SOME IDENTITIES CONNECTED WITH ALTERNANTS. 



201 



The corresponding results for (N 3 ), (D 3 ), (N 4 ) are 



1 a 1 





1 a a 





1 a 1 a 





1 b 1 

 1 c 1 



| A^C 4 i , 



1 b b 

 lee 



| A*B 2 C 4 1 , 



1 b 1 b 

 1 c 1 c 

 1 cl I d 



| A^OD* 



as may also be seen from §§ 6, 8. 



(18) The problem of finding, for determinants of a higher order than the fourth, 

 identities similar to Cayley's I do not at present enter upon : like Cayley, " je n'ai pas 

 encore trouve" la loi general e de ces equations." I content myself with stating the 

 problem in as simple a form as possible. 



To express each of the products 



For the fifth and sixth orders it is : — 



I 1 



I 1 



A aA 

 aA a 2 A 



a a 2 A aA a 2 A 



A°B 2 C 4 D 6 E 8 | 

 A°B 2 C 4 D 6 E 8 F 10 



A B 2 C 4 D 6 E 8 F 10 



as aggregates of products of a similar hind, the first factor of each product on the 

 right of the identity being formable from the corresponding factor on the left by replac- 

 ing A by an even power of A, B by the same even power of B, and so on. 

 For example, and more definitely, to determine a, /3, y, 8, e, £, t], 6, so that 



| 1 a a 2 A aA | • | A°B 2 C 4 D 6 E 8 I = I 1 a a 2 A 2 aA 2 | • | A°B a C^DvE« | 



±|1 a a 2 A 4 aA 4 | • | A B<CO>E 9 | 

 ±11 a a 2 A 6 aA 6 | • 1 A^C^E 4 |. 



