532 



BR THOMAS MTJIR ON VANISHING AGGREGATES 



Of this expansion the first term is, clearly, 





A, 



B : C, D, ■ 



• 





A a B x 



Pi 



Di 







A 2 



B 2 C 2 D 2 • 



• ■ • 





A 2 B 2 



c 2 



r> 2 







A 3 



B 3 C 3 D 3 • 







A 3 B 3 



c 3 



D 3 





= - 



A 2 



B 2 C 2 D 2 a x 



-1 Ci 



= - 







a i 



■1 ii 





A 3 " 



B 3 C 3 D 3 a 2 



«2 £ 2 









a 2 



e 2 U 





A 4 



B 4 C 4 D 4 a 3 



e 3 £3 





A 4 B 4 



C4 



D 4 a 3 



€ 3 £3 





A 5 



B 5 C 5 D 5 a 4 



6 4 £4 





A 5 B 5 



c 5 



D 5 a 4 



e 4 £4 



the second 



= - 



• 1 A x B 2 C 3 D 4 | • 



1 "1^4 1 + ! A 1 B 2 C 3 D 5 



l-l 



a l e 2 £3 | I 







= 



1 A x B 2 C 3 D 2 | • j 



a l 8 3 U 



1 - 1 A x I 



, 2 C 4 D 3 | 



• ! ° 



h S 2 £3 1 ; 





and similarly for the eight others. 

 the result will be found to be 



If the series of products thus obtained be collected, 



1 A 2 B 3 C 4 D 5 | 



{ 1 a 2 ft 74 1 - 1 a l ft S 4 | + 1 a l & € 4 1 - 1 a l ft £3 1 } 



I A 1 B 8 C 4 D 5 | 



{ - 1 a l 73 8 4 1 + 1 a l 72 e 4 1 - 1 a l 72 £ 8 1 } 



1 A 4 B 2 C 4 D 5 | 



{ 1 a 2 h 74 1 + 1 a l K € 4 1 _ 1 a l 8 2 £3 1 } 



1 A 1 B 2 C 8 D 5 | 



{ 1 a 2 € 3 74 1 - 1 a I e 3 S 4 1 ~ ! a l e 2 4 ! } 



1 A^CaD, | 



{ 1 a 2 £3 74 1 - 1 a l £3 8 4 1 + 1 a l £2 € 4 1 } 



e notation of j 



j 18, is readily seen to be 



1 A 2 B 3 C 4 D 5 



I • [ a P ; 7i ^2 € 3 £4] - 1 A i B 3 C 4 D 5 | • [a y ; 7l 8. 2 e 3 £J 



+ | A^C^Dj 



1 • [a 8 ; 7i S 2 e 3 £4] ~ 1 A x B 2 C 3 D 5 | • [a e • y t 8 2 e 3 Q 



+ A 1 B 2 C S D 4 



i • [ a £ J 7i 8 2 £ 3 £4] » 



where every one of the second factors is a Kronecker aggregate which vanishes when 

 I 7i ^2 e 3^4 I ^ s axisymmetric. The object aimed at is thus attained. 



(32) We may even yet generalise further by substituting for " the last q elements" 

 in the enunciation of § 29 the words " any q elements " ; but then the minor whose 

 axisymmetry is necessary is not the last coaxial minor but that minor of the last q rows 

 of A whose elements are in the same columns with the elements of B corresponding to 

 the particular q elements chosen at the outset. Thus 





■ B, 



B 2 B 3 



B 4 B 5 





• c, 



C2 C3 



c 4 c 5 



2 



A 2 a 2 



d 2 r> 3 



& 72 



»4 I>5 



8 2 € 2 





A 3 a 3 



ft 73 



8 3 e 3 





A 4 a 4 



ft 74 



84 e 4 



_ I 



A 2 B 3 C 4 D 5 | 

 A 1 B 3 C 4 D 5 ! 

 A X B 2 C 4 D 5 I 

 A x B 2 C 3 D 5 I 

 A^CjD, I 



[ a ; ft 73 84] 

 [ft ft 73 84] 



[7; ft 7s 84] 

 [8 ; ft 7s 84] 

 ; ft y 3 8 4 ] , 



and therefore vanishes when j fi 2 y 3 ^4 j is axisymmetric. 



Further, by equating in this the cofactor of e 4 on the left with the cofactor on the 



right we have 





