194 



DR THOMAS MUIR ON 



plications that the lines multiplied in the process shall both be columns ; — that, for 

 example, instead of multiplying 



1 a A 





1 b B 



by 



1 c C 





A 2 

 B 2 

 C 2 



A 4 

 B 4 

 C 4 



which, strictly speaking, would give 



1 + a + A 

 1 + b+B 

 l+c+C 



A 2 +aB 2 + AC 2 

 A 2 +Z>B 2 +BC 2 

 A 2 +cB 2 + C 3 



A 4 +aB 4 + AC 4 

 A 4 +6B 4 + BC 4 



A 4 +cB 4 + C 5 



we first change the multiplicand into its conjugate with the result that the product 

 takes the form 







3 

 a + b + e 

 A+B + C 



A 2 + B 2 +C 2 



aA 2 + &B 2 +cC 2 



A 3 + B 3 + C 3 



A 4 + B 4 +C 4 



aA 4 +Z>B 4 +cC 4 



A 5 + B 5 + C 5 





Doing this (N 3 ) becomes 











3 2A 2 2A 4 

 2a 2aA 2 2aA 4 

 2A 2A 3 2A 5 



— 



3 2A 2A 4 

 2a 2aA 2aA 4 

 2A 2 2A 3 2A 6 



+ 



3 2A 

 2a 2aA 

 2A 4 2A 5 



2A 2 



2aA 2 



2A 6 



and (D 3 ) becomes 











3 2A 2 2 

 2a 2aA 2 2< 

 2aA 2«A 3 2< 



A 4 



*A 4 

 *A 5 



— 



3 2A 2A* 

 2a 2aA 2aA 4 

 2aA 2 2aA 3 2aA 6 



+ 



3 2A 2A 2 

 2a 2aA 2aA 2 

 2aA 4 2aA 5 2aA 6 



= 0. 



(11) Now in the former of these although three determinants occur each with nine 

 elements, the number of different elements is only two more than the number in a 

 single determinant. Putting therefore | a 1 b 2 c 3 | in place of the first determinant and 

 £ 1( £ 2 m place of the two additional elements found elsewhere, let us investigate the 

 aggregate thus formed, viz. the aggregate 



a x c x a 3 





\ ii \ 



+ 



a 2 C 2 ?2 





a„ 



1 





Clearly the terms involving c 2 in the first determinant are cancelled by those in the 

 second, the terms involving c 3 in the first by those in the third, and the terms involving 

 £ 2 in the second by those in the third. The aggregate may thus be legitimately replaced 

 by 







«3 









£l ^3 



+ 



a i c i 



a 2 



and therefore by 



h • 





a 2 





a 3 . . 



c i 



a 2 *3 — 

 h h 1 



a 2 



C l a 3 



+ a 3 



C l a 2 





which manifestly va 



nishes. 

















