309] 
45 
309. 
NOTE ON THE THEORY OF DETERMINANTS. 
[From the Philosophical Magazine, vol. xxi. (1861), pp. 180—185.] 
The following mode of arrangement of the developed expression of a determinant 
had presented itself to me as a convenient one for the calculation of a rather complicated 
determinant of the fifth order; but I have since found that it is in effect given, 
although in a much less compendious form, in a paper by J. N. Stockwell, “ On the 
Resolution of a System of Symmetrical Equations with Indeterminate Coefficients,” 
Gould’s Ast. Journal, No. 139 (Cambridge, U. S., Sept. 10, 1860). 
Suppose that the determinant 
11, 
12, 
13 
21, 
22, 
23 
31, 
32, 
33 
is represented by {123}, and so for a determinant of any order {123 ... n). 
Let { 1 |, | 2 | , | 12 I, | 123 I, &e., denote as follows: viz. 
| 1 | = 11, | 2 | = 22, &c. 
| 12 | = 12.21, 
| 123 | = 12.23.31, 
&c., 
where it is to be noticed that, with the same two symbols, e.g. 1 and 2, there is but one 
distinct expression | 12 j (in fact | 21 | =21.12 = | 121); with the same three symbols, 
1, 2, 3, there are two distinct expressions, | 123 | (=12.23.31) and | 132 | (=13.32.21); 
and generally with the same m symbols 1, 2, 3 ...m, there are 1.2.3...(m—1) distinct