AND RELATED VANISHING AGGREGATES OF DETERMINANT MINORS. 319 



119 3 1 



When three or more of the line-numbers of ^ | 4 5 6 are dotted, ^ ^ s readily seen 

 that the aggregate so denoted cannot vanish by the imposition of axisymmetry, so that 

 all the possible cases have been considered. Looking back over them, we observe that 

 in every case evanescence of the aggregate is dependent on the axisymmetry of a four- 

 line determinant, and that this determinant is a coaxial minor of the determinant whose 

 row and column numbers are the undotted line-numbers of the aggregate. We thus 

 have the following interesting generalisation — Any aggregate of three-line determinants 

 will vanish if a four -line coaxial minor of the determinant whose row and column 

 numbers are the undotted line-numbers of the aggregate be axisipnmetric ; and it is not 

 hard to see that the theorem is not confined to three-line aggregates only. 



(7) Let us now consider the Pfaifians which differ from those of §§ 5, 6 in having for 

 elements not a 2 , a 3 , . . . . but the complementary minors of a 2 , a 3 , . , 

 and first let us take a Pfaffian of the second order, say 



in '| a 2 6 3 c 4 d 5 e 6 



a 



D5-D,' 



C 6 - C-6 



where C 4 , C 5 , 



E e - E e' 

 . . are the complementary minors of c 4 , c 5 , . . . ., and C 4 ' is what C 4 

 becomes when each element outside the first line of C 4 is changed into its conjugate ; 

 for example 



E 6 " E e' = 



h h i 



- I a, a 3 



Co 



C 4 „ 3 



Expressing each of the elements of the given Pfaffian in terms of the as and their 

 cofactors, we find the Pfaffian equal to 



W e e ~/s) - a a( b 6 "/a) + a ft(h ~ e 2 )l ■ i a 2( G i ~ d s) ~ «s( & 4 - d -i) + a i(h ~ c 2)) 

 - { a ii d 6 -ft) ~ a i(h -A) + a e( b i ~ ( h)} ■ {« 2 ( c s - e 3 ) - a #s - e i) + a b(h - C 2 )} 

 + W rf 5 - e d - a i(h - e 2 ) + a o( h i ~ d -2)} ■ {« 2 ( c 6 "/a) - a -i( h 6 -/ 2 ) + a e( b 3 - C 2 )} • 



In the expansion of this there are evidently terms in a 2 a 2 , a 2 a 3 , a 2 a 4 , a 2 a 5 , a 2 a 6 and 

 terms independent of a 2 . The cofactor of a 2 a 2 is 



6 ' 



-ft 



d e - 



-fi 



V 



~fs 







d 5 - 



" e 4 



C 5 

 C 4 



~ e 3 

 -d 3 



3 4 



which from § 5 we know to be equal t° — 2, I 5 6 I ' s i m il ar ly> the cofactors of « 2 a 3 , a 2 a 4 



a 2 a 5 , a 2 a 6 are found to be 



24| ^1231 v |23 



5 6 J, 2-| 5 6 I, 2j! 4 6 



2 3 I 

 4 5 I; 



and, manifestly, the cofactors of a 3 a 4 , a 3 a 5 ,...., a 5 a 6 all vanish. The given 

 compound Pfaffian is thus equal to 



f v I 3 4 I vlH I ^ i 2 3 I vl 2 3 I ^12 3 1) 



°»1 " a *2L I 5 6 I + a *2- I 5 6 I ~ a *2, I 5 6 I + a ^ I 4 6 I " a «2* \ 4 5 I / 



+ a s2, I 5 6 I " a *2j I 5 6 ! + a ^ I 4 6 

 TRANS. ROY. SOC. EDIN., VOL. XLV. PART II. (NO. 11). 



44 



