192 



DR THOMAS MUIR ON 



(7) Turning now to (N 4 ) and (D 4 ) we find the same alteration of their form possible. 

 From the theory of alternants it is known that 



A°B 2 C 4 D 6 

 A o B i C 2 D 6 



AOB^D 6 

 A o B i C 2 D 4 



A°B 1 C*D 6 



A B l O 2 D 3 , 



A B 1 C 2 D 3 



A B 1 C 2 D 3 



A^CPD 3 



A B1C 2 D 3 , 



(D + C)(D + B)(D + A)(C + B)(C + A)(B + A), 

 = 2A 3 + 2A 2 B + 2ABC , 

 = 2A 3 B 2 + 2A 3 BC + 22A 2 B 2 C + 32A 2 BCD , 

 - A + B + C + D, 

 = 2A 2 B 2 + 2 A*BC + 2ABCD . 



Substituting for these expressions on the right and then multiplying by | A°B 1 C 2 D 3 

 we find (N 4 ) in the form 



- 



+ 



1 a a 2 aA 



1 b b 2 bB 



1 c c 2 cC 



1 d d 2 dD 



a 



a 2 



aA* 





b 



b 2 



5B 4 





c 



c 2 



cC 4 





d 



d 2 



dD* 





and (D 4 ) in the form 



= 



1 a A aA 



1 b B bB 



1 c C cC 



1 d D dD 



1 A 2 A 4 A 6 



1 B 2 B 4 B 6 



1 C 2 C 4 C 6 



1 D 2 D 4 D 6 



1 A A 2 A 6 



IB B 2 B 6 



1 C C 2 C 6 



ID D 2 D 6 



1 A 2 A 4 A 6 



1 B 2 B 4 B 6 



1 C 2 C 4 C 6 



1 D 2 D 4 D 6 



+ 



1 a 



a 2 



aA 2 





1 



A 



A 4 



A 6 



1 b 



V 



6B 2 





1 



B 



B 4 



B 6 



1 c 



c 2 



cC 2 





1 



C 



C 4 



C 6 



1 d 



d 2 



dD 2 





1 



D 



D 4 



D 6 



1 a 



a 2 



aA 6 





1 



A 



A 2 



A 4 



1 b 



b 2 



bB 6 





1 



B 



B 2 



B 4 



1 c 



c 2 



cC 6 





1 



C 



C 2 



C 4 



1 d 



d 2 



dD 6 





1 



D 



D 2 



D 4 



1 a 



A 2 



aA 2 





1 



A 



A 4 



A 5 



1 b 



B 2 



bB 2 





. 1 



B 



B 4 



B 5 



1 c 



C 2 



cC 2 





1 



C 



C 4 



C 5 



1 d 



D 2 



dD 2 





1 



D 



D 4 



D 5 



1 a 



A 4 



aA 4 





1 



A 



A2 



A 3 



1 b 



B 4 



bB i 





1 



B 



B 2 



B 3 



1 c 



C 4 



cC 4 





1 



C 



C 2 



C 3 



1 d 



D 4 



dD* 





1 



D 



D 2 



D 3 



(8) In its new form (N 4 ) can be as easily proved and generalised as (N 3 ). Pro- 

 ceeding at once to the generalisation, we have clearty 



m. 



n. 



m 2 m 3 aA 1 aA 2 aA 4 aA 6 aA 

 n 2 n % bB 1 bB 2 bB 4 

 r 2 r 3 cC x cC 2 cC 4 



s 1 s 2 s 3 dD 1 dD 2 dD i 





B. 



A 4 



B 4 



Cj C 2 C 4 

 Di D 2 D 4 



&B 6 

 cG 6 

 dD 6 

 A 6 

 B 6 

 C 6 

 D R 



bB 



cG 



dD„ 



B, 



D„ 



= 0, 



and therefore 



= - 



m x m 2 m % aA x 



n i n 2 n 3 ^B x 



r, r 2 r 3 cCj 



h s 2 h dl) i 



A A 2 A 4 A 6 

 B B 2 B 4 B c 



C G 2 C 4 C 

 D D 2 D 4 D 



+ 



m i m 2 m 3 ftj ^2 



5B 

 cC 



■3 ^B 2 



s 3 dD 2 



^0 A-l ^-4 ^-6 



B B x B 4 B 6 

 C C x C 4 C 6 

 Do D x D 4 D 6 



