Notes on Semicirculants. 



159 



The general theorem is: If the first roiu of a semiciradant be 1, 

 2, ..., 2m, the (m + l)th row be 2m, 2m — 1, ..., 1, and the m-1 roius 

 following each of these be obtained from them by cyclical change, 

 right-handed in the first case and left-handed in the other, the semi- 

 drculant is equal to 



(_)im( m -D( 1 + 2 + 3 + 4+...) (1-2 + 3-4+...) A 5 



(IV.) 



where A is the determinant of the difference of the matrices of the 

 two per symmetric determinants 



P'(m + 3, m + 4, ..., 2m, 1, 2, ..., m-1), 

 P(2??t - 1, 2m -2, ..., m + 1, m, . .., 3). 



A very curious special case is due to Mr. A. M. Nesbitt,* viz., the 

 case where the elements are the even-placed coefficients in the 

 expansion of (a-\-by m+1 followed by m-1 zeros: the determinant is 

 then equal to 2 m(2w+11 . For example, when m — 2 we have 



2 ID . 



From this is obtainable the equally curious result that the deter- 

 minant of the difference of the tivo persymmetric matrices 



/2m + l\ /2ro + l\ /2w + l\ /2?;j + l\ 



/2?^ + l\ /2m + l\ /2w + l\ 



\ 1 / V 3 ) \2m-b) 



(2m+l\ (2m + l\ 



\ 1 / \2m-l) 



10 



1 



m 



5 



10 



1 



1 



10 



5 



10 



5 



m 



and 



(2m+l\ 

 \ ) 



(2m +1 



) 



• ( 2 "o +1 ) 



/2m + l\ /2m + l\ 

 /2m+l\ (2ra+l\ 



2w+l 

 



/2m+l\ /2ra+l\ /2w+l\ 

 \2m - 8/ \2ra - 6/ \2m - 4/ 



is equal to 2 m (m " 1) . 



* See Educ. Times (1904, Oct.), lvii. p. 419. 

 11 



(V.) 



