50 DR THOMAS MUIR ON 



But, by the previous case, the 1st, 2nd, and 3rd terms of this expansion are equal to 



e |a 3 Z> 4 l , + tfelog&sl , - c G \aJ) b \ 



respectively ; and the remainder 



= - | a A a A a A 



+ I a 6 b 3 a fi & 4 a 6 b 5 



C. Cc 



= | «6 & 3-«3 & ti «6 6 4~ a A % b b~ a A 



d< 



= ~ hMh + \ a Ah ~ l a Ai c 4 



Consequently we have the result 



a, a. a, a R 



\ 



h 



h 



h 





C 4 



C 5 



C G 







d 5 



d 6 



- I«3 & 4l e 6 + \ a 'A\ d 6 ~ \ a A\ C 6 



- \a 3 b G \d 5 + \aJ) Q \c 5 - \a b b 6 \c v 



where the first factors on the right are the set of six (C 4i2 ) two-lined determinants 

 formable from 



«3 a i a 5 «G 

 h b 4 h b 6> 



and their cofactors are the remaining six (3 + 2 + 1) elements 



*A Cr Co 



d 5 d 6 



of the Pfaffian. 



(3) Had the given Pfaffian been of a higher order than the 3rd. it is clear that the 

 cofactors of the two-lined determinants could not have been linear. It will now be seen 

 by considering another case that in general they are themselves Pfaffians, and that con- 

 sequently in the case just dealt with they have in strictness to be viewed not as elements 

 but as Pfaffian minors of the 1st order. The expansion to which we are leading may 

 thus be described as an aggregate of terms each of which is a product of a determinant 

 and a Pfaffian. 



As before we have 



«3 



«4 



«5 



H 



a 7 



a s 



\ 



h 



h 



\ 



h 



\ 





C 4 



C 5 



C 6 



c 7 



e s 







d* 



d 6 



dj 



d s 









e e 





ft 



9s 



