OF SECONDARY MINORS OF A PERSY MMETRIC DETERMINANT. 



517 



of (1+ 1 ,n) in the first and third. Lastly, the two remaining elements of the second 

 (h + 1 , 1) and (k+l , n) are manifestly equal to the two remaining elements of the 

 third (l + l ,h) and (k+l , n). The theorem (B) is thus established, the basis of it 

 being exactly the same as that of (A), viz., the proposition of § 2. 



(7) If we delete the common secondary minor from each of the determinants in (B), 

 what remains is still an identity, viz., 



(h + l,k) (h+l,n) 

 (l+l,k) (l + l, n) 



(h+l, I) (h + l, n) 

 (k+l, I) (k+l,n) 



(k + l ,h) (k + l,n) 

 (l + l,h) (l + l, n) 



On this account (B) like (A) may be viewed as an extensional of a very simple identity. 

 When in (B) we put h = 1 we obtain the theorem of § 1 due to Pascal and 

 Cazzaniga. 



(8) The conjugate determinant of P^ , a 2 , ... , a 2n _ x )— that is, the determinant 

 obtained by rotating P through an angle of 180° with the diagonal a x , « 3 , . . . , a 2n -i 

 as axis — is exactly the same in both appearance and substance as P. The secondary 

 conjugate, however — that is, the determinant obtained by the same rotation with the 

 secondary diagonal a n , a n , ... , a n as axis — is only the same in substance, being in 

 form the persymmetric determinant P(o 2n _ 1 , a 2 „- 2 , • • • > «i)> or P' sa . v - Any theorem 

 regarding the minors of the former is thus also applicable to the latter ; and. as every 

 minor of the latter is replaceable by a minor of the former, the theorems (A) and (B) 

 as applied to P(a 2n _i , a 2n _ s , . . . , a x ) must also be theorems in regard to P(a x , 

 «o , . . . , « 2 »i-i)- More definitely, since the r th row or column from the beginning of 



"2 ' 



P ; is the reverse of the (n+1— r) th column or row from the beginning of P, any 

 theorem in regard to P' will become a theorem in regard to P if we change the numbers 

 of the rows and columns accordingly. In addition to (A) and (B), therefore, we have 

 two theorems which we may call the secondary conjugates of these, viz., 



or 



V 

 k' 



V 

 k' -1 



h' 



n 



h' 

 n 



n+l-k 

 n + l - I 



n+l-k 

 n— I 



k' 

 I' 



k' 

 I' -I 



n+l ■ 



-h 



+ 



n 







n + l 



-h 



+ 



n 







li 



^ 



! 1' 



n 





i h' 



h' 



^ 



I' 



n 





\ h'- 



-1 



n + l - I 

 n + l-h 



n + l - I 

 n — h 



k' 



n 



k' 

 n 



n + l-k 

 n 



n+l -k 

 n 



(A') 

 (B'). 



(9) Again, since any identity connecting a x , a 2 , ... , a 2w -i will remain an identity 

 if the suffix of each a be diminished by unity ; and, since in the case of a minor of 

 P(a 1 , a 2 , ... , a 2) j_i) this unit-diminution of the suffixes can be made by unit- 

 diminution of the numbers of the rows or columns involved, it follows that from any 



