316 DR THOMAS MUIR ON A PFAFFIAN IDENTITY, 



Taking next the case where m = 3, we express the Pfaftian 



| a., - &j tt 3 - Cj « 4 — tZj a 5 — e l a 6 —f\ 



h~ C -> h~ d 2 6 5 _l? 2 6 G-/2 



c , - do 



5 " 



" e 3 



C 0" 



"/, 



5 " 



" e 4 



^6 



"/« 







V 



-/b 



in terms of the elements of the first frame-line and their complementary minors, thus 

 obtaining 



o 



(«2~ 6 l) 



c 4 - J 3 c 5 



e 3 C (j A 



<2 S - e. d, -f. 



~ («3- C l) 



/; 4 - (/ 2 h ~ e 2 h -/a 



e 6 -/ 6 



then, using the previous case, we change this into 



2 4 

 5 6 



thirdly, as before, we separate the portion containing the a's from that independent of 



3 4 



the a's, and find # the former, namely — a ^ I j? * I -f- a 3 2, I <5 fi I ' • ■ • • ec l ua l to 



-ri 



1 3 4 



5 6 



+ 21 



2 3 4 

 5 6 



and the latter, being the portion involving the suffix 1, equal to 

 and finally we combine those portions and obtain 



2 



1 2 3 

 4 5 6 



The case where m = 4 follows in the same way from the case where to = 3, and so on 

 generally. 



(6) The representation of the determinant aggregate 2 m +-l m + 2 ' 2m! ^ v a 

 single Pfaffian puts the whole subject of the vanishing of such aggregates on an entirely 

 new footing ; and, as may be guessed from the last step of the proof, the advantage 

 extends to all the dotted or restricted aggregates as well, for every one of those can be 

 represented in the same manner. 



By way of illustration let us examine the aggregate 



* To prove that 



a sZ 5 6 a *Z I 5 6 + a i2, 5 6 



1 2 3 

 4 5 6 



^\tl\ + ^\'il 



v I 1 2 3 



i 456 



it suffices to show that the ten three-line determinants on the right are all represented on the left, and that nothing 

 else than such representatives are there to be found. Taking, for example, the fourth three-line determinant, 



•> 4 •-' f i W( - partition it into — a 3 . - , ojf h, 

 2 4 .1... ..,w.,.„,i ;,, „ vl 23 



2 6 



3 4 



, the first of which appears on the left in 



v |2 3 



a 3 2 Kg 1 'I'" -'•'"inl in « 4 2 5 6> an< 3 the third in - %2 1 e 

 being 5 x 4, and on the right 10 x 3, the other requisite is provided for. 



The number of such parts on the left 



