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.

CHAPTER III
FORMATION OF A DETERMINANT
24. In the sequel an array will be frequently spoken of as
being a determinant, meaning however the determinant of the
array; just as in algebra x frequently means zvalzze of x. In the
same way it will be convenient to refer to rows and columns
of a determinant, meaning rows and columns of an array from
which the determinant may be developed. When no distinction is necessary the word line will be used indifferently for
row or column.
25. There are n! terms of A. Let r,r2,....r,, represent
the n1 rows and cl,c2,. c,~ the n columns of a determinant, A,
of the nith order. By (rh,ck,) -, we shall mean the constituent at the junction of the kth row and kth column. Associating any r, as r/,, with any c, as ck, we shall have,, a constit,ent of A. Associating any one of the remaining r's, as r,, T wil.L
any one of the remaining c's, as cs,, another constituent, i, is
represented which is not in the row or column of ',. Again,
associating any r still remaining, rj, with any c still remaining,
ct, we have {, a constituent not in any row or column already
chosen. Continuing this process until all the rows and columns have been exhausted, we have a set of constituents coordinated by the rule of Art. I6. The product of these constituents together with the sign-factor (-i)', i being the number of inversions in.. *, is, by definition, a term of A:
it is clear that any permutation of ri, r2...r,, combined with any
permutation of c, C2.... c,, one r with one c, will lead to a term
of Al. But there are n! permutations of n symbols taken all
at a time. Hence there are (n!)2 ways of writing terms of
A, and no more. But each term may be written in n! ways.
Therefore there are (n!)2 - n! n! terms of A, and no more.
26. It is important to bear in mind that such expressions
as - * *, where the superior components refer to rows and
aS