SOME IDENTITIES CONNECTED WITH ALTERNANTS. 



199 



A little examination of this square array brings out the fact that all its terms cancel 

 each other with the exception of those which are situated in either diagonal, and that 

 the cancellation takes place in a pleasingly symmetrical fashion, each non-diagonal term 

 and its conjugately situated fellow annihilating one another. Further, the aggregate of 

 the six terms in the principal diagonal is seen to be 



and the aggregate of the six terms in the secondary diagonal to be 



I «i & y* ^ I ; 



so that the theorem is established. 



Now turning to the identity of § 13, which, as we have noted, has no elements 

 different from the sixteen composing a single one of the determinants involved in it, we 

 see at once that if it is to be included in that just found, the elements of the second 

 parent determinant of the latter must be the same as the elements of the first. Making 

 them the same even in form we obtain the nugatory result 



2 | a&c^ | = 2 | ajb&fli \ . 



But making the consanguinity less pronounced, the second parent being of the form 

 | a^^d^ | , we find that only the 3rd and 4th of the progeny are of no account, and 

 that the 6th and 5th are the same as the 1st and 2nd respectively : so that after 

 division by 2 there results 



a x 



a. y 



a z 



a i 





\ 



h 



\ 



K 





c i 



C 2 



C 3 



C 4 





d x 



d 2 



d 3 



d, 





«i 



H 



c i 



H 





h 



\ 



4 



d 2 



+ 



% 



a t 



h 



C 4 





h 



K 



d 3 



d, 





a. y 



h K 



d 1 d 2 

 d d A 



which completely agrees with the identity of § 13. 



(17) A fourth method of investigation consists in making transformations which 

 result in the segregation of the capital letters from the small letters. As, however, 

 nothing new results from it in the case of (N 3 ), (D 3 ), (N 4 ), its application to (D 4 ) is all 

 that need be given. 



The aggregate to be considered in this case is 



1 



a A 



aA. 



1 



b B 



m 



1 



c C 



cQ 



1 



d D 



dD 



A°B 2 C 4 D 6 I - 



1 



a 



A 



aA 2 



1 



b 



B 



6B 2 



1 



c 



C 



cC 2 



1 



d 



D 



^D 2 



A B 1 C 4D5 I + 



1 a A 4 aA 4 

 1 b B 4 &B 4 

 1 e C 4 cC 4 



d D 4 dD* 



A o B i C 2 D 3 1 



Now each of the first factors of the three determinant products here appearing is of the 

 form 



1 a A w aA" 



1 b B w 5B" 



1 c C" cC» 



1 d D w dr>« 



