518 



DR THOMAS MUIR ON VANISHING AGGREGATES 



relation between those minors of P which do not involve both the first row and the 

 first column there will arise another relation which we may call the Diminutive of the 

 former. 



With this in view let us return to theorem (A), viz., 



1 1 



k i 



! 1 



I \ 



+ 



j 1 



h \ 



i A 



I i 



i h 



k : 





! * 



I \ 



The rows of P which occur in the minor on the left are those whose numbers are 



2,3, ... , k-1 , k+l , ... , n , 

 to each of which unit-diminution can be applied, the result being 



1,2,..., k-2 ,£,..., n-1 



and the minor itself becoming 



k-1 n 



h I 



Treating the other minors in the same way, we have for the Diminutive of (A) 



+ 



k - 1 n 

 h I 



l-l n 



h k 



h - 1 n 

 k I 



In similar fashion there is obtained 



k n 

 h I 



I n 

 h k 



h n 

 k I 



\ 



h 

 k-1 



n ■ 



h 



l-l 



k 

 n 



+ 



k 

 h-1 



I 

 n 



{ 



h 



k 



I j 



n ; 



h 

 Jc 



I 

 n 



+ 



k 

 h 



I 



n 



as the Diminutive of (B). 



By the change of rows into columns the two take the more convenient form 



(«) 



OS). 



(10) When the six identities (A) , (B) , (A') , (B') , (a) , (/3) are collected, a glance 

 suffices to bring out the fact that (a) is the same as (B') , and (/3) the same as (A') : in 

 other words, the Diminutive of either of the two original theorems is the Secondary 

 Conjugate of the other. 



This being the case, only one of the original theorems needs to be separately 

 established : for if (A) be proved we can obtain at once from it its Secondary Con- 

 jugate (A') , that is (/3) , and from (/3) by the principle of unit-increase we can derive 

 (B). Further, as (A) may be derived (§ 4) from a self-evident identity connecting 

 three two-line minors of an axisymmetric determinant of the fourth order by proving 

 the applicability of the Law of Extensible Minors to axisymmetric determinants, it is 

 seen that the six theorems can be traced back to a very humble origin. 



(11) Since from (A) we have 

 1 k+\ 



I 

 k+l 



1 

 k+l 





