38 Transactions of the South African Philosophical Society. 



5. If with an initial Pfaffian of still higher degree we write the 

 result of the preceding paragraph, e.g., 



a 2 



a 3 



b. 



a,.. 

 & 4 .. 









c 4 .. 



ho 



— a 2 J\ 2 — ftoAo -f- a.A., — | a 2 a-, a. 



+ 6 3 B, 



bB 



4 4 



b, b, 



+ c 4 C 4 



^6 ^7 ^8 ^9 ^io 

 flh /9/10 



+ 



. tte a§ 



• c 5 c 6 



e 6 



a, 



(h 



he 



and then again as an equivalent for the Pfaffian on the extreme 

 right, put 



"«5 A 5 + I • 



+ 6 5 B 5 



• * . ( ( f] LVj • • • 



. . . b 6 b 7 ... 



a IO 

 b 10 



-c 5 C 5 

 + d 5 T> 5 



. . c 6 c 7 ... 



. Clfj ttj ... 



^10 

 d 1Q 





e b ... 



^10 



^10 



compensating for this error by inserting 



a 2 a 3 a 5 



. compl 



a 2 a, a r 



Ms 



d 



compl + 



5 



a, 



5 



a 4 



a 5 





C 4 



d. 



3 



b. 



^4 



b. 



3 





C 4 



C 5 



d 5 



. compl 

 . compl, 



we shall obtain an identity which is of still less value when viewed 

 as giving a development of the given Pfaffian, viz., 



compl 



6^2 thn Lv. • 



b 3 6 4 . 



.. b IO 



= a 2 A 2 - a 3 k, + & 4 A 4 - a 5 A 5 - 

 + b 3 B° 3 - 6 4 B 4 + b 5 B 5 



1 d 2 a 

 ' b 



3 ^4 

 3 K 



.cc 



C 4' 



. . c IO 



+ C 4 C 4 ~ C 5 C 5 



C 4 







^10 



1 ^5-1-' 5 









+ \ a 2 a 3 a, 



b, bl 



3 



.compl - | a 2 



a 4 a^ 



. compl 







c s 





^4 









+ 1 ffl, ft, a. 



1 1 J 4 



.compl - | b 3 



h h 



. compl 







c 4 c 5 





C 4 C 5 



d 5 









+ 1 a 6 Z 



> 7 c 8 cl 9 c lo 1 , 









(\) 



