AND RELATED VANISHING AGGREGATES OF DETERMINANT MINORS. 315 



has for its constituent determinants the three-line minors of the four-by-six array 



d 1 d 2 d, 



which have no zero elements, and is equal to 



«3 



a 4 



a 5 



a 6 



h 



h 



h 



h 





C 4 



C 5 



C 6 



d s 





^ 



d 6 



- a„ 



a„ a. 



f> 6 



h 



«5 



a 6 



h 



h 



h 



% 



C 5 



C 6 





<h 



d 6 



a form which shows that it vanishes when j a l b 2 c B d i | is axisymmetric, and becomes 



b z b 4 



«5 «rt 



h h 



■4 C 5 C 6 



d 5 d R 



when | a 1 b 2 c 3 d i | is skew. 



The three-line minors of | otib^c^l^^f j which contain none of the diagonal elements 

 are 6'5"4/3'2*l in number, namely, one for every set of three rows: and the sum of 

 the aggregates derivable from the rows is the same as the corresponding sum derivable 

 from the columns, namely, 3^ I 1 ^ r I • 



(5) The general theorem which is at the basis of all such results is — If the elements 

 on one side of the principal diagonal of a zero-axial determinant of the 2m"' order be 

 removed to the other side, each being attached to its conjugate by the sign — , the 

 Pfaffan of the matrix so formed is equal to the aggregate of the m-line minors of the 

 original determinant tvhich contain no zero elements. The same stated in symbols and 

 without any reference to an originating zero-axial determinant is 



*2 ~ 2 1 ' 2 3 - 3 2 ' 3 4 _ 4 3 » • • . ( m - !)m - ( m )m-l j = ^ 



1 , 2 , 3 , . . . , m 



m + 1 j m + 2 , m + 3 , ... , 2m 



To establish this, let us begin with the case where m = 2. Evidently we then have the 

 given Pfaffian 



a 2 - ij a 3 — c x a 4 — d l 

 b 3 -c 2 b 4 - d 2 



= «0 #3 



b 3 - c„ b 4 — d 2 



c 4 - d s 



- I I 



l 1 "i 



b 3 - c 2 b 4 — d 2 

 c 4 - d 3 



c 4 - d 3 

 and therefore, from § 3, equal to 



{\a,c 4 \ - \a 3 b 4 \ - \a 2 d 3 \} - f!& x c 4 | - | b 3 d x I ! c i rf 2 l} 



=-z 



1 2 

 3 4 



v | 3 4 

 2j | 1 2 



y I * 2 



Zd 3 4 



