An introduction to the summation of differences of a function; an elementary exposition of the nature of the algebraic processes replaced by the abbreviations of the infinitesimal calculus, by B. F. Groat.

12

INVERSIONS AND DETERMINANTS

ov choice from rows and columnis is always even or always odd irrespective of the order of factors.
The general square array may be represented by
I I I I
1 2 3 I
2 2 2 2
I 2 3 n1
3 3 3.....
1 2 3 n
At..........
1 2 3 n
the natural orders of superior and inferior components corresponding respectively to natural orders of rows and columns.
'Iake any product of the array, as acy....0, represented by
i - ''; then the orders of arrangement of superior and ininferior components, corresponding to the order of factors, is
the same as the orders of choice from rows and columns respectively. But (Theorem II) the total number of inversions
among the complex symbols in the product is always even or
always odd irrespective of the order of factors. Therefore, the
total number of inversions of natural order, in the order of
choice from rows and columns, is always even or always odd
irrespective of the order of factors.
Cor. If the sign-factor(-I)i be attached to any product of,
the kind mentioned, i being the total mumber of inversions of
natural order in the orders of choice from rows and coluzmns, tzhen
thie sig1n ziti which thle product is to be ultimately affected is always
l, or always-, irrespective of thie order offactors.
Lo. The determinant of a square array of the nth order is
the algebraic sum of all the possible products of the n2 constituents taken n together, limited by the condition that one
and only one constituent is taken from each row and one and
only one from each column; the sign with which any product
is ultimately affected being +, or -, according as there is an
even or odd total number of inversions of natural order in the
orders of choice from rows and columns.
21. The determinant of a square array is frequently represented by A. Hence the definition of a determinant, expressed in mathematical symbols, is — '(-I)i (aZb2c3.....mn);