158 Transactions of the South African Philosophical Society. 



we introduce 



~P'(a, b, c, d, e) for 



c 



d 



e 



b 



c 



d 



a 



b 



c 



that is to say, for the determinant which is got from the former by 

 changing the order of the rows and which therefore is symmetric 

 with respect to the second diagonal instead of the first, we may 

 enunciate our result as follows : The semicirculant of the (4m + 2)£/i 

 order whose elements are 1, 2, ..., 4m + 2 is equal to ( — ) m_I (w + e) 

 (w - e) A 2 , where w, e are the sums of the odd-placed and even-placed 

 elements respectively, and where A is the determinant of the difference 

 between the matrices of the two per symmetric determinants * 



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



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



In connection with this it is interesting to note that in every case 

 there are three elements not included in the first persymmetric 

 matrix, viz., m, m+1, m + 2, and likewise three not included in 

 the second, viz., 3m+3, 3m+2, 3777- + 1 : further, that the element 

 which fills the univarial diagonal in the first matrix is 3m +2, and 

 in the second matrix m+1. 



7. If instead of having, as in the foregoing semicirculants, both 

 the cyclical changes right-handed, we make one the opposite of the 

 other, the resulting determinant will be found still resolvable into 

 factors. In this case, however, no distinction is necessary between 

 the orders 4m and 4m+2, centrosymmetry being now always 

 attainable by merely reversing the order of the last m of the 2m 

 rows. Thus, when m=3 we have 



« / 



d e 

 c d 



f e 

 a f 

 b a 



c — d 

 b — c 

 a — b 



a—e b—d 

 f—d a—c 



a 



b c 



d e f 







a 



b 



c d 



f 



a b 



c d e 





f 



a 



b c 



e 



f a 



bed 





e 



f 



a b 



/ 



e d 



c b a 





d 



c 



b a 



e 



d c 



b a f 





e 



d 



c b 



d 



c b 



a f e 





/ 



e 



d c 





a+f 



b-\-e c-\-d 



• 



a- 



-/ 



b — e 





f+e 



a-\-d b-\-c 





f- 



-e 



a — d 





e-\-d 



f+° 



a+b 





e- 



-d 



f-'o 



==- (a+b + c + d+e+f) (a-b + c-d+e-f), 



* In P to find the variables we run our eye along the first row and down the 

 last column, in P' up the first column and along the first row. If the order be 



the wth P and P' are connected by the sign-factor ( — )i'U«-i). 



