196 



DR THOMAS MUIR ON 



where, be it observed, the number of different elements in the three determinants is only 

 1 6, that is to say, exactly the number in any single one of them. Replacing therefore 

 the first of them by | a l b 2 c 3 d i | we obtain for investigation the more general aggregate 



Oj a 9 a 3 a i 



b 1 b., b 3 Z> 4 



a* a. 



a x 



H 



\ 



\ 



a 3 



a 4 



h 



h 



c i 



C 2 



*i 



d 2 



h 



h 



d 3 



** 



\ \ d x d 2 



d x d 2 d 3 d i i b s b i d 3 d i 



i 



Expressing each of the three determinants as an aggregate of products of complementary 

 minors formed from the first two columns and the last two columns we see that 



the 1st product of the 1st determinant 

 2nd 1st 



are cancelled by 



. 5th 



. 6th 



. 1st 



. 6th 



.. 1st 

 .. 1st 

 .. 2nd 

 .. 2nd 



the 2nd product of the 2nd determinant 

 2nd 3rd 



. 5th 



. 5th 



. 1st 



. 6th 



.. 3rd 



.. 2nd 



.. 3rd 



. .. 3rd 



respectively. Only six products are thus left, viz., the 3rd and 4th of each determinant, 

 the aggregate of the six being 



a x 



a 2 





h 



\ 



+ 



[ h 



\ 





<h 



a 4 





a, 



a ? 





C 3 



C 4 



<h 



d 2 





C 3 



c i 



\ c x 



C 2 





d 3 



d A 



+ 



h 



\ 





d x 



d 2 



H 



« 4 





C l 



C 2 





! a x 



«2 





K 





a s 



a i | 



h 



K 



\ 



\ 





d 3 



d, 





1 c 3 



C 4 





d x 



d 2 





c i 



C 2 





d 3 



d i 



+ 



which is nothing more than the double of zero in the form of the well-known vanishing 

 aggregate 



I cti£ g I • I a 3 /3 4 | - ! ai /3 3 I • I a 2 /3 4 | + | ai /3 4 I • I aj3 3 \ . 



(14) The new form of (N 4 ) is 



+ 



4 



2A 2 



2A 4 



2A 6 





4 



2A 



2A 4 



2A 6 



2a 



2aA 2 



2aA 4 



2aA 6 





2a 



2aA 



2aA 4 



2aA 6 



2a 8 



2a 2 A 2 



2a 2 A 4 



2a 2 A 6 





2a 2 



2a 2 A 



2a 2 A 4 



2a 2 A 6 



2aA 



2aA* 



2aA 5 



2aA 7 





2aA 2 



2aA 3 



2aA 6 



2aA 8 



4 



2A 



2A 2 



2A 6 





4 



2A 



2A 2 



2A 4 



2a 



2aA 



2aA 2 



2aA« 





2a 



2aA 



2aA 2 



2aA 4 



2a 2 



2a 2 A 



2a 2 A 2 



2a 2 A 6 





2a 2 



2a 2 A 



2a 2 A 2 



2a 2 A 4 



2aA 4 



2aA 5 



2aA 6 



2aA 10 





2aA 6 



2aA 7 



2«A 8 



2aA 10 



= o, 



where now the number of different elements is 20, — that is to say, four more than the 

 number in any single determinant. Replacing these four elements by £ x , £ 2 , £ 3 , £ 4 , 



