OF SECONDARY MINORS OF A PEKSYMMETRIC DETERMINANT. 



521 



; l 



1 h 



k \ 



11 i 



= 



1 

 h 



n 

 k 



+ 



1 h 

 k n 



> 









= 



1 



n 

 k 



+ 



1 



h- 1 



k+1 



n 



by§ 







= 



1 



h 



n 

 k 



+ 



1 



k+1 



+ 



1 

 k+1 



h - i 



11 



by (A) , 



1 n 

 h k 



1 n 



h-\ k+1 



1 n 



ft-2 k + 2 



+ 



In connection with these results the last sentence of S 2 should not be forgotten. 



(15) A second point to be observed when tabulating instances of (A) , (B) t (A') , (B') 

 is that the secondary conjugate of an identity may not be a different identity. 

 Taking (A) and (A'), viz., 



1 



k 



! 1 



I 



+ 



1 



h 



h 



I 



// 



k 





k 



1 



I' 



K 



i k ' 



h' 



+ 



V 



U 



lc 



n 



\ V 



ii 





K 



n 



we see that as there is one integer fixed in the first, viz., I, and one integer in the second, 

 viz., n, the two cannot have a common instance unless V = 1 and l — n. In the next 

 place, the secondary conjugate of 



1 k 



h n 



being 



1 n+l-h 



n + 1 - k n 



the condition that the two may be identical is clearly 



h+k = n+1 . 



Lastly, it is seen that if this condition be fulfilled, the minors on the right of (A) will 

 also be their own secondary conjugates. It follows therefore that the instances common 

 to (A) and (A') are included in 



1 n - m 



1 + rn n 



1 n 



1+m n -in 



+ 



1 1+m 



n - in n 



(a) 



and since in the selection of in here all that is necessary is that 1 + m < n — m, i.e., that 



2m < n - 1 , 



we conclude that — 



In the case of an axisynvmetric determinant of the (2n) tA or (2n + l) th order the 

 number of instances in which the identity (A) is its own secondary conjugate is n — 1, 

 the form then being (a). 



A similar examination of- (B) and (B') leads to the conclusion that no instance of 

 (B) is its own secondary conjugate. 



(16) Let us now return to the fundamental theorem of § 2, the establishment of 

 which was there made dependent on a previously known theorem for the expression of 



