(35) 



DEVELOPMENTS OP A PFAFFIAN. 



By Thomas Muie, LL.D. 



(Bead December, 1903.) 



1. The fundamental development of a Pfaman is, of course, that 

 which Jacobi used for the purposes of a definition, viz., in umbral 

 notation, 



'|123456.| = 12- 13456| - 13-i|2456| + 14-'|2356| 



- 15*'|2346J + 16-'|2345|, 



or, more concretely, 



a 2 a? a, Oc ae 



b 3 b 4 b 5 b 6 



c 4 c 5 c 6 



— <x 2 .| c 4 c^ Ce 



dr de 



e 6 



- a y \ b 4 b 5 b 6 



d 5 d 6 



e 6 



+ 



a A .\ b 3 b s b 6 



c 5 c 6 



e 6 



d s d 6 



e 6 





- a 5 .\ b 3 b 4 b 6 



C 4 C 6 



d 6 



+ 



a 6 .\ b 3 b 4 b s 



C 4 C 5 

 d, 



the fifteen terms of the development of the Pfaman of the third 

 degree being got in five sets of three terms each. The analogous 

 development of a determinant, it may be remembered, viz,, 



I aj) 2 c 3 d 4 \ = a T \ b 2 c 3 d 4 \ - a 2 \ b z c 3 d 4 \ 4- a 3 \ b z c 2 d 4 \ - a 4 \ b x c 2 d 3 



has also been used for the purposes of a definition, the first to do so 

 being Vandermonde (1771). 



2. In the second place, since 



a 2 a 3 a 4 a 5 ae 

 b 3 b 4 b 5 b 6 



— 



®2,\ ^4 ^5 ^6 



de de 



c 4 c 5 c 6 

 de de 





e 6 



e 6 







+.1 



Cvt a 4 de 

 63 b 4 b 5 



a 6 



be 



c 4 c 5 

 d 5 



c 6 

 d 6 





e 6 



and since the Pfaffian on the extreme right can be expressed as an 



