512 



DR. THOMAS MTJIR ON VANISHING AGGREGATES 



the minor obtained by deletion of the a th , /3 th , y th , .... rows, and J*f, 17 th , 6 th , . 

 columns, — in other words, the complementary minor of 



a, /?, y, 



Z,v,0, •••• 



the two results will be found readily comparable. The first of them, relative to an axi- 

 symmetric determinant of the 5 th order, would then become 



2 4 



3 5 



2 5 



3 4 



or, with an appearance of greater generality, 



a y 



P 8 



a 8 



j8 y 



where a, /3, y, 8 are a set of four integers chosen from 1, 2, 3, 4, 5 ; and the second, 

 relative to any axisymmetric determinant, would become 



( 1 r ] 





j 1 



r + 1 | 



+ 



: 1 



2 i 



lis: 





1 



s-1 i 





1 s-1 



r ; 



As neither of these is included in the other, I have been induced to make a fuller 

 investigation, which has resulted, among other things, in the generalisation of both, and 

 the discovery of the nature of the connection between the two as thus generalised. 



(2) It is seen that in both theorems one determinant is expressed as the sum of two 

 others, and the presumption is thus raised that they are cases of the general theorem 

 published in 1888, which shows how any determinant of the n tli order can be expressed 

 as the sum of n determinants of the same order, the specialisation introduced having the 

 effect of reducing the n determinants to 2. As regards 4 th order determinants the said 

 general theorem gives 



h b 2 



Cfc-1 W/Q (A/Q f^A 



Qi 



Q 2 



Q 3 



C 2 



d 2 

 P„ 



d^ d 2 



Qi 



Q 2 



Q 3 





?! 



«2 



P 4 P B 



+ 





Q 2 



H C 4 





d, 



Qb 



Cvq Ct A 





P 3 



p 5 



P 6 a A 



+ 





C 2 



C 3 ^62 





d x 



d 2 



a 3 Q 3 



where there occur on the right nine elements which do not appear on the left, viz., 

 P x , P 2 , . . . , P 6 and Q l , Q 2 , Q 3 . Now it is manifest that the fourth determinant on 

 the right will vanish by making 



and the third by making 



"3 » " 5 » 6 > ^&3 — **1 ' ^2 > f ' 3 > a 4 ) 



"2 > 4 > ^2 > "^ 6 ™ C l ' C 2 ' a S > C 4 





