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309] 
NOTE ON THE THEORY OF DETERMINANTS. 
47 
ossible manner 
be the product 
{1234}, {12345}, 
of a set of n things; and the formulae for the several determinants up to the deter 
minant of a given order are all of them obtained by means of the form 
jn — for each 
expression for 
I 1 2 3 4 I the 
Lgn is + Or — 
wise when n is 
3 the partitions 
which is carried up to the order 7, but which can be further extended without any 
difficulty whatever. 
It is perhaps hardly necessary, but I give at full length the expressions of the 
determinant of the third order: this is 
11231 = 
11 
1 2 
1 3 
- 
11 
2| 
1 3 
-1 
2 
3 1 
1 
-1 
1 3 
1 1 
2 
+ 
1 1 
2 
3 
+ 
1 1 
3 
2 
and by writing down in like manner the expression for the twenty-four terms of the 
determinant of the fourth order, the notation will become perfectly clear. 
The formula hardly requires a demonstration. The terms of a determinant {123...%}, 
for example the determinant {1234}, are obtained by permuting in every possible 
manner the symbols in either column, say the second column, of the arrangement 
1 1 
2 2 
3 3 
4 4