OF SECONDARY MINORS OF A PERSYMMETRIC DETERMINANT. 



529 



The general theorem involved may without symbols be formally enunciated as 

 follows : — 



A vanishing Kronecker aggregate of m-line minors may have each minor 

 extended to any order lower than the (2m)' A without impairing the identity, 

 'provided the introduced column-numbers be included among the invariable row- 

 numbers previously existing. 



(27) The need for the two restrictions, (1) that the order of the extended minors be 

 lower than the (2/u) th , and (2) that the introduced column-numbers be not different 

 from numbers found among the original set of invariable row-numbers, will be 

 tolerably apparent from the consideration of one special case, say the case of 



^ | 12 4 5 6 |. 



Here if we put a , j3 in place of the column numbers 1, 2 we have a more general 

 aggregate for investigation. By expansion in terms of minors of the first two rows 

 and the complementaries of these there is obtained 



* yl23 

 q/3456 



xy 

 a/3 



123 



456 



xy 

 a4 



xy 

 J34 



z 



Z 



1 23 

 (3 5_6 | 



123 

 a56 



M 



xy 

 a5 



z 



1 23 

 £46 



- 



xy 



a6 



z 



1 23 



(3i5 



1 * y 

 1 IjSB 



Z 



1 23 



o46 



+ 



xy 

 (36 



z 



123 

 a4 5 



1 x y 



+ ]45 



Z 



123 

 06 



- 



xy 

 46 



^ j 1 23 

 ^ | a/35 



[ x y 



^ 1123 • 



^ la/34: 



123 



456 



va 



nish 



38, bu1 



: the 



sim 



tilar 



factors 



Now in the first line of this development ^ 



of the other terms do not unless ft = 1 or 2 : in the second line arises the like condition 



a=l or 2 ; and in the remaining lines both conditions at once. Our only possible 



result of the 5 th order is thus 



x y 1 2 3 



12 456 



= 



(28) As one might expect from a remembrance of the opening lines of § 20, the 

 theorem may also be looked upon as a special case of Kronecker's. Thus from 



Kronecker as amended we have 



x y 1 2 3 I 



7 8 4561 







provided that 

 gives us 



378456 

 378456 



be axisymmetric ; and the change of 7 into 1 and 8 into 2 



xy 1 2 3 

 12 4 5 6 



i.e. 



x y 1 2 3 

 12 456 



= 



when 



2 



3 12 4 5 6 



3 12 4 5 6 



123 



456 



is axisymmetric. As the condition for the vanishing of 



3456 



is merely that 3455 be axisymmetric, we thus see that the extension 



