320 1)R THOMAS MUIR ON A PFAFFIAN IDENTITY, 



and therefore (§ 5) is equal to 



^Z 



1 2 3 I 



4 5 6. 



(8) The result of the preceding paragraph, which may also be written 



I 1 2 1 

 3 4 



z 1 



1 2 

 3 5 



2| 



1 2 I 

 3 6 





2| 



1 2 i 



4 5 1 



Z ! 

 2| 



1 2 



4 G | 



i 2 



5 6 



1 2 3 

 4 5 6 



is the identity which accounts for the alternative forms of the denominators in Pfaff's 



1 2 3 



auxiliary equations. It is also interesting as giving an expression for ^ I ^ t jj | in terms 



of similar aggregates of lower order. Further, by comparison of it with the second 

 result of § 6 we deduce the curious identity 



U 



1 2 

 3 4 



YJ 12 



^35! 



VI 1 2 1 

 <"| 3 6 1 



= a 2 

 1 



y\ 1 2 \ 



^ 1 4 5 1 



V 1 1 2 1 

 *-> 1 4 6 | 



y ji 2 1 



Zj ! 5 6 1 





a., 



b 4 -d. 2 

 c, - d„ 



a 5 



a 6 



& s - e 2 



»e-/« 



6 '5 - e 3 



C 6-/ 3 



d 5 -e 4 



^6-/4 





*c ~/s 



(9) As for the Pfaffian corresponding to the full adjugate of '| a 2 b s c i d 5 e 6 |, namely 



or 



A/ A 5 - A 5 ' A 6 - A 6 ' 



B3-B3' b 4 -b; b 5 -b 5 ' b 6 -b 6 ' 



Art ~~ -"-O Ao An 4 



E 6 -E 6 ' 



2 



3 4 j 

 5 6 1 



2 



2 4 I 

 5 6 1 



z 



2 3 1 



5 6 1 



2 



2 

 4 



3 I 

 6 ! 



z 



2 3 1 



4 5 1 







2 



1 4 1 

 5 6 1 



z 



1 3 1 

 5 6 1 



2 



1 



4 



3 1 

 6 



z 



1 3 1 

 4 5 ! 











z 



i 2 



5 6 



2 

 2 



1 

 4 



i 

 3 



2 | 

 6 1 

 2 

 6 



z 

 z 

 z 



1 2 1 

 4 5 | 



i 2 1 

 3 5 | 



i 2 1 

 3 4 1 



it is of course equal to 



(A 2 -A 2 ')j c 4 -c; c 5 -c/ 



a-o/ d;-d ( ; 



E 6 -E,/ 



- (A, -A 3 ') I B 4 -B 4 



B 5 -B 5 ' 



B 6 -B 6 ' 



D5-IV 



A3-D0' 





E 6 - E 6 ' 



