SOME IDENTITIES CONNECTED WITH ALTERNANTS. 



191 



This transformation makes a totally different mode of proof possible : for, seeking 

 for a vanishing determinant of the sixth order which can be expanded into the left-hand 

 member of (N 3 ) by Laplace's theorem, we easily obtain 



Similarly the manifest identity 



gives at once (D 3 ). 



1 



a 



A 



A 2 



A 4 



1 



1 



b 



B 



B 2 



B 4 



1 



1 



c 



C 



C 2 



C 4 



1 



. 



. 



A 



A 2 



A 4 



1 





. 



B 



B 2 



B 4 



1 



• 



• 



C 



C 2 



C 4 



1 



1 



a 



aA 



aA 2 



aA 4 



a 



1 



b 



6B 



&B 2 



SB 4 



I 



1 



c 



cC 



cC 2 



cC 4 



c 





. 



A 



A 2 



A 4 



1 



. 





B 



B 2 



B 4 



1 



m 



m 



C 



C 2 



C 4 



1 



= 



(6) Here, as is usual, a really appropriate proof makes generalisation easy. It i& 

 readily seen that the vanishing of the latter determinant of the sixth order is not 

 dependent on the values of the elements in the first half of the first and second columns. 

 Substituting therefore h, h, I for 1, 1, 1 and m, n, r for a, b, c in these columns we 

 still have 



h 



m 



«A 



aA 2 



«A 4 



a 





h 



m 



. 



. 



. 





k 



n 



bE 



6B 2 



6B 4 



b 





k 



n 



. 





. 





I 



r 



cC 



cC 2 



cC 4 



c 





I 



r 



. 





. 



, 



. 



. 



A 



A 2 



A 4 



1 



— 







A 



A 2 



A 4 



1 







B 



B 2 



B 4 



1 





, 





B 



B 2 



B 4 



1 







C 



C 2 



C 4 



1 









C 



C 2 



C 4 



1 



and therefore 



= 0, 



+ 



h m aA 



k n bB 



I r cC 



h m aA 4 



k n &B 4 



I r cC 4 



1 



A 2 



A 4 





1 



B 2 



B 4 



— 



1 



C 2 



C 4 





1 



A 



A 2 





1 



B 



B 2 



— 



1 



C 



C 2 





h 



m 



aA 2 





k 



n 



&B 2 





I 



r 



cC 2 





h 



m 



a 





k 



71 



b 





I 



r 



c 





1 A A 4 

 IBB 4 



ICC 4 



A A 2 A 4 

 B B 2 B 4 



C C 2 C 4 



On putting h, h, I = 1, 1, 1 and m, n, r = a, b, c, we have (D 3 ), and on putting 

 h, k,l = 1, 1, l and a, b, c = 1, 1, 1 we have (N 3 ). 



A still further generalisation lies in the change of the exponents of A, B, C into 

 suffixes, the vanishing of the determinant of the sixth order being independent of the 

 meaning assigned to A n . 



