A DEVELOPMENT OF A PFAFFIAN HAVING A VACANT MINOR. 



51 



= 9%\ 



a 3 a 4 a 5 a 6 



b 3 Z> 4 % b G 



c i c 5 c 6 



d 5 d 6 



A 



a 3 a 4 a 5 a~ 



h h h b r 



d 5 d r 



+ c s 



a, a K a R a 7 



d 6 d 7 



A 



ft 8 j a 3 a 4 



a 5 



«6 



a 7 



C 5 



C 6 



C 7 



<*5 



^6 



rf 7 





e 6 



e 7 



A 



a 8 l 6 3 5 4 



*5 



*6 



6 7 



C 5 



C 6 



C 7 



^5 



^6 



^7 





e 6 



e 7 

 A 



where, again, all the terms on the right except the last two can be dealt with by using 

 the preceding case, the result of such use being an expression consisting of 30 {i.e., 5x6) 

 terms of the form 



On examination, however, it will be found that these can be grouped in sets of three 

 by reason of the fact that each of the ten (i.e., C 5j2 ) two-lined determinants appearing in 

 the expression occurs three times. For example, the determinant | a 3 6 4 1 occurs the first 

 time, as we have just seen, with the cofactor —g%e 6 , the second time withf s e 7 , and the 

 third time with —e & / 7 ; its full cofactor thus being 



(^6-/s e 7 + e s/7) 01 ' 



- e.. e* e Q 



A /■ 



ffs 



By condensation in this way, therefore, there is obtained from the first five terms of the 

 development with which we started an expression consisting of ten terms of the form 



- J a._ i b i J • J e e e 7 e s 



A A 



9 S 



As for the remaining two terms of the said development, the same reasoning as before 

 gives their aggregate 



= j a sh- a A «8 5 4-«A a S h 5~ a 5 b S a $h- a 6 b S «8 & 7 " V's 



t 7 

 A 



= ~ I a S h 8 I • I d 5 d 6 d 7 



e 6 e 7 



A 



+ I a 4 6 8 J . J c 5 c 6 c 7 

 e 6 e 7 



A 



a A I • I C 4 C 5 C 6 



d 5 d 6 



