 

 (that is, the middle row of the determinant) must be 3m + 2, 3m + 3, 

 ... ; and as the elements below are the same, the column in which 

 these elements stand, viz., the mth., must be centrosymmetric. 

 Similar reasoning makes clear that the column having m + 1, m + 1 

 for consecutive elements, viz., the (377£ + l)th, is also centrosym- 

 metric when deprived of its top element. 



If the order-number be 4m + 3, the middle row and the row 

 immediately following it are 



2m + 3, 2771 + 4, ...., 4771 + 3, 1, 2, ..., 2m + 1, 2m + 2 



4m + 3, 4m + 2, ..., 2m + 3, 277t + 2, 2m + l, ..., 2, 1 ; 



and these, if we only write the parts which are pertinent to the 

 argument, are 



, 3m+3, f m-f 1, m+2, 



, 3771 + 3, , 771+2, m+1, 



so that it is now the (377i + 2)th column which is perfectly centro- 

 symmetric, and the (77i+l)th which is approximately so. 



10. If the perfectly centrosymmetric column thus proved to exist 

 in a semicirculant of order 2771+1 be advanced by cyclical change of 

 columns to the first place, the operations performed in § 8 can be 

 followed in exact detail. These are 



row 2m+I - row,, rpw am - row 2 , , row w+2 - row m 



and then 



col 2 + col 2m+I , col 3 + col 2m , ; 



and the value found for the determinant is 



rr.\ M T -M 3 |.| M 2 -M 3 I , (VII.) 



