SOME IDENTITIES CONNECTED WITH ALTERNANTS. 



197 



and the first determinant by | a 1 b 2 c s d i | we obtain for investigation the more general 

 aggregate 



a i «2 a 3 a i 



\ b 2 b 3 5 4 



C l C 2 C 3 C 4 



d x d. 2 d 3 d i 



ii a s 



\ d 1 



\ d 2 



C 4 



a i ii a 2 

 \ d x b 2 & 4 



h d 3 



K L 



£ 



a 2 



«3 



d x 



h 



h 



A 



C 2 



C i 



d 4 



& 



£ 



Now it is readily seen that 



the cofactor of d„ in the 1st determinant 



are cancelled by 



d 3 

 h 



1st 



1st 



2nd 



2nd 



3rd 



the cofactor of d 2 in the 2nd determinant 



"3 



h 



3rd 

 4th 

 4th 

 3rd 



4th 



respectively. All therefore that is left in each determinant is one element accompanied 

 by its cofactor, viz., the element in the place (41) ; so that the aggregate with which we 

 started is reduced to 



-d, 



a 2 



H 



a 4 



+ \ 



\ 



h 



\ 





h 



c s 



C 4 





€l «3 a i 



£Z C 3 C 4 



I di K 



1 fc 2 C 2 



+ K 



ii a 2 



di \ 



fc2 C 2 



which again is clearly equivalent to 



fcl a 2 a Z a i 



and therefore vanishes. 



d x b 2 



d 1 b 2 



(15) This identity is not new, being simply a special instance of the fourth-order 

 case of the theorem above referred to as having been used in proving Kronecker's 

 relation between the minors of an axisymmetric determinant. (N 3 ), (D 3 ), (N 4 ) are thus 

 seen to be dependent on the same theorem. The full expression for | a 1 6 2 c 3 c? 4 I is 



a x 



a 2 



a 3 a 5 





«1 



\ 



\ 



^3 & 



+ 



\ 



c i 



H 



c 3 y 5 



C l 



»U 



D 2 4 



D 34 d i 





D K 



c 2 y 5 



D„„ d Q D 



+ 



u 



a x 



a 5 



a s 



a 4 





«5 



a 2 



H 



a i 



\ 



& 



h 



K 



+ 



A 



\ 



\ 



h 



c i 



75 



h 



c i 



75 



C 2 



C 3 



c i 



D 12 



d 2 



I>23 



E»24 





*1 



D12 



D» 



D M 



