OF SECONDARY MINORS OF A PERSYMMETRIC DETERMINANT. 



519 



and from (B) 



1 k + 1 

 h I 



1 l+l 

 h It 



1 h+l 

 k I 



it follows that In any -per symmetric determinant the aggregate 



1 I 

 h k+1 



l l+l 

 k It 



l h 

 k,+ 1 / 



l h + l 

 k I 



vanishes, it being remembered that l>k+l,k>h>L. 



Along with this, of course, goes its 'secondary conjugate,' viz., In any per symmetric 

 determinant of the n th order the aggregate 



It -I h 

 I n 



k h 

 l-l n 



+ 



I k-l 



h n 



I k 

 h-1 n 



vanishes, it being remembered that l<k— 1 , k < h < n . 



(12) In tabulating the various instances of the theorems (A) , (R) , (A') , (B') in 

 connection with any particular determinant, caution is sometimes necessary to prevent 

 the appearance of the same instance a second time in a different form. This is due to 

 the fact that more than two secondary minors of a persymmetric determinant may be 

 equal. On account of the axisymmetry, every minor that is not coaxial is equal to its 

 conjugate, and therefore may be said to occur twice in the determinant : but this 

 occasions no difficulty in performing the tabulation referred to if in specifying minors 

 we agree to make the lowest-numbered line always a row. To avoid the difficulty 

 which does arise when the same minor happens to be found in more than two positions 

 the following theorem is worth noting: — 



In any 'persymmetric determinant of the u th order the secondary minor got by 

 deleting the I st and h. th rows and k th and n th columns is the same as the minor got by 

 deleting the I st and, (k+ l) th roivs and (h— l) th and n th columns : that is, in symbols 



1 h 



k n 



1 k+1 



h-1 n 



By vvay of proof we note (l) that the deletion of the 1 st row and n th column of 

 P(a 1 ,a 2 , ... , a 2ft _!) gives us P(a 2 , a 3 , ... , a 2 _ n2 ) : (2) that this latter determinant 

 being axisymmetric the deletion of its (h— l) th row and Jc th column produces the same 

 effect as the deletion of its k th row and (h— l) th column : and (3) that the (h— l) th row 

 and k th column here are the h th row and k tb column of the original, and the k th row and 

 (h— l) th column the (k+ l) th row and (h- l) th column of the original. 

 Obvious extensions of this, not needed for our present purpose, are 



1 k+1 q+1 



h - 1 p - 1 n 



1 2 k + 2 



h - 2 n - 1 n 

 TRANS. ROY. SOC. EDIN, VOL. XL. PART III. (NO. 22). 4 i 





1 h 

 k q 



P 

 n 



= 



: 1 



2 



h 





; k 



n-1 



n 



= 



