520 



DR THOMAS MUIR ON VANISHING AGGREGATES 



and the general identity based on the persymmetry of 



is 



1,2,3, , m 



n - m + 1 , n - m + 2 , . . . . , n 



1 , 2 , . . . . , m , h x , h 2 , ; : , h s 



A'j ,/)•., , . . . . , k s ,n-m+l,n — m + 2 1 . . . , n 



1,2 m , k x + m , k 2 + m , 



h, - m , hg -vt., . • • , h s - in , n — m + 1 , n - m + 2 , 



. , k s + m 



. . . . , n 



As, however, there are other ways of preserving persymmetry (§ 5, e) than by 

 deleting the first m rows and the last m columns, there . must . be similar identities not 

 herein included. 



(13) The secondary minors of a per symmetric determinant of the r\. th order which 

 have equivalent felloiv minors other than their conjugates are ^(n — 2)(n — 1) in number. 



If in giving values to h and Jc we have counted the cases where h~^k , no fresh case 

 will be got by taking h > k , = Jc + a , say ; because the identity 



1 Jc + a 

 k n 



1 k+\ 



k + a - 1 n 



would already have been reached in the form 



1 k + 1 



k + a - 1 11 



1 k + a 

 k n 



Further, it is clear that h>\ and k<n. The possible cases are thus- 



h = 3, 



so that the total number of them is 



(n-2) + (n-S) + 



k = 2, 3, . .., n-\ 

 k= 3 ,...,»— 1 i 



+ 1 



i.e., J(n-2)(»-l). 



(14) When h=l the identity (B) can be applied to the first minor of its right-hand 

 member, the said minor being previously replaced by its conjugate ; and the repetition 

 of the operation leads finally to 



i 1 



k + 1 ; 





[ I 



2 ; 



+ 



: 1 



/ 





; k 



t ■. 





1 2 



/.• - 1 I + 1 



1 2 



k-2 1+2 



as is already known. 



A similar action is possible in the case of the identity (A) when I = n, if the last 

 minor on the right be replaced by an equivalent fellow minor other than its conjugate 

 in accordance with § 12. We have in fact 



