1 34 Prof. Sylvester on Elimination and Derivation 

 Positive Permutations. 



1 



2 



3 



1 



2 



3 



2 



1 



3 



4 



4 



4 



2 



3 



1 



4 



4 



4 



1 



3 



2 



2 



1 



3 



3 



1 



2 



2 



3 



1 



4 



4 



4 



1 



3 



2 



4 



4 



4 



3 



1 



2 



3 



2 



1 



3 



2 



1 



and again 



Negative Permutatio7is. 



1 



2 



3 



4 



4 



4 



2 



1 



3 



2 



1 



3 



2 



3 



1 



1 



2 



3 



4 



4 



4 



1 



3 



2 



4 



4 



4 



2 



3 



1 



1 



3 



2 



3 



2 



1 



3 



1 



2 



3 



1 



2 



3 



2 



1 



4 



4 



4 



I reject from the permutations of each species all those 

 where 1 or 3 or both appear in the 4th place, and also those 

 where 2 or 4 or both appear in the 1st place, for these will 

 be presently seen to give rise to diagonal products which are 

 zero. 



The permutations remaining are 



Positive effectual permutations. 



1 



3 



3 



] 



2 



1 



4 



3 



3 



2 



1 



4 



4 



4 



2 



2 



Negative effectual permutations. 



3 



1 



1 



3 



1 



4 



3 



2 



4 



3 



2 



1 



2 



2 



4 



4 



I now accordingly form four positive squares, which are 

 a b c o I m n o I m n o a h c o 



a b c a b c o o I m n I m n a 



1 m n o a b c a b c a o I m n 



I m n o I ju n o a b c o a b c 

 Drawing diagonal lines from left to right, and taking the 



sum of the diagonal products, I obtain a-w^ + 11''-^ + V c^ 

 -\-am-c. Again, the four negative squares 



1 m n o a b c o a b c o I m n o 

 a b c o o I m n I m n o o a b c 

 o I m n I m n o o a b c a b c o 

 o a h c a a b c o I m n o I m n 



