36 Transactions of the South African Philosophical Society. 



aggregate of products, each of which has for its first factor one 

 of the determinants of the set 



a 3 a. a? $6 

 b 3 b A b 5 b 6 



and . for its second factor one of the remaining elements of the 

 Pfaffian,* we obtain 



a 2 a 3 a 4 a 5 a 6 

 b 3 b 4 b 5 b 6 



— ft 2 | C 4 C$ Cft 



a$ d,6 



e 6 



- | a 3 6 4 |.e 6 + 



1 a 3 b 5 \.d 6 - 

 | a 4 b 5 \.c 6 + 



| a 3 b 6 \.d 5 



I a 4 b 6 \.c 5 



<?6 



a 5 bsl.c^ 



the fifteen terms of the development being now got in one set of 

 three and six sets of two. (E x ) 



3. In the third place, since neither a 3 nor b 3 can occur in the 

 same term with a 2 , we have from the second line of § 2 



a 2 a^ a, a? a^ 

 b 3 b 4 b 5 b 6 



e 6 



a. 



C 4 C 5 Cfr 



d 5 d 6 



a- 



+ b. 



h 



d 5 



be 

 d 6 



ft 4 



a 5 

 d 5 



a 6 

 d 6 



+ 



06 



« 4 



a 5 

 b 5 



a 6 

 b 6 



C 4 



c s 



d 5 



c 6 

 d 6 



c 6 



or, if we use A 2 , A 3 , B 3 for the complementary minors of a 2 , a 3 , b 3 , 

 and put for the Pfaffian on the extreme right its equivalent the 

 determinant - | a 4 b 5 c 6 | , there results 



a 2 a 3 a, a? a^ 

 b 3 b 4 b 5 b 6 



£6 



ft,/!, 



03A3 



+ 6A 



a, b s c 6 I 



(K) 



This, however, by reason of the low degree of the Pfaffian on the 

 left, is a defective illustration of the form of development now 

 reached. Taking, instead, the Pfaffian of next higher degree we 

 have 



+ 63B3 



b 3 b 4 . 



.. a s 





a 2 i\ 2 





9z 









+ 



a 2 A 2 

 6,B, 



b, b 5 .. 



c 4 c 5 .. 



. a 8 



• h 



■ c 8 





98 



a 3 A 3 - ^(\a 4 b 5 c 6 \.g 8 ), 



See Trans. Boy. Soc. Edinburgh, xl., pp. 49-58. 



