SOME IDENTITIES CONNECTED WITH ALTERNANTS. 



195: 



(12) The identity thus obtained is of course more general than (N 3 ), because it holds 

 when the eleven different elements involved in it are quite independent of one another. 

 Passing over (D 3 ) let us at once try the process on the very general theorem of § 6 which 

 includes both (N 3 ) and (D 3 ). The aggregate then to be considered is 



Xh 2&A 2 



2&A 4 





2m 2mA 2 



2mA 4 



— 



2aA 2aA 3 



2aA 5 





+ 



ZA 



2&A 



2AA 4 





2m 



2mA 



2mA 4 





2aA 2 



2aA 3 



2aA 6 





2h 



2AA 



2&A 2 





2m 



2mA 



2mA 2 



— 



2aA 4 



2aA 5 



2aA 6 





2AA 



2AA 2 



2/iA 4 



2mA 



2mA 2 



2mA 4 



2aA 



ZaA 2 



2«A 4 



where the number of different elements is now five more than the number in any one of 

 its four determinants. Replacing these five elements by £j, £ 2 , £ 3 , £ 4 , £ 5 , and the first 

 determinant by | ctib 2 c 3 j we obtain 



\ h 



and see that the terms in c x , c 



£i 





«I £l 



+ 



h £2 





£5 H 



£i 



«2 



£ 2 



\ 



c i 



£s 



c 3 in the first determinant are cancelled by those in 

 Cj, c 2 , c 3 in the 4th, 2nd, 3rd determinants respectively ; so that the aggregate reduces to 



ii a s 





a i £l a 2 



& 



£2 h 



+ 



h £2 \ 



- \ £2 



• & 





& • £* 





&3 €5 



which by reason of the terms in £ 3 , £ 4 , £ 5 cancelling themselves reduces further to zero. 



Strange to say, the general vanishing aggregate to which we have thus been led for 

 the purpose of establishing Cayley's identities (N 3 ), (D 3 ) is essentially the same as that 

 to which a study of Kronecker's theorem regarding the minors of an axisymmetrie 

 determinant brought me in 1888 (see Proc. Roy. Soc. Edinb., xv. pp. 96-98). As a 

 foundation of two important theorems of so diverse a character the said vanishing 

 aggregate becomes of considerable interest. 



(13) Turning now to (N 4 ) and (D 4 ) let us take the latter first, because although it 

 proved recalcitrant to the previous method, it ought to yield more readily than (N 4 ) to 

 the present method, if it yield at all. The new form of it is 



4 2a 



2A 



2aA 





4 



2a 



2A 4 



2aA 4 



2A 2 2aA 2 



2A 3 



2aA 3 





2A 



2aA 



2A 5 



2aA 5 



2A 4 2aA 4 



2A 5 



2aA 5 



+ 



2A 2 



2aA 2 



2A 6 



2aA 6 



2A 6 2«A 6 



2A 7 



2aA 7 





2A 3 



2aA 3 



2A 7 



2aA 7 









4 



2a 



2A 2 



2aA 2 









2A 



2aA 



2A 3 



2aA 3 









2A 4 



ZaA 4 



2A 6 



2aA 6 











2A 5 



2aA 5 



2A 7 



2aA 7 



= 0, 



VOL. XL. PART I. (NO. 9). 



2 F 



