454 Transactions of the South African Philosophical Society. 



— 



- d 2 



146 

 146 



. 2 '■ 156 



"' ; 156 



+ 2de 



156 ; 

 146 ; 



s = 



- d 2 



145 

 145 



f2 ; 156 

 7 ; 156 



+ 2r// 



146 : 

 145 ': 



p = 



- e 2 



145 

 145 



7 ; 146 



+ 2e/ 



146 ; 

 145! 



If, therefore, we perform on the determinant the operations 



row 2 + 



col 2 + 

 there is obtained 



156 

 156 

 156 

 156 



roWj, row 3 + 

 col„ coL + 



146 

 146 



146 

 146 



roWu 

 col I} 



# 



N = 



d 2 2d 2 

 e 2 2de 



I/" 1 2d/ 



156 

 156 

 156 

 146 

 156 

 145 



2de 



2e 2 

 2ef 



156 

 146 

 146 

 146 

 146 

 145 



P 



2df 



2ef 

 2f 2 



156 

 145 

 146 

 145 

 145 

 145 



M*e*p 



. 



d 



e 



/ 



d 



156 : 



156 ! 



156 : 



156 ; 



146 ; 



145 i| 





146: 



146 ; 



146 : 



e 



156! 



146 ; 



145 ; 



f 



145 ; 



145: 



145 : 



156 ; 



146! 



145 i| 



= M 2 e 2 f 



g h 

 h I 



by §8. 



We thus have the theorem : In any axisymmetric determinant A 

 which has (1, r) equal to zero for all values of r except n - 2, n - 1, n, 

 the cofactor of A in the norm of 



(1, n-2) J[n-2, n-2] + (1, n-1) sl[n-l,n-l] + (1, n) J[n, 7i] 



is 



where 



4(1, ^2)»(I, n-iy{l,ny. I \, \ 



71-3 



n-3 



(XL) 



2 3 

 2 3 



n-3 



n-3 



isHhe minor of A occupying the rows and 



columns luhose numbers arc 2, 3, ..., n-3. 



10. It is easy to see from §3 why the norm in the preceding case 

 should vanish when any one of the three elements (l,?i-2), (l,n-l), 



