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.

ARRAYS AND DETERMINANTS II
may be taken to represent the value and position of the corresponding quantity of the array represented, by having the orders
of every pair of associated simple components correspond to
the two orders of position referred respectively to rows and
columns as has been explained.
I6. Coordination of products. The product of any lt constituents of a square array of the nth order, so selected that
one and only one constituent is taken from each row and one
and only one from each column, we shall refer to as aproduct
of the array. There are other ways of coordinating products,
but this is the only one with which we shall be concerned.
17. The diagonal row of constituents through the top left
hand corner of the array is called the principal diagonal, and
the product of the constituents of this diagonal is the principal
product of the array.
I8. In considering any such product, take the factors in
order to the right, beginning at the left. Then, corresponding
to this order of factors, there is an order in which they were
taken from columns and an order in which from rows. \We
shall speak of such an order as the order of choice, or order of
selection, from columns or rows as the case may be; when such
choice has been made in the natural order of rows or columns
we shall refer to it as the natural order of choice.
Illustration. What is the order of choice from rows and cola b c d(
e f ~ h
umns in the product,f /b I e, of the ar:a- I f k /I,'Z n p:
The order of selection from columns is 324I, or CZBDA; the
order of choice from rows is 4132, or DACB; the order of
choice from columns and rows is represented by c4bldsa2, or by
324 I
4132
19. Theorem. In any product of anz array, coordinated as in
Art. 16, the total number of inversions of natural ordter in the ordclrs