182 Mr. A. Cayley on the Theory of Determinants. 
which is carried up to the order 7, but which can be further ex- 
tended 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 
{128} = 11/213] 
—|[1 213] 
—|}2 3{1| I 
—|8 142] 
+]1 2 3] 
+]1 3 2]5° 
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...n}, 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 
hI 
22 
3.3 
A 4 
