134 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 Permutations. 
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 
1 2 
1 
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 
1 
2 
1 
4 
3 
3 
2 
1 
4 
4 
4 
2 
2 
Negative effectual permutations. 
3 
1 
1 
8 
1 
4 
3 
2 
4 
3 
2 
1 
2 
2 
4 
4 
I now accordingly form four positive squares, which are 
a h c 0 1 m n 0 
1 m 11 0 
a h c 0 
0 a h c a b c 0 
0 1 m n 
1 m n 0 
1 m n 0 0 a h c 
a b c a 
0 1 m n 
0 1 m n 0 1 711 n 
0 a b c 
0 a h c 
Drawing diagonal lines from left to right, and taking the 
sum of the diagonal products. 
I obtain cr 
Ib^ n 4- c^ 
4- a m^ c. Again, the four negative squares 
1 m n o a h c 0 
a b c 0 
1 m n 0 
a b c 0 0 1 m n 
1 m n 0 
0 a b c 
0 1 m n 1 m n 0 
0 a b c 
a b c 0 
0 a b c 0 a b c 
0 1 m n 
0 1 m n 
