190 



DR THOMAS MUIR ON 



identities, but the labour of writing out the products is very considerable. For example, 



the multiplier 



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

 or 



2A 3 B 2 C + 22A 3 BCD + 22A 2 B 2 C 2 + 42A 2 B 2 CD 



on the left-hand side of (N 4 ) gives rise to 24 + 4 + 4 + 6 determinant terms, which how- 

 ever reduce to 22, viz. : 



A 3 aA a 2 aA 3 



+ I A 2 aA 



aA 4 



+ 2 1 A 2 a a 2 A 2 aA 3 | 

 + | A aA 2 a 2 «A 4 | 

 + | A aA a 2 A 3 aA 2 I 



+ | A 3 a a 2 A 2 aA 2 | + | A 3 a a 2 A aA 3 | 



+ | A 2 a a 2 A aA* | + \ A 2 a a 2 A 3 aA 2 \ 



+ 2 | A 2 aA a 2 A 2 aA 2 \ + 3 | A 2 aA a 2 A aA 3 1 



+ | A a a 2 A 3 aA 3 | + I A a a 2 A 2 aA 4 | 



+ 2 | A aA a 2 A a A* | + 2 | A aA 2 a 2 A aA 3 \ 



+ | A 3 aA a 2 A aA 2 

 + |A 2 aA 2 a 2 aA 3 



+ 3 | A aA a 2 A 2 aA 3 



+ |1 «A 2 ft 2 AaA 4 | + |1 aAa 2 A 3 aA 3 | + |1 aA a 2 A 2 aA 4 | + | 1 aA 2 a 2 A 2 aA 3 |; 



and the result of multiplication on the right-hand side is 64 determinant terms which 

 reduce to the same 22. 



In the case of (D 4 ) still further reduction on both sides is possible, viz., to 13 terms ; 

 but it is quite clear that little is to be hoped for from this method when determinants 

 of higher order than the fourth come to be dealt with. 



(5) A most important simplification of the form of the identities is suggested on 

 noticing that the multipliers in (N 3 ), 



(B + C)(C+A)(A+B) and A 2 + B 2 + C 2 + BC + CA + AB, 



are each expressible as the quotient of an alternant by the difference-product of 

 A, B, C, viz. : 



and 



(B + C)(C + A)(A+B) = 



A 2 + B 2 + C 2 + BC + CA + AB = 



1 



A 2 



A 4 



1 



B 2 



B 4 



1 



C 2 



C 4 



1 



A 



A 4 



1 



B 



B 4 



1 



C 



C 4 



1 



A 



A 2 



1 



B 



B 2 



1 



C 



C 2 



1 



A 



A 2 



1 



B 



B 2 



1 



C 



C 2 



When these new expressions are substituted, multiplication of both sides by the common 

 denominator J A°B 1 C 2 | gives (N 3 ) in the form 



= 0, 



1 a A 





1 A 2 A 4 





1 a A 2 





1 A A 4 





1 a A 4 





1 A A 2 



1 b B 





1 B 2 B 4 



— 



1 b B 2 





IBB 4 



+ 



1 b B 4 





1 B B2 



1 c C 





1 C 2 C 4 





1 c C 2 





1 C C 4 





1 c C 4 





ICC 2 



and (D 3 ) in the form 



1 a a A 1 



1 A 2 A 4 





1 a aA 2 





1 A A 4 





1 a aA* 





1 A A 2 



16 6B . 



1 B 2 B 4 



— 



1 b bB 2 





IBB 4 



+ 



1 b bB* 





IBB 2 



1 c cC 



1 C 2 C 4 





1 e cC 2 





1 C C 4 





1 c cC 4 





ICC 2 



= 0. 



