528 



DR THOMAS MUIR ON VANISHING AGGREGATES 



where the members of the aggregate \ p ,q ,r; 4,5,6,7,8 ; are minors of the deter- 

 minant | a 1 (3 2 7 3 ^ 4 e 5 £ 6 ^ 7 g | , and therefore all vanish when the latter is axisymmetric. 

 The general form of the development of the aggregate of § 18 is thus apparent. 



(26) The case where the quadrate array is of the (2m) ttl order and the non-quadrate 

 array consists of m — 1 columns is specially interesting. We have then to increase the 

 latter array by w — 1 columns repeated from the square array in order to make § 17 

 applicable, and the theorem takes the form of an extensional of Kronecker's without any 

 apparent corresponding extension of the condition of axisymmetry. Thus Kronecker's 

 four-line identity is 



= I a 5^6 77 S s I - I a 4^e77 e s ! + ! a J 3 577^ I - I a 4 /* 5 y 6 i7 8 I + I a 4 A>y 6 7 I 



with the condition that | a x /3 2 . . . & | be axisymmetric, and, as the preceding paragraph 

 shows, we can extend each minor to the 7 th order by prefixing x x y 2 z 3 without any 

 condition in regard to the 24 introduced elements 



*1 I 3-2 J 



"1 > "2 > 



y s 



Equating cofactors of a^ in the two sides of this extended identity we have a similar 

 identity, and equating cofactors of y 2 in the two sides of the result thus derived we 

 have a third. 



If we use a different notation for Kronecker's aggregates, denoting the aggregate 



say, by 



1 2 3 



4 5 6 





1 2 4 

 3-5 6 



"2 



+ 



1 2 3 

 4 5 6 





1 2 5 

 3 4-6 





1 2 6 

 3 45 



where underlining and overlining indicate variability, the matter may be put more 

 satisfactorily. The proposition then is that besides the vanishing of 2, 



the determinant 



123456 



123456 



123 



456 



when 



is axisymmetric, we also have 



z 



a; 1 2 3 

 145 6 



x 1 2 3 



2 45 6 



xy\ 23 

 12456 



= o, 

 = o, 



= 0, 



apart altogether from any condition in regard to the elements 





x 6 



y 6 





