Developments of a Pfaffian. 



39 



for here, instead of the 945 terms of the proper development, we 

 have 10 sets of 105 terms each, 5 sets of 45 terms each, which 

 merely cancel 225 terms of the preceding sets, and 1 final set of 120. 



6. Using, as above, the contraction " compl " to stand for the 

 complementary minor of the determinant or Pfaffian which precedes 

 it," we may formulate as follows the series of general identities 

 thus reached : — 



61 o CJu-2 



a n 



= a 2 A 2 - 2 { | a 3 6 4 | . compl j- , 



= a 2 A 2 - a 3 A 3 - 2 { | « 4 b s c 6 j . compl } , 



(E.j 



+ 63B3 



= a 2 A 2 - a 3 A 3 + a 4 A 4 - \ a 2 a 3 <x 4 



+ 63B3 - 6 4 B 4 b 3 6 4 



+ c 4 C 4 c 4 



compl 



+ 2] I a- b c 7 d 8 |. compl} (E 3 ) 



it being remembered that only the first two identities are un- 

 exceptionably effective, giving, as they do, the final development 

 of the Pfaffian without superfluous terms. 



It may be noted in passing that the triangular mode of disposing 

 the terms of the first kind on the right is not without advantage, 

 in that it is a help to the formation of the terms of the second and 

 following kinds. Thus in (E 4 ) having to commence with 



a 2 .co( + « 3 .cof + a 4 .cof + a..cof 



o t o 



+ & 3 .cof + Z> 4 .cof + b-.coi 



+ c 4 .cof -f c 5 .cof 



+ aLcof 





it is only necessary to leave out in succession the first, second, 

 frame-lines of this quasi-Pfaffian, and we obtain 



+ 



b 3 6 4 b 5 



c 4 c 5 



d< 



.cof + 



a, a, a~ 

 c 4 c s 



d< 



.cof + 



a 2 a, a. 



"■=4 3 



b, b 5 



d c 



.cof + 



in (E 5 ), the triangular matrix being larger, it is possible to form from 



* If strict uniformity of notation were more important than brevity, such 

 a term as " a 2 k 2 " would have to be replaced by " a 2 . compl." Perhaps the 

 best uniform notation, however, would be got by using a contraction of the word 

 cofactor, say the contraction " cof " ; the difficulty of indicating what signs are 

 to be +, and what — , would also then be avoided. 



