516 



DR THOMAS MUIR ON VANISHING AGGREGATES 



The three minors, if written in the other mode of notation, take the form 



2,3, , k , k + 2 , , n 



1,2, ..., h - 1 , h + 1 , ..., 1-1,1 + 1 , ..., re 



2,3, ,1,1 + 2, ..., re 



1,2, ..., h-1 ,h+l , . . . , k-1 ,k+l , ..., re 



2,3,..., h , h + 2 , , re 



1,2, ..., ib-1 Jb + 1, ..., Z-l.Z+1, ..., n 



and by the transposition of certain rows and of certain columns become 



h + 1 , 1 + 1 , 2 , 3 

 * , n ,1,2 



h+l,k+l, 2,3 

 Z , re ,1,2 



k+l,l + l ,2,3 

 Zi , n ,1,2 



., re 



, re- 1 



, re 



, re- 1 



, re 



, re - 1 



., h,h+2, . . ., &,& + 2, . . ., 1,1 + 2, 

 . , h - 1 , h + 1 , ..., k-1 ,k+l , ..., Z - 1 , 1 + 1 



.,h,h + 2, ...,k,k + 2, ..., 1,1 + 2, 

 ., h-l,h + l , ..., /t-1,^+1, ..., Z - 1 , I + 1 



., Jc, h + 2, . . ., k,k + 2, . . ., 1,1 + 2, 

 . , h-l,h+l , ..., 7c -1, k+1, ..., Z - 1 , Z + 1 , 



if the common sign-factor ( — iy+ k + l + n - 10 be deleted. In the matter of their last 



secondary minor they are now identical ; and the said minor, which in the shorter 



notation is 



i l,A+l,Jb+l,Z + l 



; h , k , I , re 



is seen to be axisymmetric. To apply the proposition of § 2 for the purpose of 

 establishing our theorem (B), it therefore only remains to take the three comple- 

 mentary minors of this common secondary minor, viz., the minors 



h + 1 , 1 + 1 



k , n 



h+1 , k+1 



I , re 



k+1 ,1+1 



h , re 



and prove that the cofactors of the elements in the first are equal to the cofactors of 

 the same elements in the second or third, and that the two remaining elements of the 

 second are the same as the two remaining elements of the third. Now the first element 

 in the first, (h+1 , h), has the cofactor 



Z+1,2, 3, 



re ,1,2,.... , 



and the first element in the third (k+1 , h), which because of the persymmetry is equal 

 to (h+ 1 , k), has the same cofactor : similarly, it is seen that the element (1+ 1 , k) in 

 the first has the same cofactor as the element (k+1 ,1) in the second. Again, the 

 cofactors of (h+1 , n) in the first and second being 



Z+l, 2, 3, 



k ,1,2, 



A;+l ,2,3, . 

 I ,1,2, . 



are by reason of the persymmetry the conjugates of each other : similarly with the 

 cofactors 



ft+1,2,3, 

 k ,1,2, 



k+1 ,2,3, 

 h ,1,2, 





