SOME IDENTITIES CONNECTED WITH ALTERNANTS. 



193 



m x m 2 m s aA 4 

 n x n 2 n s &B 4 

 r i r 2 r s cC i 



S-i So So CvUi 





A A l A 2 Ag 

 B B x B 2 B 6 

 C C x C 2 C 6 

 D D, D 2 D 6 



+ 



+ 



m 1 m 2 m 3 «A 6 

 % x n 2 n % 6B 6 

 r i »"« r s c Cg 



S l S 2 S 3 ^Dg 



w^ m 2 m 3 «A 



«1 » 2 W S &B 

 *1 ^2 r 3 CC 



S l S 2 S 3 °^0 





A Q Aj A 2 A 4 

 B B x B 2 B 4 

 C C x C 2 C 4 

 Do Di D 2 D 4 



A x A 2 A 4 A 6 

 B x B 2 B 4 B 6 

 Cj C 2 C 4 C 6 

 Di D 2 D 4 D 6 



Changing the suffixes of the capital letters into exponents, and putting 



we obtain (N 4 ). 



m 1 ,n 1 ,r 1 ,s 1 = 1,1,1,1, 

 ra 2 , n 2 , r 2 , s 2 = a , b , c , d , 

 m 3 , n 3 , r 3 , s 3 = a 2 , & 2 , c 2 , d 2 , 



(9) The insertion of particular values in the same general identity will not however 

 give (D 4 ), oi' anything resembling it. In fact it would seem that (D 4 ), although from 

 other points of view simpler than (N 4 ), cannot be proved in this way at all, — that is to 

 say, from ai vanishing determinant of the 8th order by the use of Laplace's expansion- 

 theorem, — the first factors of the determinant products containing as many as eight 



columns, 



1 a A aA A 2 aA 2 A 4 aA i 



1 b B SB B 2 6B 2 B 4 6B 4 



1 c C cC C 2 cC 2 O cC 4 



1 d D e/P D 2 rfD 2 D 4 eZD 4 ; 



and the second factors as many as seven, 



1 A A 2 A 3 A 4 A 5 A 6 



1 B B 2 B 3 B 4 B 5 B 6 



1 C C 2 C 3 C 4 C 5 C 6 



1 D D 2 D 3 D 4 D 5 D 6 ; 



so that if a determinant of the 8th order were formed with these, the number of resulting 

 products instead of being fewer than in the case of (N 4 ) would be far greater. 



(10) There is still a third form which the identities may be made to take, and from 

 which it was reasonable to expect something in the way of suggestion for a new mode 

 of proof. This is obtained from the previous form, — that is to say, the form which we 

 have just been considering, — by performing the determinant multiplications there 

 indicated. In the original form the identities consisted of terms each of which was the 

 product of a determinant and a symmetric function : then they were changed into 

 vanishing aggregates of products of pairs of determinants : and now by a further change 

 they appear as vanishing aggregates of single determinants. 



A little examination shows that it is essential in performing the determinant multi- 



