514 



DR THOMAS MUIR ON VANISHING AGGREGATES 



By applying the theorem to one of the determinants on the right, the original 

 determinant may be expressed as the sum of three: e.g., in the case of the 4 tA order 

 we have 



«1 



a.. 



a 3 



«4 





\ 



\ 



h 



\ 







c l 



C 2 



c s 



C 4 





eZ, 



d. 2 



c % 



d, 





«1 



P 



C l 



^ 





P 



X 



c i 



£?! 





X 



a. 



«3 



«4 



Q 



&2 



&3 



h 



+ 



Y 



Q 



&R 



^4 



+ 



Z>1 



Y 



C l 



*1 



«s 



C 2 



C 3 



c i 





C 2 



a s 



C 3 



C 4 





&3 



C 2 



C 3 



C 4 



«4 



^ 2 



C 4 



d i 





d, 



a, 



f 4 



^4 





&4 



^2 



C 4 



<*4 



(3) We are now in a position to enunciate and prove the first generalisation above 

 referred to, which is — 



Secondary minors of any axisymmetric determinant of order higher than the 3 rd are 



connected by the relation 



.lift! | 1 I ;• , j 1 h \ 



' ; h l\ ~ \ Ji h\ + \ k l\ 

 where 1 > k > h > 1 . 



If we indicate the minors by means, not of omitted lines but of those retained, the 



identity takes the form 



2, ..., ... ,k-l,k + l, 



1, ...,h-l,h+l, ... ,1-1,1+1, 



2, , l-l, l + l,.. 



1 , . . . , h - I , h + 1 , . . . , k-1 , k + 1 , . .. 



2,...,h-l,h + l, 



1,..., k-1, k+1, ..., l-l, l+l, .. 



or, by transposition of certain rows and of certain columns, and the omission 

 common sign-factor ( — ]_)*+*+*-? } 



+ 



of the 



h, 



I, 



2 



3, . 



. , h - 1 , 



h+1, . . 



., k-1, 



k + 1, . . . 



l-l, 



l + l 



1, 



k, 



2 



3, . 



., h-1, 



h+1, 



., k-1, 



k+1, . . . 



l-l, 



l+l 



h, 



k, 



2 



3, . 



.,h-l, 



h + 1, 



., k-1 



k + 1 , ... 



l-l, 



l + l 



1, 



I, 



2, 



3, • 



., h-1, 



h+1, . . 



., k-1, 



k + 1, ... 



l-l, 



l + l 



k, 



1, 



9 



3, • 



., h-1, 



h+1, 



., k-1 , 



*+l, . . . 



/-I, 



l+l 



1, 



h, 



2 



3, • 



., h-1, 



h+1, 



., k-1, 



k+l , ... 



l-l, 



l + l 



As thus written the three determinants have the same final secondary minor, viz.. 

 2, 3, . . ., K-1, h + 1, . . ., k-1, k+1, .. ., l-l, l + l, . . ., i 



2 , 3 , . . . , h-1, h + 1 



., k-1, k+1, 



l-l, l + l, . . ., n , 



which is clearly axisymmetric ; the cofactors of (h , l) and (I , 1) in the first determinant, 



viz., 



I, 2, 3, 

 k, 2,3, 



and 



h , 2 , 3 , 

 k, 2,3, 



are the conjugates of the cofactors of the same elements in the second determinant and 

 third determinant respectively : the cofactors of (h , h), (I , k) on the left are exactly the 

 same as their cofactors on the right ; and the two new elements introduced into the one 

 determinant on the right, viz. (k , 1), (h,l), are the same as those introduced into the 

 other. It follows therefore from the theorem of the preceding paragraph that (A) is 

 true. 





