318 



DR THOMAS MUIR ON A PFAFFIAN IDENTITY. 



the dot beneath it indicates that it is permanently appropriated as such, and that 

 therefore no element with 1 as a roiv-mimber appears in the aggregate. Thus, as we 



have seen in § 5, 2? ^ j t is equal to 



-b, 



- C a - £, 



& s - Co &„ - £?o 



h ~ H 



c i d s c 5 e 3 

 d 5 ~ h 



-A 



C 6 — /s 



which shows that T ? ? * vanishes under the same circumstances # as T I ] \ \ I 

 ■^1561 " 456|' 



The fourth aggregate, 2, 4 5 6 I ' nas ^ 01 ^ s ec l u i va l ent the Pfaffian obtained 

 from \ a 2 — b x , b B — c 2 , . . . . , e 6 — y* 5 1 by deleting all the elements having either the 

 suffix 1 or the suffix 2, and therefore is 



- J . a 3 a 4 a 5 a 



bo b, b. b e 



d 5 - e i 



J 6 



H -ft 



d 6 ~fi 



-/. 



It consequently vanishes when | c B d A e b f & | is axisymmetrict It is known as a Kronecker 

 aggregate, being one of the aggregates which, according to Kronecker, all vanish when 

 I c&i&2 c 3^4 e 5./(5 ! i s axisymmetric. 



To obtain the Pfaffian equivalent of the fifth aggregate, 2, 4 5 r I ' we m ^e manner 



strike out from l| a 2 — b^ , b s — c 2 , . . . . , e 6 — f 5 | all the elements having the column- 

 number 1 (here a suffix) and all the elements with the row-number 4 (here a d), the 

 outcome being 



- «0 



«r> 



h 



h 



-e 2 



h 



"A 



C 4 



V 



" e 3 



C 6 



-/s 







e 4 





-/« 



This will vanish if each of the elements of the complementary minor of a 4 , namely, 

 the minor 



I h~ e 2 h :>~ e 2 h t;-f-2 



e 6 ~fb ' ' 



vanishes : in other words, if | b 2 c B e 5 f 6 j be axisymmetric. 



* Observe that the result deducible regarding the evanescence of 2 \ i ? I from making use of the fact that 



v I 1 2 3 I _ v I i 2 3 



^456 — Z 4 5 6 



4 5 6 



v I 2 3 4 i 



~ Z ll56| 



is less extensive than that before obtained. In this connection it is well also to note that if three selected four-line 

 coaxial minors of I a^cd^fg \ be axisymmetric, all the twelve others must be axisymmetric ; for example, the axi- 

 symmetry of | a x hj ■//, , | a^e/e I , | c 3 rf 4 e 5 / G | implies the axisymmetry of | a-fifadfafz I . 

 tSee Proceedings Roy. Hoc. Edinburgh, xxiii., p. 147, § 5. 



