[ 180 ] 
XXVIII. Note on the Theory of Determinants. 
By A. Cayiny, Esq.* ’ 
f dasa following mode of arrangement of the developed expres- 
sion of a determinant had presented itself to me as a con- 
venient one for the calculation of a rather complicated determi- 
nant 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. 189 (Cambridge, U. 8., Sept. 10, 1860). 
Suppose tnat the determinant 
1t tae. ie 
21. 80 438 
S12 e824 BS 
is represented by {123}, and so for a determinant of any order 
4123 .-.n}. 
Let 11], ]2], [12], [123], &c. denote as follows: viz. 
fl] = 11, [2] = 22, &c. 
Pie 12 ek, 
} 123] = 12.23.31, 
&e., 
where it is to be noticed that, with the same two symbols, e. g. 
1 and 2, there is but one distinct expression |12] (an fact 
}21 | = 21.12 ={12]); with the same three symbols 1, 2, 3, 
there are two distinct expressions, |123]}(=12.23.381) and 
, 182 | (=13.32.21); and generally with the same m symbols 
1, 2, 3...m, there are 1.2.3...m—1 distinct expressions 
}123...mj, which are obtained by permuting in every possible 
manner all but one of the m symbols. 
This beg so, and writing for greater simplicity }1}2] to 
denote the product |1 | x { 2], and so in general, the values of 
the determinants {12}, {123}, {1234}, {12345}, &e. are as 
follows: viz. 
* Communicated by the Author. 
