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.

INVERSIONS OF ORDER

7

Example. Tile number of inversions in alb2:3d4, d4c:^a.b2,
d4c3b2al, or any other permutation of the group, is respectively
0, 10, 12, even.
Scholium. The greatest possible number of inversions in
such groups is n(n-I), where n is the number of complex
symbols.
7. Theorem III. n anzy line of doubly complex symbols, we/lose
letters and suffixes are tie members of corresponding scqzlueces, tike
total nlu,,iber of inversiolns dzle to tIe presence of anill spccifilcd coizplex sym;:bol, H,, is even or odd according as t1e order of thlat sylbol is eveCz or odd.
If the order of H, is even, then H and x are both even or
both odd; consequently there must be present an evcnz number
of letters and suffixes together which are of lower orders than
H, x respectively. If Hx is odd, then h, x are one odd the
other even, and there must be. present an odd number of
letters and suffixes together of lower orders than H, x respectively. Therefore the number of inversions due to AILx in
the firt position of the line is even or odd according as Hx is
even or odd, since the only inversions due to H, in that position are with letters and suffixes of lower orders than H, x respectively. But the number of inversions due to H,7 is always
even or always odd, independent of the arrangement of the
line, since in any one interchange, and consequently in any
succession of interchanges, it is impossible for IHx to gain or
lose an odd number of inversions with any other doubly complex symbol. The theorem follows.
Cor. Thie lnmber of inversions due to tihe symbol H, is evenz?o
odd accordilng as (-I )r is + or -; r being tite order of If.
Cor. If there be a line of simple symbols arranfged In any
order, then the lnumber of inversions in the line (d/e to the jresence o'f any specified sJymbol of the sequence will be e.,en or odd
accordingz as the sum of the ortders of the specified sjymbol alnd
its positionz is even or o((d.
For a line of doubly complex symbols may be written with
its letters in natural order and its order of suffixes corresponding to the order of symbols in the given line. Then the order
of position of any suffix is the same as the order of its literal
partner, and the number of inversions in the given line is the