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.

o0 INVERSIONS AND DETERMINANTS
Hence we may represent any n1 quantities in square array by
any one of
/l a2 (73... a. i / b1 Ci.... l i (I ) (12) (I3)....(In) *
bi b2s b....., b 2 b2 C2 6....12 (21) (22) (23).... (21i)
Ci C2 C3....C, a3 b C3....n3 (31) (32) (33). * (3)
MIZ 712 773... Z., I,, b,l C7n... 7, (?II) (122) (72 3)....(?17n )
I X I I 1 |
I 2 3.,
i 3 3 33
I 2 3 n
Sl 11 el 71.
13. In the first array just written the order of any letter is
the same as the order of the row containing it, and the order
of any suffix is the same as the order of the column containing
it. In the second array letters correspond to columns and suffixes to rows. The modes of arrangement in the remaining
two arrays are apparent. It is evidently immaterial which one
of these, or other similar forms, is used, simply for the purpose of representing a square array.
14. DEFINITIONS. The i2 quantities of an array are called
constituents, or elements, of the array. The order of a constituent with respect to rows or columns is respectively the same
as the order of the row or column containing it. The order of
a constituent with respect to rows and columns is the sum of
the orders of the row and column containing it. A square array of n rows and n columns, and consequently containinig i/'
constituents, is said to be of the nth order. Care must be
taken to distinguish different classes of order.
I5. Hence we may not only represent an array of quantities by a similar array of complex symbols, but each symbol
* Here of course the figures, and the numbers resulting from their combination, do
2
not have their usual signification; but any symbol, as - or (32), represents the quantitywhich
stands in the psitio in te arra represented.
stands in the saize position in the array represented.