Devciojwiieiits of a Pfaffian. 



31 



the 105 terms of the Pfaman being got in 3 sets of 15, and 10 sets 

 of 6 each. (E 2 ) 



4. We may pursue the process further and obtain a fourth 

 development of a like character, but a drawback then attaches 

 to the result, the fact being that it is impossible now to say that 

 the Pfaman 



a 4 



a, . 



.. a s 



K 



b 5 . 



• • bg 



C 4 



c, . 



. c 8 





d 5 . 



- d 8 



08 



with its three vacant places is equal to 



a 4 A + + 

 & 4 B 4 

 + c 4 C 4 



-& 4 B 4 



a, 



a 6 



a 7 



a 8 



b 5 



be 



A 



bg 



c, 



c 6 



C 7 



c 8 



d 5 



d 6 



d 7 



dg 









9s 



the cofactors of a ¥ - b v c 4 in the said Pfaman being not A 4 , B + , C 4 , 

 but 



| . b 5 b 6 b 7 bg | . a 5 a 6 a 7 a 8 



C 5 Cs C ? Cg C 5 Cg C ? Cg 



e 6 e 7 e 8 e 6 e 7 e 8 



ii h ii h 



r/s , us 



If, therefore, we insist on retaining a 4 A 4 - & 4 B 4 + c 4 C 4 , we shall be 

 repeating 9 terms already included in <% 2 A 2 - ci 3 A 3 + & 3 B 3 , and must 

 rectify the error by bringing in the product 



a, 



a 6 



a 7 



a 8 



b 5 



be 



b 7 



h 





£e 



G 7 



e 8 







A 



9s 



the result being 



6^2 ttn Cva 



e 6 e 7 e a 

 f 7 f» 



93 



a 2 



a. 



a 4 



a s . 



.a 8 



^ 2 A 2 



- 



a 3 A 3 



+ 



<x 4 A 4 





b 3 



h 



h- 



-h 





+ 



6-B, 



o 5 



- 



W 







«4 



c. . 



5 



...c 8 









+ 



cfi< 



98 



CI2 Cv-j CI, 



b 3 6 4 



e 6 e 7 e 8 



98 



+ I a 5 b 6 c 7 dg | , 



(* 3 ) 



where, instead of the 105 terms of the proper development, we have 

 123, viz., 6 sets of 15 each, 1 set of 9, cancelling a previous 9, and 

 1 set of 24. 



