46 NOTE ON THE THEORY 
OF DETERMINANTS. [309 
expressions J123...m|, which are obtained 
all but one of the m symbols. 
This being so, and writing for greater simplicity | 1 | 2 | to denote the product 
|1 | x | 21, and so in general, the values of the determinants {12}, {123}, {1234}, {12345}, 
&c. are as follows: viz. 
by permuting in every possible manner 
No. of terms. 
+ 
{12} = + | 1 | 2 | .... 
1 
-|1 2| .... 
1 
1+ 1 = 
{123} = + | 1 | 2 | 3 | ... 
1 
~ 1 1 2 | 3 | ... 
3 
+ | 1 2 3 | ... 
2 
3+ 3 = 
{1234} = + I 1 | 2 | 3 | 4 | . . 
1 
-|1 2| 3|4 | . . 
6 
+ | 1 2 3 | 4 | . . 
8 
+ | 1 2 | 3 4 | . . 
3 
- | 1 2 3 4 | . . 
6 
12+12 = 
{12345} = + | 1 | 2 | 3 | 4 | 5 | . 
1 
-|1 2 1 3 | 4 [ 5 | . 
10 
+ | 1 2 3 | 4 | 5 | . 
20 
+ | 1 2 | 3 4 | 5 | . 
15 
- | 1 2 3 4 | 5 | . 
30 
- | 1 2 3 | 4 5 | . 
20 
+ | 1 2 3 4 5 | . 
24 
60 + 60 = 
where, as regards the signs, it is to be observed that there is a sign — for each 
compartment | | containing an even number of symbols; thus in the expression for 
{1234}, the terms | 1 2 | 3 4 | have the sign = +, and the terms | 1 2 3 4 | the 
sign —. Or, what comes to the same thing; when n is even, the sign is + or — 
according as the number of compartments is even or odd; and contrariwise when n is 
odd. As regards the remaining part of the expression, this merely exhibits the partitions