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.

6

INVERSIONS AND DETERMINANTS

positions obviously causes a loss or gain of a single inversion.
If there are m symbols intervening between the two to be
interchanged, then the transfer of each of the two to the position of the other obviously causes a loss or gain of one inversion with each intermediate; moreover there is also a loss
or gain of one inversion between the two symbols interchanged.
Therefore there are 2m+I, an odd number, of losses and gains
together. But the change in the number of inversions is the
difference between losses and gains, and if the sum of two integers be odd their difference is odd. Therefore the change i;
-he number of inversions is odd in any case.
4. A complex symbol can be formed by uniting into a single
symbol of two simple parts any two symbols chosen from as
many different sets of sequences of the kind we have been considering. A triply complex symbol may be formed by choosing
from three sets of sequences; and so on. As illustrations,
/I b
(II),, b3, A4, 0a, a BC, are doubly, and A, mY, Xn, triply complex. We shall confine our remarks to the first kind.
5. DEFINITION. The order of a complex symbol is the snum
of the orders of its simple parts.
6. Theorem II. If to any motuber of consecutive letters takenc/ i/i
any order, the same number of consecutive mzmerical sziizxes be attacked, one szuffx to each letter, thezn upon writing these complex syjlbols inz lile in any order at pleasure, the total nituber of invezrsiovns
among both letters and suffixes will be either always odd or alwcys
evCZn.
In any one interchange of two symbols the number of inversions among either letters or suffixes is changed by an odd
number (Theorem I); and the sum or difference of two odd
numbers is an even number. Hence the total number of inversions after one interchange remains either odd or even as
it was. But by successive interchanges two at a time the symbols can be brought into any prescribed order one at a timnThe truth of the theorem is apparent.