AND KELATED VANISHING AGGREGATES OF DETERMINANT MINORS. 321 



and therefore by the main result of § 7 



i 2 3 i 



] 9 o I 



V A 2 - A 2 ) • a 2 .2L | 4 5 6 I + ^ 3 ~ 3 ) ' a 3 . | 4 5 6 



^> I 1 2 3 



< - (a 2 A 2 - a 3 A :j + • • • ) + (a 2 A 2 ' - a 3 A 3 ' + ••■)> , 



= z 



1 2 3 

 4 5 6 



{ - 1 a a 3 a 4 a 5 a 6 



1 I h h h h 



C 4 C S C <3 



(/ 5 ^6 



+ I a. 2 a 3 a 4 a 5 a ( 

 63 Z> 4 6 5 b 6 



(10) In dealing with the identities of §§ 5, 6, the only special case considered was that 

 which arises from making conjugate elements identical : the results, however, are equally 

 interesting when conjugate elements are made to differ only in sign. Doing this, we 

 find that 



Z j 4 5 6 ' = ~ 8 '' a 2^ c i c h e 6 1 when | a^^d^fg | is skew : 

 Z I 4 5 6 = ~ ^ I a A c 4^5 e « I when | b^dg^f^ | is skew : 



y 1 1 

 ^ 4 



1 2 3 



5 6 



z 



1 2 3 

 4 5 6 



- 2 '| a 2 & 3 c 4 rf 3 e 6 I when 



- 2 '| a 2 6 3 ^ 4 rf 5 e 6 | a2=0 when 



hr-iPhft, i is skew : 



c 3 f? 4 e 5 / 6 I is skew . 



On the other hand, the identities of §§ 7, 8 do not in like circumstances give any- 

 thing new, the results then obtained being simply the theorem regarding the adjugate 

 of a Pfaffian and the theorem regarding a minor of the adjugate. 



